New Franklin Arithmetic - Forgotten Books

N EW FRANKLIN

A R I T H M E T I C

SECOND B OOK

EDWIN P . SEAVER, A.M. , LL.B .

SU P ERIN TENDENT OF P U BLIC SCHOOLS , BOSTON

AND

GEORGE A. WALTON, A.M.

AG ENT O F MASSACHUSETTS BOARD OF EDUCATION ; AUTHOR OF

WALTON ’S -ARITHMETICS , AN THMETICAL TABLES , ETC .

NEW YORK AND CHICAGO

SHELD O N AND COMP ANY

“KL

HARVARDCOLLEGELIBRARYGIFTOF

GEOIGEMU HUR PLl'

J i'

l J .‘

“HUM" 26 , 1 92 4

Carm en , 1 895,

B! SHELDON AND COMPANY.

PREFACE.

Tms book is notmerely a revised edition of the Frankl in WrittenArithmetic publ ished by the authors seventeen years ago. In its

general plan and method, it preserves al l the essential characteristics of that favorably known and widely used text-book , but it

has been constructed out of new material . Al l the i l lustrations are

fresh, nearl y al l the examples are new, and the text has been rewrit

ten throughout. The principles of arithmetic do not change, neitherdo the essential elements of good teaching change , but there is infinite room for variety in the appl ication of principl es and in the

exempl ification of method . In these respects the authors trust thattheir new book may be found ful ly up with the times.

The method of the book is thoroughly inductive . Every new topic

is introduced, as it should be in good teaching, by easy oral questionsdesigned to bring the subject matter clearl y before the learner’s mind.

Then fol low il lustrative examples, definitions, principl es and rules in

true inductive order.

The number of examples provided for practice and dril l is unusual lylarge, particul arl y in the earl ier hal f of the book. There appears tobe a demand for an abundance of such material , which this book aims

to satisfy. But the authors are very far from recommending that al lthe examples under one topic Should be worked before a pupil or a

class of pupils is permitted to take up the next topic.

Opinions may differ as to whether topics l ike taxes, insurance,foreign exchange, stocks and bonds, equation of payments , and averageof accounts shoul d be included in the course of study for common

schools, but there seems to be no doubt that such topics should be

included in a book that aims at ordinary compl eteness. In this book

the treatment of the topicsmentioned is unusual ly simpl e and practical ,the il lustrations being not the old-fashioned imaginary transactions,but the actual ones occurring dail y in the course of mercantile busi

ness. By means of them a considerable amount of accurate and use

ful information is conveyed

4 P REFACE.

An important feature of the book is the anticipation of topics . In

the earl ier sections are found wel l-chosen examples which serve to

fami l iarize the l earner with matters that are to be presented in a

more formal and compl ete shape in the later sections. Thus a knowl

edge oi compound numbers is conveyed by the exampl es given to

il lustrate the appl ications of mul tipl ication and division of simplenumbers ; many examples in percentage and interest are given amongthe appl ications of decimal fractions ; and the simpler examples inmensuration are introduced long before the systematic treatment ofthat subject in the last section is reached.

It is this same principl e of anticipation that has l ed the authors to

make a somewhat free use of algebraic notation and methods in the

treatment of problems in interest, proportions, square and cube root,

and some other topics.

P roportion has received a somewhat broader treatment than hasbeen usual in arithmetical text-books of l ate years . It is bel ieved

that proportion affords a straightforward, Simpl e, and cl ear form of

analysis, the training in which is highly useful for general purposesand cannot be too strongl y recommended for al l who are to be con

cerned with the further study of mathematics or with the physical

sciences . In this book the range of examples is enlarged so as to

include some of the Simpler appl ications of proportion to similar

geometrical figures.

The section on mensuration has been enl arged so as to contain a

considerable amount of practical geometry. This, with the practicalexercises suggested for the purpose of making the principles objec

tively cl ear, wil l , it is hoped , make mensuration a less dry and for

bidding topic than children usual l y find it.

In conclusion , the authors desire to express their hearty acknowl edgments for the favorabl e reception accorded to their former book, and

venture to hope that the present one may be found no less worthy

of publ ic favor.

TABLE OF CONTENTS.

P AGE

NUMBERSAND THEIR ExpnEsSION

Notation and NumerationOf Integral Numbers 8

Of Decimal s 1 3

Of U nited States Money 1 5

ADD ITION 1 6

SUBTRACTION 30

MULTIP LICATION 45

D IVISION 72

MISCELLANEOUS AND REV IEW S

Symbol s of OperationU NITED STATES MONEY.

Notation

Coins and P aper MoneyAccounts and Bil ls

TEST ExEEC ISES

FACTORS.

D ivisibil ity of NumbersCancel lation

COMMON FRACTIONSReduction

Addition

Least Common Denominator

SubtractionMul tipl icationDivision

To find the part one num

bet is of another

P AG E

To find the whole when a

part is given

Al iquot P artsDEC IMAL FRACTIONS .Reading and Writing 1 3, 1 47

Reduction 1 49

Addition 20 , 1 50

Subtraction 36 , 1 51

Mul tipl ication 56 , 1 52

Division 79 , 1 54

WEIGHTS AND MEASURES59 , 6 1 —6 7 , 384-389

COMPOUND NUMBERS 1 60

Reduction 1 66

Addition and Subtraction 1 7 1

Mul tipl ication and D ivision 1 75

Longitude and Time 1 77

Standard Time 1 78

MENSURATION O F SURFACESAND SOL IDS

Squares and other Rect63, 1 83

G overnment Lands 1 84

Triangles 1 85

Circl es 1 86

Rectangular Sol ids 66, 1 87

Wood 1 88

Lumber . 1 89

Laths, Shingl es, etc. 1 90

P apering 1 9 1

CarpetingStone and BrickWorkBins, Cisterns, etc .

P ERCENTAG EP rofitand LossCommission

Commercial D iscountInsurance

Taxes

Customs or Duties

INTEREST.Simpl e InterestAccurate InterestP artial P ayments

P robl ems in InterestCompound InterestBank D iscount

Exchange

Stocks and Bonds

Average of P ayments

Average of Accounts

P ROP ORTIONP artnership

P OWERS AND ROOTSInvolution

Evolution

Square RootCube Root

CON TEN TS.

P AGE

1 9 1 MENSURATION OR PRACTICAL

1 92 G EOMETRY 325

1 94 Triangl es 1 85, 33 1

202 Quadrilaterals 63, 1 83, 334

2 1 3 Regul ar P olygons 337

220 Area of P olygons 338

222 Circles 1 86 , 338, 34 1 , 343

224 Right Triangl es 346

227 Sol ids 66 , 1 87, 351

231 Volume and Surface Areaof Sol ids

236Similar P olygons

247Similar Sol ids

248 SUP P LEMENT.254 U se of Dril l Tabl es260 Contractions in Mul tipl ica264 tion 376

269 Divisibil ity of Numbers 377

2 76 G reatest Common Factor 378

282 Least Common Mul tipl e 379

285 Circulating Decimals 381

290Annual Interest 382

304Ver. ,N .H., and Conn .Rul es

307for P artial P ayments

307TABLES or WEIGHTS AND

308MEASURES

3 1 0 METRIC SYSTEM3 1 8

DRILL TABLES, 29 , 44, 7 1 , 89 , 1 46 , 1 59 , 1 8 1 .

MISCELLANEOUS ExAMP LEs AND REVIEW S, 4 1 , 67, 87 , 90, 1 03, 1 35,1 42 , 1 56 , 1 94 , 233, 289 , 365, 394 .

NEW

FRANKLIN ARITHMETIC.

SECOND BOOK.

SECTION I.

NU MBERS AND THEIR EXP RESSION .

ARTICLE 1 . A col l ection of simil ar objects suggests theideas of one and more than one . One of any kind is a unit .

A unit or a col lection of units is a number.

2 . Arithmetic is the science of numbers and the art ofusing them in computation .

INTEG RAL NUMBERS.

3. An integral number is a number whose unit is a whol eor undivided thing, as seven apples, the unit of which is oneappl e ; ten days, the unit of which is one day ; fifteen dozen,the unit of which is one dozen .

4 . Smal l numbers are reckoned by ones and by tens,

l arger numbers by tens of tens, or hundreds, and stil l l argernumbers by tens of hundreds, or thousands. Above thousands, numbers are reckoned by ten-thousands, hundred

thousands, mil l ions, ten-mil l ions, hundred-mil l ions, and so on .

5. One, ten, a hundred, a thousand, etc., are al l cal l ed

units, because each is used as a singl e thing in reckoningother numbers.

8 N UMBERS AND THEIR EXP RESSION .

6 . To distinguish these units, one is cal l ed a unit of thefirst order, ten a unit of the second order, a hundred a unitof the third order, a thousand a unit of the fourth order,

ten thousand a unit of the fifth order, and so on . Whena unit is Spoken of without the order being named, a unitof the first order, or one, is meant.

7 . These units form a scal e ; and because ten units ofany order make a unit of the next higher order, the scal eis cal l ed a scal e of tens , or a decimal scal e.

8 . A system of numbers whose successive units form a

scal e of tens is a decimal system of numbers. The systemof numbers in common use is a decimal system.

NOTATION AND NUMERATION.

9 . Notation is the art of writing numbers, and numera

tion is the art of reading them.

1 0 . There are two systems of notation in common use,

the Roman and the Arabic. The Roman notation empl oysthe characters I . V. X. L. C. D . and M. and combinationsof these to express numbers. It is not used in arithmeticalcomputation .

1 1 . The Arabic notation empl oys ten charac ters cal l ed

figures.

0 i 2 3 4 5 6 7 8 9

! ero One Two Three Pour Five Six Seven Bight Nine

1 2 . The first of these figures, 0, is cal l ed zero or cipher ,

and stands for no number. The others stand respectivelyfor the numbers Whose names are written beneath.

1 3 . Numbers larger than nine require two or more figures for their notation. These are written Side by Side

,

and so the p lace occupied by each figure becomes important

N OTATION AND N UMERATION . 9

1 4 . Tens areexpressed by writing a figure to denote howmany tens, and then pl acing a zero at the right of. it. The

figure so written is said to occupy the second pl ace, or thetens

’

pl ace, whil e thefi rst or unitS’

pl ace is fil l ed by the zero.Thus

,

Ten (one ten) , 1 0. Forty (four tens) , 40. Seventy (seven tens) , 70.

1 5 . Numbersmade up of tens and units are expressed bywriting a figure in the second pl ace for the tens

,and a

figure in thefirst pl ace for the units. Thus,

Eleven (one ten and one) , 1 1 . Twenty-four (two tens and four) , 24.

1 6 . Hundreds are expressed by writing a figure in the

third pl ace, the second and first pl aces being fil l ed withzeros. Thus,

ode hundred, 1 00. Five hundred, 500 .

1 7 . Numbers made up of hundreds, tens, and units are

expressed by writing a figure in the third pl ace for the

hundreds, a figure in the second pl ace for the tens, and a

figure in thefirst place for the units. Thus,Three hundred twenty-seven (3 hundreds, 2 tens, 7 units) , 327.

Four hundred thirty (4 hundreds, 3 tens, 0 units ) , 430.

Five hundred six (5 hundreds, 0 tens, 6 units) , 506 .

1 8 . Thousands are expressed by writing a figure in the

fourth pl ace, tens of thousands by writing a figure in the

j iflh pl ace, and hundreds of thousands by writing a figure

in the sixth pl ace . The figures in these three pl aces takentogether form a group cal l ed the thousands

’

group ; whil ethe figures in the hundreds’, tens’, and units’ pl aces formthe units

’

group . These groups are usual ly separated bya comma. Thus,One thousand, Fifty thousand,

Ten thousand, One hundred thousand,

Ten thousand six hundred twenty,Nine hundred seventy-eight thousand eight hundred six,

1 0 N UMBERS AND THEIR EXPRESSION .

1 9 . Mil l ions, tens of mil l ions, and hundreds of mil l ionsare expressed by writing figures in the seventh, eighth,and ninth

'

pl aces. These figures taken together form the

mi l l ions’ group . Thus,Fivemil l ions,Fifty mil l ions,Five hundred fifty-five mil l ions ,

20 . The above il l ustrations Show the principl es uponwhich al l numbers are written, which are( 1 ) Units of any order are expressed by writing a figure

in the place corresponding to that order, and fi l l ing vacant

p laces with zeros.

(2 ) The va lue exp ressed by a figure is increased tenfold

by each remova l one p lace to the left, and decreased tenfold byeach remova l one p lace to the right.

2 1 . The general method of writing integral numbers isshown by the fol l owing

TABLE.

ORDER NAMES.

d d F P LACES .m N

w

4 8 0 , 0 3 4 , 5 0 8 , 6 7 2 m s.

5th group, 4th group, 3d group, 2d group, l et group,Trll llons. Bll llons. U ll llons. Thousands. Units. iG nom

’s‘

ETC

.

Nora .— The groups above bil l ions are seldomused. Their names

are tri l l ions, quadri l l ions, quinti l l ions, sexti l l ions, septi l l ions, octi l

l ions, nonil l ions, deci l l ions, undeci l l ions, duodeci l l ions, tredeci l l ions,

quatuordeci l l ions, quindeci l l ions, sexdeci l l ions, sep tendeci l l ions, octo

deci l l ions, novemdeci l l ions, viginti l l ions, etc.

N OTATION AND N UMERATION . 1 1

Exercises upon the Tabl e .

22 . a . What pl aces are occupied by the units’

group the

thousands’ group ? the mi l l ions’ bil l ions’ tril l ions’

b. Write 265 as units ; as thousands ; as mil l ions.

In reading and writing large numbers, the first thoughtShoul d be of the groups.

23. To read integral numbers .

Il lustrative Examp le. Read the number 46372509 .

( 1 ) Beginning at the right, point ofi the figures into groups of

three figures each, thus

(2) Beginning at the left, name the number expressed by each

group in order, thus

“ Forty-six mil l ion three hundred seventy-two thousand five hun

dred nine.

”

NOTE .— In reading any number , the pupil shoul d make a sl ight

pause after naming a group. The name of the units’ group is omitted

in reading.

write

Exercis es .

In words the fol l owing

1 4 . 40762

1 5 . 9006

1 6 . 300708

1 7 . 1 094

1 8 . 703800

1 9 . 500087

20 . 84000

2 1 . 51 051 0

22 . 90007

2 3 . 70001 6

2 4 . 347 1 6300

2 5 . 80040704

2 6 . 389049 1 8

1 2 N UMBERS AND THEIR EXP RESSION .

25. To w ri te integral numbers .

I l lustrative Examp le. Write in figures four hundredforty-seven mil l ion one hundred ninety-eight thousand six

hundred ninety-four.

( 1 ) Think of the groups. These are

447 mil l ions, 1 98 thousands, 694 units .

(2 ) Arrange the groups in their proper order, thus

Exercises .

26 . Write in figures the fol l owing, supplying vacant

pl aces with zeros40 . Seven thousand three hundred forty-one .

4 1 . Eighteen thousand nine hundred eighty-four.

42 . Thirty-seven thousand three hundred nineteen.

43 . Nine hundred thousand five hundred sixteen .

44 . Four hundred twenty thousand Six hundred eighty .

45 . Eight hundred ten thousand two hundred seven .

46 . TWO hundred fifty-nine thousand seventy.

4 7 . One mil l ion one hundred thousand one hundred one .

48 . Five mil l ion thirty-six thousand Six hundred ten.

4 9 . Forty-five mil l ion seven hundred twenty-four thousand two hundred forty-seven.

50 . Eighty mil l ion seventy thousand five hundred seven.

5 1 . N ine hundred one mi l l ion two hundred eighteenthousand five hundred twenty-two .

52 . Three bil l ion thirty-seven mil l ion nine hundred Six

thousand two hundred.

53 . Two hundred thirty-four mil l ion eight hundred sixtythree thousand three hundred eighty-nine.

54 . Seventeen bil l ion seven hundred fifty-nine mil l ion

ninety thousand sixty-seven.

DECIMAL FRACTION S. 1 8

DEOIMAL FRACTIONS.

27 . As a hundred is made up of ten equal parts, eachof which is a ten, and as a ten is made up of ten equal

parts, each of which is one, so we may consider one to bemade up of ten equal parts, each of which is a tenth ; a

tenth to be made up of ten equal parts, each of which isa hundredth ; a hundredth to be made up of ten equal parts,each of which is a thousandth ; and so on .

Tenths, hundredths, thousandths, etc., are fractiona l units

and, as they form a decimal scal e (Art. col l ections ofsuch units are cal l ed decimal fractions .

The above-named units are writ One hundred, 1 00 .

ten as in the margin, each unit One ten, 1 0 ‘

of a l ower order being expressed 32? tenth 3}by writing the figure 1 , one place One hundi‘edth , (1 0 1

further to the rl ght. One thousandth ,

28 . The dot put at the right of the units’

pl ace is cal l edthe decimal point .

29 . The method of writing decimal fractions is Shownby the fol l owing

TABLE.

P LACE NAMES.UNITS

.

DEOIMAL

PO

m0 7 4 8 9 6 3

NOTE.— In writing decimal fractions it is wel l to fi l l the units’

place with a zero when there is no other figure to be written there.

1 4 N UMBERS AND THEIR EXP RESSION .

30 . To read and w rite decimal fracti ons.

Il lustrative Examp les. Read

First read the number as if it were an integral number, thusSixty-three then add the name of the units of the l owest order(thousandths) , thus, Sixty-three thousandths.

”

Write nine hundred sixty-three thousandths.

First write the number as if it were an inten number, thus : 963,then place the decimal point so that the right-hand figure shal l expressthe required denomination , thus,

To read or write integral numbers and decimal s together,read or write first the i ntegra l number, then the decimal . In

reading,use the word and before the decimal .

Exercises .

3 1 . Read the fol l owing :55 . 59 . 63 . 67 .

55 50 . 54 . 68 .

57 . 61 . 55 . 69 .

58 . 52 66 . 70 .

Write the fol l owing7 1 . Fourteen hundredths. Nine hundredths .7 2 . Twenty-seven thousandths. Four ten-thousandths.

7 3 . 3 and 33 hundredths. 1 04 and 25 thousandths.

74 . 7 and 29 thousandths. 45 hundred, and 5 tenths.

7 5 . One hundred seventeen thousandths.76 . Four hundred and four ten-thousandths.77 . Six hundred seven, and sixty-seven thousandths .

78 . Five hundred sixty-eight hundred-thousandths.

32 . It is frequently convenient to separate a numberinto parts, each part containing onl y the units of a singl eorder. Thus, the number may be separated into 7hundreds, 3 tens, 4 units, 2 tenths , and 5 hundredths . Such

parts are cal l ed tonne of the number.

UN ITED STATES MONEY. 1 5

U NITED STATES MONEY.

33 . The units of U nited States money form a. decimalscal e, shown by the fol l owing

TABLE.

1 0 mil l s 1 cent, marked 15, ct.

1 0 cents 1 dime.

1 0 dimes 1 dol lar, marked 3.

1 0 dol lars 1 eagle.

34 . Dol lars and cents are the units chiefly used in reckoning money. Cents are hundredths of a dol l ar. Dol l ars arewritten as integral numbers at the l eft of the decimal point ;cents are written in the first two pl aces at the right, andmil l s in the third pl ace.

The number 2 1 dol l ars, 43 cents, 8 mil l s is written thus852 1 438. The number 2 1 dol l ars, 5 cents, 3 mil l s is writtenthus :

Exercises .

35. Read the fol l owing79 . as. 5 87 . 3so. 84 . 88 .

8 1 . $ 0 724 85 . 89 .

82 . 86 . 90 . 851 909 293

Write in figures

9 1 . Fifteen dol l ars thirty-six cents.

92 . N inety-three dol l ars sixty-two cents five mil l s.

93 . Seventeen dol lars five mil l s.94 . Forty-five cents four mil l s.

95. N ine do l l ars five cents two mil l s.

95 . Five hundred dol l ars five cents.

97 . N ine thousand seventeen dol l ars two cents.

98. Sixty-eight thousand four hundred eighty dol l ars.

SECTION II .

ADDITION .

36 . I l lustrative Example. How many books are 5 books,

3 books, and 2 books ?SOLUTION . The numbers 5, 3, and 2 together make 1 0 .

37 . P utting numbers together to find what number theymake is adding them.

38 . The numbers to be put together are addends . The

number found by adding two or more numbers is their sumor amount .

39 . Addition is the process of putting two or more numbers together to find their sum or amount .

40 . Only numbers of the same kind,name

,or denomina

tion can be added together, as appl es with appl es, poundswith pounds, un itswith units, tenswith tens, hundreds withhundreds, etc.

The Operations of arithmetic are indicated bysigns. The Sign of addition is an upright cross, It

is read “and

”or

“pl us.

” The Sign indicates equa l ity,

and is read “equals

”or

“ is equal to.

” The expression5 3 2 1 0 means that 5 and 3 and 2, added, make 1 0 ;it is read 5 plus 3 plus 2 equal sWhat is the sum in the example given above What are the

addends Define addition .

Oral Exercises in Addition.

42 . I. Add each pair of numbers expressed in the columnsa, b, 0, etc.

, on the Opposite page, til l you can give the sums

rapidl y at sight.

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1 8 ADDITION .

Add the fol l owing by columns and by l ines as indicated.

a b c d e f g h i j

VI . Add the fol l owing by col umns upward and downward ; thus, in column a

,say 20, 50, 60, 80, 1 00,

VII. Add by l ines from l eft to right, and from right tol eft.

a b c d e f g h i

p

q

r 300 + 600+ 700 + 300

m

t 400+ 500 + 800 + 700

u 300 + 700 + 300 + 500

VIII . Add the above again and disregard the zeros tillafter adding ; thus, in col umn a , say

“ 2, 5, 6 , 8, 1 0, 1 1 tens,

or 1 1 0 ” in column f, say “ 3, 7 , 9, 1 2 , 1 3, 1 5 hundreds,or 1 500 .

IX. v . How many times does the hammer of a commoncl ock strike from one to ten o’cl ock inclusivew. A maid counting the Sil ver of a house, found that

there were 22 teaspoons, 1 2 coffee spoons, 6 tabl e spoons,9 dessert Spoons, 4 sal t spoons and 2 sugar spoons, andthat 2 teaspoons were missing . What number of spoonsbel onged to the house

WRITTEN EXAMP LES. 1 9

Exampl es for W ritten W ork .

43 . I l lustrat ive Example. What is the sum of 824, 57 1 ,278, and 636

WRITTEN WORK.For convenience , the numbers are written so

that units of the same order shal l be expressed in824 the same column , and a line is drawn beneath .

57 1 Beginning with the units to add (thus , 6 , 1 4, 1 5 ,278 there are 1 9 units 1 ten and 9 units. The

636 9 is written in the units’ place, and the 1 ten is

kept to add with the tens. Adding the tensSum,

2309(thus, 1 , 4, 1 1 , 1 8, there are 20 tens = 2

hundreds and no tens . A zero is written in the

tens’ pl ace, and the 2 hundreds are kept to add with the hundreds.

Adding next the hundreds, the sum of al l the numbers is found tobe 2309 . Ans. 2309 .

Keeping a number to add with the numbers expressed in the next

column is cal l ed carrying.

In working exampl es , use as few words as possibl e ; thus, in the

il lustrative exampl e, say merely“ 6

,1 4, 1 5, 1 9 1 , 4, 1 1 , 1 8, 20 ;

2 , 8, 1 0 , 1 5, 23 ; sum,

Add the fol l owing by columns ; keep the answers and findtheir sum.

1 . 2 . 3 . 4 . 5 . 6 . 7 .

8 .

9 . 41 6 + 592 — 9

1 0 . 87 + 64 +

1 1 . 428 + 53 +

1 2 .

Add the above numbers by l ines ; keep the answers and

find their sum. How shoul d the sum comparewith the sumOf the answers found in adding the numbers by col umns

“ Do not stop to say“ write 9 , carry but do it.

20 ADDITION .

Add the fol l owing numbers by columns ; al so add themby l ines.

1 3 . 1 4 . 1 5 1 6 . 1 7 . 1 8 .

3072 5932 4320 1 243 3201 9876 2

5629 8275 7869 5687 5674 594 1 2

8900 3908 954 1 32 1 9 91 28 9099 2

2387 762 4376 1 245 5463 8907 2

998 541 4 2408 987 91 87 8796 2

5734 9632 32 1 4589 342 9768 = 2

ADDITION OF DECIMALS.

44 .“I l lustrative Examp le. What is the sum of

and

WRITTEN WORK The numbers are written so that units of the

same order shal l be expressed in the samecolumn . The adding begins with the units of

the lowest order, and proceeds briefly, thus :2071 36

Thousandths, 9 , 1 5 ; write 5, carry 1 .

31 -95 Hundredths, 1 , 3, 8, 1 1 , write 4, carry 2 .

Tenths, 2 , 1 1 , 1 2 , 1 6 , 1 7 ; write 7 , carry 1 , etc.

Ans.

Add the fol l owing by columns and by l ines.

2 5 2 6 . 27 28 .

29 .

30 .

31 .

32 .

33 .

FORMS OF EXAMP LES IN ADDITION . 21

45 . From the preceding exampl es may be derived thefol l owing

Rul e for Addition .

(1 ) Write the numbers to be added so that uni ts of the same

order sha l l be eap ressed in the same column . Draw a l ine

beneath.

(2 ) Add the units of each order separately, beginning with

those of the lowest order.

(3 ) When the sumof the units of any order is less than ten,

write it under the l ine in its p rop er p lace ; when ten or more,write only the un its of the sum,

and carry the tens to the next

column .

(4) Write the whole sum of the last addition .

P roof .

Repeat the work, adding downward instead 0f upward.

46 . Forms of Exampl es in Addi ti on.

34 . How many are and 9872 , and

35 . Add the numbers from 95 to 1 04, inclusive.

36 . Find the sum of $ 1 8, $ 695, and

37 . How many days are there in six years, two of whichhave 366 days each, and the others 365 days each38 . 3287 feet feet feet 6 thousand

48 feet what number of feet39 . What is the amount of 8435 pounds, 463 pounds,

pounds, 4099 pounds, and pounds40 . Find the popul ation of Boston in 1 890, the number ofinhabitants in 1 850 hav ing been and the increasesince 1 850 being4 1 . What is the sum of four hundred thousand eight hun

dred sixty ; sixteen thousand nine hundred ; twenty-sevenmil l ion ninety-five thousand eight hundred ten ; and one

hundred forty-seven mil l ion four hundred eighty-seventhousand three hundred twenty-six

22 ADDITION .

42 . and ninetyfour dol l ars more are howmany dol lars43 . What is the total of the fol l owing sums of money

97 cents, and 8 cents44 . what45 . Five thousand and five tenths, plus three hundred

and seventy-five hundredths, plus eight thousand seven and

sixty-five thousandths equal s how many

46 . 47 . 48 .

47 . Mental additi on tons and

I l lustrative Examp le. Add 27 and 44 .

SOLUTION .— 27 and 40 are 67 , and 4 more are 7 1 . Ans. 7 1 . Or

say, simply,“ 2 7 , 67 , 7 1 . Ans .

a Add 1 7 and 36 . d Add 52 and 37 .

b Add 26 and 1 8. e Add 64 and 28.

6 Add 34 and 29 . f Add 46 and 35.

k l m n o p g

g r

h s

i

j u

NOTE .— Accountants often add at once the numbers expressed in

two, three, or four columns, by using the method il lustrated above.

ORAL EXERCISES. 23

Oral Exerci ses .

48. a . To a cabl e road running 1 7 mil es northerly fromthe post office, 8 mil es were added in the opposite di rection .

What was the entire l ength of the roadb. Sarah has 1 5 l il ies, John has 8 more than Sarah, and

Annie has 9 more than John . How many has Anni ec. It took 24 yards to carpet a sitting-room,

1 3 to carpeta hal l , and 9 to carpet the stairs of a house . How manyyards were requi red for al ld. How many rods of fencing woul d be

requi red to fence the l ot of l and here repro

sentede. It requires 37 feet of pipe to go from

a spring to a l ane, 9 feet to cross the l ane,1 2 feet more to reach the yard,and 30 feetmore to reach the house . How many feet are required toreach from the spring to 20 feet within the house

f . A flag was suspended over a street by a rope stretchingfrom houses on opposite Sides, one 1 5 feet back, the otherflush w ith the street, which consisted of two sidewal ks,each 9 feet wide, and a roadway between, 30 feet wide.

Al l on 1 0 feet for tying, what l ength of rope was re

quired

g. General U lysses S. Grant was born in 1 822 ; he enteredWest P oint at the age of 1 7, and 9 years l ater he wasmarried ; 1 2 years after his marriage, he engaged in the

leather business with his father, in 4 years more he wasappointed l ieutenant general of the U nited States Army,and in an other 4 years he was el ected P resident. How

old was he then, and in what year was he el ected P residenth. If you add 7 to 1 7 and then add 30 and 8 more to

the sum, how many wil l you havei . A deal er bought turnips at 45 a bushel . For howmuch must he sel l them to gain 1 8 53 a bushel

24 ADDITION .

j . A merchant sold a hat for 55fi, and by so doing l ost20 How much did it cost himk. Having sol d 1 9 bushel s of appl es and 1 5 bushel s of

pears, a marketman has 6 bushel s of appl es and 6 bushel sof pears l eft. How many bushel s of appl es had he at

first Of pears Of bothl . What must you pay for 1 pound of sugar, 1 pound

of coffee, 1 dozen eggs, and 6 l emons at the present priceswhere you l ive Make an exampl e stating prices.

49 . Exampl es for W ri tten W ork .

50 . There are 92 days in the spring months, 92 in the

summer, 9 1 in the fal l , and 90 in the winter months . Howmany days are there in the year5 1 . The distance from Boston to Al bany is 202 mil es ;

fromAl bany to Buffal o, 297 mil es ; fromBuffal o to Tol edo,296 mil es ; and from Tol edo to Chicago, 243 mil es. Whatis the distance fromBoston to Chicago52 . In the upper grade of a grammar school there are 29

boys and 56 girl s ; in the second grade, 38 boys and 60 girl s ;in the third grade, 42 boys and 65 girl s ; and in the fourthgrade, 48 boys and 59 girl s. How many boys are there inthe school ? How many girl s HOW many pupil s

‘7

53 . After taking from a bank I have remainingin it How much had I in the bank at first54 . If a history cost a dictionary an arith

metic ink 1 3 cents, and stationery 29 cents, howmuch di d al l cost55 . A surveyor measures four fiel ds, and finds in the

first acres ; in the second,

acres ; in the third,acres, and in the fourth, acres. How many

acres does he find in al l

56 . If bricks are required to buil d a house,to build a stabl e , and 2364 to l ay a Sidewalk

,how

many bricks are required in al l ?

26 ADDITION .

Find the amounts in the

6 7 . G rocer’s B il l .

For January,

February,

March,

April ,May,J1 1 1 1 6

,

69 . Mrs. Benson Spent for a dress, for l ace

and trimmings, and 1 6 formaking . She al so spentfor a Shawl , for a bonnet, for gl oves, for

boots, and has l eft. How much money had she at

first70 . My cow, Daisy, gave 1 388 pounds of milk in April ,

1 456 pounds in May, 1 440 pounds in June, 1 31 7 in July,and 1 1 75 in August. Jumper gave, during the same time,1 42 1 pounds, 1 485 pounds, 1 398 pounds, 1 362 pounds, and1 228 pounds. How many pounds did each cow give Howmany did both give7 1 . New York contains square mil es ; New Jersey,

781 5 square mil es ; P ennsyl vania, Del aware, 2050 ;

Maryl and, the District of Columbia, 70 ; Virginia,and West Virginia, Texas contains

square mil es more than al l these put together. How manysquare mil es has Texas72 . The expenses of the Worcester Normal School for

1 892 were as fol l ows : for sal aries, for janitor,600 ; repairs, fuel , stationery,

printing, etc.,

tel ephone, water, apparatus, books

,

and miscel l aneous, What was the

amount of the expenses

fol l owing l ists of items

68 . Bank Deposits.

January bal ance,February deposit,MarchAprilMay

June

WRITTEN EXAMP LES. 27

73 . A,B, C, and D engaged in trade ; A put in

~

$ 1 275,and B it 1 350 ; C put in $ 2580 more than B

,and D put

in as much as A and B together. How much did al l

put in

74 . For the nine months preceding March 1,1 894, the

receipts of the U nited States were from cus

toms ; from internal revenue, and miscel l a

neous, What were the total receipts75 . Find the amount of the duties paid to the U nited

States government upon imported textil e manufactures in1 893 ; these being as fol l ows

Cotton, 3 Wool , s

Silk,

Fl ax, etc.

,

76 . The fol l owing were the values of the mineral productions of the U nited States in 1 893 : metal l ic,non-metal l ic, unspecified, Whatwas the total val ueIn the report of the Commissioner of Education for 1 892

are the fol l owing items :In the North Atl antic Division there are mal e

teachers, femal e, and worth of school

property ; in the South Atl antic Division, mal e

teachers, femal e, school property ; inthe South Central Div ision, mal e teachers,femal e, 15 school property ; in the North Central ,

mal e teachers, femal e, and

school property in the Western Division, 4034 mal eteachers, 8887 femal e, and school property.

77 . How many mal e teachers are empl oyed in the U nitedStates How many femal e teachers78 . What is the value of the school property79 . Furnish yoursel f with a good outfit of such school

books as you use , with stationery, etc., at regul ar prices, and

give their cost.

28 ADDITION .

Add the fol l owing l edger col umns

Name the different orders of un its from un its to tril l ions ; from

tril l ions to units . Name the groups from the units’ group to tril l ions ;

from tril l ions to units. What is the effect of removing a figure one

pl ace to the l eft to the right

What is Addition 7 What is the amount Add oral ly 64 and 87 .

How do you write numbers to be added Why 7 Add five numbers

expressed by four figures each, and explain. G ive the r ul e for addi

tion the proof . Il lustrate adding at once numbers expressed in two

ormore columns.

DRILL TABLE. 29

DRILL TABLE NO . 1 . (See Suppl ement, Art.

50 . For suppl ementary practice in addition .

0 p q t

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 20

Exercises upon the Tab l e .

854 04 . Add the numbers expressed in columns 1 to 20.

1 05—21 1 4 . Add the numbers expressed in the doubl e col

umps m to v.

1 1 5—21 1 9 . Add the numbers expressed in the fourfoldolumns fromw to 33 .

Copy and add the numbers expressed from w to 33 (unitsto thousands), in l ines

1 20. a . 1 2 2 . c. 1 24 . e. 1 26 . g. 1 28 . i . 1 30 . k.

1 2 1 . b. 1 23 . d. 1 2 5 . f . 1 2 7 . h. 1 2 9 . j . 1 3 1 . L

SECTION III.

SU BTRACTION .

51 . I l lustrative Examp le. There are 1 50 plum trees inan orchard ; if 50 shoul d be taken away, how many woul dbe l eft

SOLUTION .— To find how many are l eft, we take 50, a part of 1 50,

away, and by counting or otherwise, find that there are 1 00 l eft.

52 . Taking part of a number away to find how many arel eft is subtracting .

53 . The number part of which is to be taken away isthe minuend. The part of the minuend to be taken away isthe subtrahend. The part of the minuend l eft after a part

has been taken away is the remainder.

54 . Subtraction is the process Of taking part of a num

ber away to find how many are l eft.

55 . The Sign of subtraction is a horizontal l ine, The

expression 1 5 5 1 0 means that if 5 of the 1 5 are takenaway 1 0 are l eft ; it is read “ 1 5 minus 5 equal s or

1 5 l ess 5 equal s 1 0 .

What is the minuend in the exampl e above What is the subtrahend Define subtraction. Read the expression 1 2 7 5.

Oral Exercises in Subtraction.

56 . I. Give the remainders in each of the fol l owing pairsof numbers expressed in columns and in l ines til l you can

give them rapidly at sight.

ORAL EXERCISES. 31

“

a

H

OD

O'R

N

OG

H

b

_ 2

_ 2

— 8

_ 4

— 5

— 3

- 9

- 6E

'Q

'3

0

3

3

N

R"

H

co

ma:

II. Subtract3 . By 2

’s from 30 ; from 29 . w. By 6

’s from 1 00 ; from 99 .

t. By 3’s from 40. x. By 7

’s from 1 00 .

u . By 4’s from 50 ; from 49 . y. By 8

’s from 1 00 ; from 99 .

v . By 5’s from 50 ; from 49 ; 3 . By 9

’s from 1 00.

from 48 ; 47 ; 46 .

III. Froma . 1 1 take 2 , 4, 7, 6, 3, 8, 5. e. 1 5 take 9 , 7 , 8, 6 .

b. 1 2 take 3, 5, 8, 9 , 4, 6, 7 . j : 1 6 take 9, 8, 7 .

c. 1 3 take 5, 4; 8, 6, 7, 9 . g. 1 7 take 8, 9 .

d. 1 4 take 6 , 5, 9 , 8, 7 . h. 1 8 take 9 .

IV. From each number expressed in columns 72, j , k bel ow,subtract 2 , and notice the units of the remainders.

In the same way subtract 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 .

i j k l m n o p q

r 1 0 1 1 1 2 1 3 1 4 1 5 1 6 27 28

s 20 2 1 32 33 24 45 46 67 48

t 40 41 42 43 54 35 66 57 88

u 70 6 1 62 63 74 65 56 77 78

v 90 81 52 83 94 85 96 87 98

V. From the numbers expressed in column k above, sub

tract any number from 3 to 9 ; in col umn l from 4 to 9 ; incolumnm from 5 to 9, and so on.

32 SUBTRACTION .

VI. a 49 7 3 5 — 4 — 9 — 8 — 7 = how many 9

b 54 5 2 3 — 7 — 9 — 8 — 6 = h0w many‘7

c 60 4 7 6 — 2 — 3 — 9 — 9 — 8 — 5 — 2

d 72 — 8 — 5 4 9 7 — 3 6 — 7 — 9

VII. Give the remainders in the fol l owing exampl es, bothin the col umns and in the l ines.

g h k l m 71

50 — 30 80 — 50 90 — 60

20 — 1 0 40 — 30 80 — 30

57 . Mental subtracti on tens and units .

I llustrative Examp le . From 42 take 25.

SOLUTION .— 42 l ess 20 are 22 , l ess 5 more are 1 7 .

VIII. In the fol l owing, subtract first by columns, thenby l ines

, giving resul ts only.

a b c d

k 50 — 41 6 1 — 37

l 28 — 1 9 38 — 1 9

m 62 — 33 72 — 65

n 34 — 1 5 36 — 28

0 83 — 47 9 1 — 44

p 66 — 1 9 52 — 37

IX. q. From 30 take 20 ; 1 0 ; 1 5 ; 27 ; 3 ; 1 2 ; 1 8 ; 2 1 ;9 ; 26 ; 5 ; 23 ; 1 9 .

r. From 60 take 40 ; 20 ; 1 5 ; 45 ; 1 2 ; 48 ; 24 ; 36 ; 54 ;6 ; 37 ; 49 .

s. From 1 00 take 70 ; 30 ; 65 ; 35 ; 25 ; 75 ; 1 7 ; 83 ; 33 ;66 ; 45 ; 87 ; 54 .

t. From 1 25 take 1 5 ; 1 7 ; 1 3 ; 1 2 ; 5 ; 8 ; 7 ; 1 6 ; 9 ; 1 1 ;6 ; 1 4 ; 1 6 ; 1 8.


59 . I l lustrative Examp le. Mr. Graves received a sal aryof 3000, and hi s son received it268. What was the difference in their sal aries

WRITTEN WORK ,To find the difference , a part of 3000 equa l to

(2) (9) (9) (1 0)268 must be taken away.

as3 O 0 0As there are no units , no tens , and no hundreds

2 6 8in the minuend, one of the 3 thousands is taken

(l eaving 2 thousands) and changed to hlmdreds2 7 3 2 one of the 1 0 hundreds is taken (l eaving 9 hun

dreds) and changed to tens ; and one of the 1 0

tens is taken (l eaving 9 tons) and changed to units. 3000 is thus

changed to 2 thousands 9 hundreds 9 tens and 1 0 units, fromwhich,

taking 2 hundreds 6 tens and 8 units, 2 732 remains . Ans . 8 2732 .

In practice simply think 8 from 1 0, 2 6 from9 , 3 2 from9 , 7 0 from 2 , 2 and write the resul ts in order.

P erform and prove the fol l owing exampl es

73 . 7 4 . 7 5 . 7 6 .

1 407 4000 9081 501 5

— 769 - 1 92 — 483 — 364

36 SUBTRACTION .

SUBTRACTION OF DECIMALS.

60 . I l lustrative Examp le. From mil es takemil es .

WRITTEN WORK The numbers are written as before, so that

1 6 .5units of the same order shal l be expressed in the

same column . Beginning with the units of the38 75

lowest order to subtract : 5 from 1 0, 5 ; 7 from 9 ,

2 ; 8 from 1 4 , 6 ; etc . Ans . miles.

98 . From take 1 00 . Subtract from 1 .

99 . From take 1 01 . Subtract from 1 0.

1 02 . Find the difference between 1 6 and 1 6 thousandths .Subtract the fol l owing in l ines and in columns

1 03 . 1 04 . 1 07 . 1 08 .

1 05 . 1 09 .

1 06 . 1 1 0 .

From the preceding exampl es may be derived the

fol l owing

Rul e for Subtracti on.

( 1 ) Write the minuend and underneath write the subtra

hend, so that un its of the same order sha l l be expressed in the

same column . Draw a l ine beneath.

(2 ) Begin with the un its of the lowest order to subtract, and

proceed to the highest, writing each remainder under the l ine

in its proper p lace.

(3 ) If any term of the minuend is less than the correspond

ing term of the subtrahend, add ten to it and then subtract ;

but consider that the next term of the minuend has been

diminished by one.

P roof .

Add the rema inder to the subtrahend the sum shoul d

equa l the minuend.

FORMS OF EXAMP LES IN sUBTRACTION . 37

62 . Forms of Exampl es in Subtracti on

1 1 1 . From 9460 take 5466 ; from the remainder take1 284.

1 1 2 . From ten thousand one hundred subtrac t fOur thousand fifteen ; from the remainder subtract 397 .

1 1 3 . How many more is 3685 than 1 8571 1 4 . What number added to 1 763 wil l make 1 8221 1 5 . The sum of two numbers is 745 ; the greater one is

569 . What is the smal l er one

1 1 6 . Find the difference between and twentythousand two hundred two .

1 1 7 . A minuend being 1 828 inches, a subtrahend 934

inches, what is the remainder1 1 8 . 2789 plus what number equal s 42801 1 9 . 7280 291 .

1 2 0 . A minuend being 5280 feet, and a remainder 2789feet, find the subtrahend.

1 2 1 . After a certain number had been taken out of 500 ,

there remained 1 84 . What was the number taken out

m From 980 take 98 ; from the remainder take 98, andso continue til l nothing remains . How many 98

’s . are

subtracted1 2 3 . l ess equal s what1 2 4 . rods rods how many1 2 5 . Subtract 1 tenth from 1 0 .

1 26 . Take 426 thousandths from 1 .

1 27 . added to What wil l make 500

1 28 . If 9 1 and 7 hundredths is the minuend and

the remainder, what is the subtrahend1 2 9 . Mr. Lee bought a farm for and sol d it

for 4525. How much did he gain1 30 . Mr. Fitz bought a house for 975. After spending

upon it 15 he sol d it for How much did helose 7

88 SUBTRACTION .

Exampl es for Oral or W ritten W ork.

63. a . In Jul y, during 1 7 days the wind bl ew from the

southwest, during 6 from the west ; for the other days, thewinds were variabl e. How many days were they variabl eb. Mr. Wood had 38 tomato pl ants, and sol d 1 2 to one

man and 8 to another. How many had he l eft 7c. Al vin had 27 peaches and 25 pears . He gave 9

peaches to Bennie and 8 pears to John. Howmany peacheshad he l eft How many pearsd. Having 80 mil es to travel , I rode 20 mil es in the

morning, 1 5 in the afternoon, and 1 2 in the evening. Howmany mil es of the journey remained

e. Harry was to be away from home 63 days. After25 days had passed, how many more days had he to remain

f . If James buys a kni fe for 35 cents and l oses 1 2 centsby sel l ing it, for how much does he sel l itg. Nel l ie’s cash account for a week was as fol l ows

On hand, 20 cents ; al l owan ce, 25 cents ; earned, 1 5 centsspent, 1 3 cents . What was the bal ance of her accounth. If you have 23 cents, how many more cents must you

get to buy a sl ed worth 62 centsi . Aman bought a horse for 75 and sol d him for 54 .

How much money did he l ose

j . Out of 50 butterflies observed by a boy, al l but 1 7

al ighted upon Objects approaching their own col or. Howmany al ighted on such protecting surfaces How manymore did al ight on such surfaces than did not

h. A caterer bargained to pay for a set of fixtures 90

in board. The board for four successive months was 1 2,

1 0, 1 5, and 1 1 . What stil l remained dueh. The roadway over the Natural Bridge in Virginia is

1 22 feet high from the creek bel ow, and 40 feet above thetop of the arch. What is the di stance from the top of thearch to the creek bel ow


m. The first Sunday school in New Engl and was establ ished in 1 793. How many years is it since then

n . Frankl in was born in 1 706 and died in 1 790 . Whatwas his age at the time of his death

O. In 1 830 the population of the U nited States was 1 3mil l ions ; in 1 890 it was 63 mi l l ions. How many mil l ionsdid the popul ation increase in that time 9

p . If sound travel s 4768 feet a second in sal t water,and 47 1 4 feet a second in fresh water, how many more feeta second does it travel in sal t water than in fresh

How much money would be l eft, and what might the coinsbe

,if a dol lar were given in payment for

q. A hat, a Sl ate, 1 5¢ ink,8¢

r. A dictionary, 67 3 dozen pencil s, 1 8¢s. Amicroscope, a case to put it in, 1 3¢t. 50 one-cent l etter stamps ; paper, 7¢ envel opes, 8¢u. A col l ar, 1 3¢ tie

,a handkerchief, 1 9¢

v.

“Robinson Crusoe,” 27 Grandfather’s Chair,” 36¢Black Beauty,” 24¢

64. Exampl es for W ritten W ork .

1 31 . The first sav ings bank was establ ished in 1 778 at

Hamburg ; the first establ ished in New Engl and was in

1 81 6 . What time el apsed between1 32 . The year’s earnings of a family were $ 1 925. If

their expenses were $ 1 1 36, how much was saved

1 33 . A and B together invested 5740 in business. If B

put in 2575, how much did A put in1 34 . The earni ngs of a coOperative bank for the second

Six months after its establ ishment were 1 874 .53,

'

whichwas 35 more than the earnings for the first sixmonths.

What wei'e the earnings for the first six months1 35 . TheAmazon River is 3750mil es l ong ; theMississippi

is 4300 mil es l ong. What is the difference in their l engths

40 SUBTRACTION .

1 35. The sai l ing distance fromNew York to Queenstownis 2890 mil es. When a steamer has run 1 368 mil es on hercourse fromNew York, how far has she stil l to go ?1 37 . Iron mel ts at a temperature of 3980 degrees Fahrenheit, and gold at 2590 degrees. How much higher .tempera

ture is required to mel t iron than gold1 38 . In one week a grain el evator received bushel sof grain ; of this bushel s were del ivered. Howmuch remained in the el evator1 39 . A pound of l ead, avoirdupois weight, equal s 7000

grains ; a pound of gold, troy weight, equal s 5760 grains .How many more grains are there in a pound of l ead thanin a pound of gold1 40 . What is the difference in value between a hal f eagl e5) of U nited States money and a pound of Engl ish

money valued at

1 41 . On the 1 st of January I had in a bank ;on the 3d I drew out and on the 4th,

HOW much remained in the bank1 42 . Aman having 9 1 8 kept 381 and bought with the

remainder a horse and carriage, paying for the horse 387 .

How much did he pay for the carriage1 43 . Two vessel s sail ing towards each other are 1 08 mil es

apart. If one sail s mil es and the other mil es, howfar apart wil l they be1 44 . The depth of rainfal l for the summer months was

found to be inches ; for June and July it wasinches. What was the depth of rainfal l in August1 45 . The several items of an account amount toof this amount has been paid. Find the bal ance.

Some Mountain Heights in Feet.

Mt. Blanc, Switzerl and, Mt . Olympus, G reece, 9745

Mt. Everest, Hindostan , P ike’s P eak, Col orado,

Mt. Hercul es, New Guiana, Mt. Vesuvius, Italy, 3948

42 SUBTRACTION .

1 61 . Having $ 675 in the bank, Mr. Black drew as fol

l ows : and How muchmoney was l eft in the bank1 62 . P revious to a fire, the contents of a store were esti

mated to be worth after the fire to be worththe difierence was to be paid by an insurance

company . How much was the company to pay1 63 . By making part payment of a note

, the interestupon it, which had been $ 1 25 a year, was reduced to

How much interest was saved by the payment1 64 . A specul ator bought a house for $3400, paid $ 1 800

for repairs, and 545 for grading the l ot, after which hesol d it for $8000 . Did he gain or l ose

, and how much1 65 . The pol ar diameter of the earth is 7 mil es.

How much greater is the equatorial diameter, which is7 mil es1 66 . In 1 892 P aris contained inhabitants, and

London more than P aris. Howmany inhabitantshad London1 67 . In 1 890 the U nited States numbered

inhabitants ; of these, ma l es and femal eswere foreign born . How many were native born1 68. At the Worl d’s Fair in Chicago in 1 893, the total

attendance was this was l ess thanthe attendance at the P aris Exposition in 1 889 . What wasthe attendance at the P aris Exposition1 69 . The sal aries of cl erks at the Chicago Fair amounted

to of officers, of musicians,of architects, What was the

sum of al l the sal aries1 70 . The l argest day’s attendance at each of three greatWorl d’s Expositions was as fol l ows : P aris, October 1 3,1 889

, P hil adelphia, September 28, 1 876,Chicago, October 9 , 1 893, What was the excessof Chicago

’s attendance over each of the others

MISCELLANEOU S EXAMP LES. 43

1 71 . The sum of three numbers is 4875 ; one of the num

bers is and another is What is the thirdnumber1 7 2 . From the sum of and take theirdifference .

1 73 . A cubic inch of mercury in a sol id form weighsof a pound, in l iquid form,

of a pound l ess.

What is the weight of a cubic inch in l iquid form

Find the bal ance of the fol l ow ing accounts

1 74 . 1 7 5 .

What is Subtraction 5’ What is the minuend‘

1’ The subtrahend

The remainder Take oral l y 36 from 84 . Find the difierence between

469 and 5007 , and expl ain. G ive the rul e ; the proof . When the

minuend and difference are given , how can you find the subtrahend ‘

1’

When the subtrahend and difference are given , how can you find the

minuend

‘Dr. is read debtor,”and means that the sums written beneath

are owed.

ICr. is read creditor,”and means that the sums written beneath

have been paid by the debtor or are due from him.

44 SUBTBACTION .

DRILL TABLE NO . 2 . (See Suppl ement, Art.

66 . For suppl ementary practice in subtraction .

q t

1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 20


1 78- 1 89 . In each l ine a , b, 6,

etc .,from 1 00 take

m+ n + a

From each number in w, take 1 728.

From 9234 take each number in w.

Take each number in x fromTake each number in y fromIn w,

take b from a ; d from c, etc .

In x, take b from a ; d from c, etc.

In y, take b from a ; d from 6, etc.

In each l ine a , b, c, etc.

, from x plus y take w.

SECTION IV

MU LTIPLICATION .

6 7 . I l lustrative Examp le. Find the number of days in 6months of 31 days each.

(1 ) Added. (2 ) Mul tip l ied. The number of days in 6

31 Mul tipl icand, 31 months of 3 1 days each can

31 Mul ti 1.

6be found by adding, as in the

p l er, margm, or by taking 5 1x 1 ’s,31 P roduct

,1 86 which is six, for the un its

31 of the number, and six 3’s,

31 which is 1 8, for the tens of

31 the number. The resul t is

1 861 86 . Ans . 1 86 days .

68. When equal numbers are united by un iting at oncethe units of the same order, the numbers are said to be

69 . One of the equal numbers to be united is the

mul tipl icand. The number that shows how many equal

numbers are to be un ited is the mul tipl ier. The resul tobtained by mul tipl ying is the product .

70 . Mul tipl ication is the process of uniting equal numbersto find their product.

7 1 . The mul tipl icand and mul tipl ier are cal l ed factors

(makers) of the product. The product is a mul tipl e of

the fac tors making it ; thus, 1 5 is a mul tipl e of 5 and of 3 .

Now .-The uni ts of the product are of the same kind as the

units of the multipl icand. Thus, if days are mul tipl ied, the product

is a number of days if tens are mul tipl ied, the product is a number of

tens ; if tenths are mul tipl ied, the product is a number of tenths .

45

46 MULTIP LIC’ATION .

72 . The sign of mul tipl ication is an obl ique cross x .

The expression 40 x 5 200 means that five 4o’s are 200.

It is read “ 40 mul tipl ied by 5 equal s It may al so beread five 40’s equal or

“ five times 40 equal sIn the exampl e above, what is the mul tipl icand the mul tipl ier

the product Name two factors of 24 . Name two other factors.

Name another two. Read the expression 1 5 x 3 45.

73 . How many are 5 x 3 ? 3 x 5 ? 2 x 4 x 3 ? 4 x 2 x 3 ‘

?

The product of two or more factors is the same, wha tever

the order in which the factors are taken .

74 . To find the product of two factors in the Mul tipl ication Tabl e, find one of the factors in the l eft-hand col umn,the other in the top l ine ; the product wil l be found in the

same l ine with the first factor and in the "

same column withthe second.

MULTIPLICATION TABLE .

pQ

N

N

H

H

H

H

H

no

m

o

m

m

a-

w

o

m

en

»

OO

NJ

M

N

H

H

H

E

17 37—

35

— a_ _

84 96 1 08 1 20 1 32 1 44

ORAL EXERCISES. 47

75. The numbers 1 , 4, 9, etc. , in the diagonal of thetabl e are the products of two equal factors, and are cal l edthe squares of the numbers 1 , 2 , 3, etc.,

respectively.

76 . I. a . Repeat the .mul tipl ication tabl e of 2’s ; thus,

Two 2 ’s are 4, three 2

’s are 6, four 2 ’s are eight,” to

“ twel ve 2 ’s areb. Repeat the same in the reverse order.

c. In the same way repeat the mul tipl ication tabl e of3’s ; 4

’s ; 5

’s ; 6

’s ; 7

’s ; 8

’s ; 9

’s ; 1 0

’s ; 1 1

’s ; i 2

’s.

II. d. Name the squares of the numbers from 1 to 1 2 ;thus, 1 , 4, 9, etc. (Art.

III. e. Turn to page 44 and mul tipl y the number ex

pressed by each figure in l ine a by 2 ; by 3 ; by 4 ; 5 ; 6 ;7 ; 8 ; 9 ; 1 0 ; 1 1 ; 1 2 .

IV. Give at sight the products in the fol l owing exam

ples in the columns and in the l ines .*

a b e f g h

m 2 x 5 4 x 7 7 x 4

n 3 x 4 4 x 8 7 x 9

o 7 x 3 3 x 9 9 x 7

p i x 5 8 x 6 8 x 8

9 2 x 7 9 x 2 6 x 7

r 4 x 3 5 x 7 4 x 8

s 5 x 4 6 x 5 9 x 6

t 3 x 8 6 x 7 4 x 7

Thus in columns a, b, etc. ,do not say three 2 8 are 6 ; four 5

’s

are 20 ” but simpl y 6 , etc .

v: j7 x 5

6 x 9

8 x 9

1 1 x 6

7 x 1 2

1 1 x 1 0

1 1 x 1 2

48 MULTIP LICATION .

V. a . Mul tiply each of the fol l owing numbers by 2 and

add 1 to the product, naming onl y final resul ts.

1 3 6 4 7 2 8 5 9

Mul tiply the numbers above

b. By 3 and add 2 . f . By 7 and add 5 ; 6 .

c. By 4 and add 2 ; 3 . g. By 8 and add 6 ; 7 .

d . By 5 and add 3 ; 4 . h. By 9 and add 7 ; 8.

e. By 6 and add 4 ; 5. i . By 1 2 and add 8 ; 9 .

How many cents equal in val ue

j . 3 two-cent pieces 1 centk. 8 ten-cent pieces 2 centsl . 6 five-cent pieces 3 centsm. 9 ten-cent pieces 5 cents

77 . I l lustrative Examp le. Mul tiply 60 by 4 .

SOLUTION .— 4 times 6 tens is 24 tens, or 240 . Am.

Naming resul ts only,

Mul tiply 40, 70, 20, 60, 50, 30, 80, 90,each by 2 ; by 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 1 0.

0 . Mul tiply 600, 400, 200, 700, 500, 300, 800, 900,each by 2 ; by 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 1 0.

78 . Il lustrative Examp le. What is the cost of 4 wagonsat 8 70 each

Sow rrom— Since 1 wagon costs $ 70, 4 wagons wil l cost 4 times

8 70, or $ 280 . Ans. 0 280 .

p . What is the cost of 2 horses at 200 eachq. The original val ue of shares in the Washington mil l s

was 1 000 each. What was the value of 1 0 sharesr. At 60 pounds to a bushel , how many pounds are there

in 6 bushel s of wheat In 9 bushel s of potatoes

50 annzauruwnnarrosn


80 . Il lustrative Examp le. How many days are there in3 years of 365 days each

WRITTEN WORK .Writing the mul tipl icand and mul tipl ier as

in the margin, mul tiply each term separatelMul tl pl icand, 365beginning with the units.

Y

Mul tl pher, 3 Three 5’s are 1 5 ; 1 5 units equal 1 ten

P roduct 1 095 and 5 units ; write 5 under the l ine in the

units’ place, and keep the 1 ten to add to the

product of the tens .

Three 6 ’s are 1 8 1 8 tens with the 1 ten kept, equal 1 9 tens, or 1hundred and 9 tens. Write 9 under the l ine in the tens

’

place , and

keep the 1 hundred to add to the product of the hundreds .

Three 3’s are 9 . 9 hundred and the 1 hundred kept equal 1 0 hundred, or 1 thousand and no hundreds. Write 0 under the l ine in the

hundreds’ pl ace, and 1 in the thousands’ place . The entire product is

1 095. Ans. 1 095.

7 .

4829

x 2

1 2 . 1 3 .

4689 7529

x 3 x 6

1 8. 6838 x 9 20 .

1 9 . 6249 x 8 2 1

For the sake of rapid working, use as few words as possibl e.

Thus, in the exampl e above, say fifteen ,

” “eighteen,

” “nineteen,

”

nine,” “ ten.

” While saying fifteen, write 5 ; while saying nineteen,write 9 ; and whil e saying ten, write 1 0.


24 . 1 257 x 9 26 . 3798 x 1 2 as x 1 0 30 . x 1 1

25 . 2806 x 2 7 . 7381 x 1 1 29 . x 1 2 31 . x 1 2

Mul tiply How many are32 . 2581 days by 4 36 . 7682 sheep x 733 . 3428 hours by 5 37 . 3085 books x 934 . 7398 mil es by 8 38 . 7692 rods x 1 135 . 9402 eggs by 6 39 . 8548 feet x 1 2

Find the sum of the first two products in each of thefol l owing exampl es, and see if it agrees with the third

product.

40 .

2468 x 3

2468 x 7

2468 x 1 0

81 . Mul tipl icati on tens and units .

I l lustrative Examp le. At 8 36 each, how much wil l7 coats cost

SOLUTION. For oral work mul tiply first the tens, then the units ;

thus 7 times 8 30 is 8 2 1 0, and 7 times 8 6 is 8 42 , which, added to

82 1 0, makes 8 252 . Ans. 8 252 .

Find the cost48 . Of 6 mugs at 1 5 55 each . 50 . Of 1 0 hats at 64 55 each.

49 . Of 4 caps at 75 each. 51 . Of 9 coats at 8 34 each.

52 . Of 7 cows at 8 56 each, and 4 oxen at 8 1 00 each .

53 . Of 1 0 sl eighs at 8 75 each, and 1 0 robes at 8 8 each.


54 . If 3 boys can weed a row of beets in 1 8 minutes, howl ong wi l l it take 1 boy to weed it55 . How many l ines are there in two magazine articl es,

one of 7 , and the other of 5 pages of 48 l ines each4 56 . If Charl es and Wil l iam earn 35 cents an hour each,how much wi l l both earn in 8 hours

82 . I l lustra tive Examp le. How many pounds of butterare there in 47 boxes of 5 pounds eachSOLUTION .

— 47 boxes of 5 pounds each , wil l contain 47 times 5

pounds . But 47 times 5 equals 5 times 47 , which is 235 (Art.Ans. 235 pounds.

NOTE . In expl aining probl ems, fol l ow the l ogical process in doing

the written work, take the shortest way , disregarding the denomination ; but with the resul t, give the denomination required by the

statement of the probl em.

What must be paid for the fol l owing purchases57 . 32 spool s of cotton at 4 60 . 76 feet whitewood at 858 . 50 yards ofmusl in at 9 61 . 80 feet pine at 7

59 . 48 yards of gimp at 8¢ 62 . 1 00 feet walnut at 1 163 . A deal er bought two horses at 8 1 35 each. The next

week he sol d the Span for 8 348. Did he gain or l ose, andhow much64 . How much is gained by a broker in exchanging for

eign coins for 8 235 U nited States money, if he makes 2cents on a dol l ar

83 . In 1 whol e thing of any kind, how many hal ves are

there How many thirds fifths eighths etc.

65 . 1 2 whol es are how many hal ves sixths ninths66 . 30 whol es are how many thirds fifths tenths ‘7

67 . How many hal ves are 1 6 and 1 hal f ? 62 and 1 hal f ?68 . How many thirds are 33 and 1 third 66 and 2

thirds 1 6 and 2 thirds69 . How many fourths are 31 and 1 fourth 1 8 and

3 fourths


84 . Hu mpl es for W ritten W ork .

70 . How many mi l es can be travel ed in 6 days, the ratebeing 345 mil es per day7 1 . Three heirs inherited 8 5468 each. How much did

al l inherit72 . How many kernel s of corn

, 5 kernel s to a hil l , are

pl anted in 8 rows of 1 27 hil l s each 973 . What must be paid for 4 barrel s of sugar, each con

taining 331 pounds, at 5 cents a pound74 . In one year Texas produced pounds of rice.

What was its value at 6 cents a pound75 . What wi 1 1 825 sheep cost at 8 9 apiece at 8 8 apiece7 6 . What must be paid for the use of 8 3695 for a year,

if 7 cents is paid on every dol l ar77 . What must be paid for insuring a house worth

at the rate of 2 cents on a dol l ar

78. When 7 1 8 bricks are required for each rod of sidewalk

,how many w il l be required for 1 1 rods ? for 1 2 rods

85 . I l lustrative Example. At 56 pounds to a bushelhow many pounds are there in 1 0 bushel s of corn ? in

1 00 bushel s in 1 000 bushel s

SOLUTION .— Ten 56 ’s equal 56 tens (Art . or 560 one hundred

b6 ’s equal 56 hundreds, or 5600 one thousand 56 ’s equal 56 thou

sands , or Ans. 560 pounds 5600 pounds pounds.

In mul tiplying by 1 0 , 1 00 , 1 000 , etc.,the product may be immedi

ately expressed by annexing to the expression for the mul tip l icandas many zeros as the mul tip l ier has .

Find the sum of the products in exampl e bel ow

79 . 80 . 8 1 .

42 x 1 0 689 x 1 00 4692 x 1 009 x 1 00 83 x 1 000 4692 x 1 000

25 x 1 00 1 25 x 1 0 4692 x


Find the total value of

82 . 75 town bonds at 8 1 00, 83 . 1 00 cords of wood at 8 8,1 00 R R. tickets at 8 9, 1 0 saddl es at 8 42 ,450 eagl es at 8 1 0 . 600 acres of land at 8 1 00.

86 . I l lustrative Examp le. There are 640 acres in 1

square mil e . How many acres are there in 200 square mil es

x 1 0, and 200 2 x 1 00 ; hence

WRITTEN WORK.

.640 x 200 is the same as 64 x 2 x 1 0 x 1 00

(Art.

640 When the mul tipl icand and mul tipl ier, one or

200 both , have zeros at the right, the zerosmay be dis

regarded inmul tip lying, but theremustbe annexed1 281000 to the expression for the product as many zeros

as were disregarded.

84 ..Mul tiply 475 by 20 by 300 ; by 4000 ; and add the

products .85 . Mul tiply 7500 by 500 ; by 6000 ; by 70 ; and add the

products.

86 . Mul tiply 6740 by 8000 ; by 1 1 0 ; by 1 200 ; and add

the products.

87 . Mul tiply 869 by by 9000 ; by 900 ; and add

the products.

88 . There are 60 seconds in a minute,and 60 minutes in

an hour. How many seconds are there in an hour89 . To al l ow 50 cubic feet of air per minute for eachchild

,how many cubic feet of air per hour should pass

through a school hal l containing 840 pupil s90 . During the World’s Fair in Chicago there were

paid admissions. At 50 cents each,what amount

was paid for admissions9 1 . Mul tiply 282 by 8 ; by 70 ; and by 900 ; and add the

products.

92 . Mul tiply 31 2 by 3 ; by 40 ; by 200 ; and add the

products.


87 . I l lustrative Example. Mul tiply 31 2 by 243 .

WR ITTEN WORK .For convenience, the mul tipl i

cand, 3 1 2 , is mul tipl ied by each

(1 ) 3 1 2 (2 ) 3 1 2 term of the mul tipl ier separatel y .

243 24 The onl y difference between forms936 936 ( 1 ) and (2 ) of the written work

1 2480 1 248 is, that in form (2 ) the zeros in62400 624

the partial products are omitted .

7581 6 7581 6In the second form, care must

be taken to place the first or

right-hand figure of each partial product directl y under the term of

themul tipl ier used.

Mul tiply the fol l owing as indicated in l ines and columns,and prove each exampl e by mul tiplying the mul tipl ier bythe mul tipl icand (Art. 7

1 20 . 1 2 3 . 1 24 . 1 2 7 . 1 28 .

Mul tiply 1 1 7 . 596 x 639 1 21 . 788 x 928 1 25 . 79 1 2 x 395by 842 x 978 1 2 2 . 837 x 842 1 2 6 . 6998 x 876

88. Il lustrative Examp le.

WRITTEN WORK .

7643

604

30572

45858

461 6372

99 . 1 00 .

9 7 . 243 x 54298 . 576 x 342

1 09 . 1 065 x 9251 1 0 . 5670 x 573

Mul tipl y 7643 by 604 .

1 2 9 . 7338 x 305

1 30 . 9560 x 7080

1 31 . 30972 x 289

1 32 . 44002 x 6007

1 33 . 1 3487 x 1 060

1 34 . 69080 x 7508

56 MULTIP L'

ICATION .

MU LTIPLICATION OF DECIMALS.

89 . I l lustrative Examp le. Mul tiply by 9 .

WRITTEN WORK . Since the lowest order of units in the mul ti

307 48pl icand is hundredths, the l owest order in the

product must be hundredths, which is expressed9 by putting the decimal point two places from the

27673 2 right. Ans.

1 35 . Mul tiply by 8. 1 38 Mul tiply by 40.

1 36 . Mul tiply by 43. 1 39 . Mul tiply by 1 31 .

1 37 . Mul tiply by 39 . 1 40 . Mul tiply by 325.

-1 4 1 . There are feet in a rod. How many feet arethere in 23 rods1 42 . If an express train goes mil es an hour for 1 5

hours,What distance does it go

90 . From the preceding exampl es may be derived the

fol l owingRul e for Mul ti p l icati on .

(1 ) Write the mul tip l ica nd, and underneath write themul ti

p l ier. Draw a l ine beneath.

(2) If the mul tip l ier consists of one term on ly,mul tip ly each

term of the multip l icand by the mul tip l ier, beginn ing with the

term of the lowest order, and carrying as in addition .

(3) If themultip l ier consists of more than one term,multip lyby each term of the mu ltip l ier sep arately, writing the p artia l

products so that units of the same order sha l l be expressed in

the same column .

(4) Add thepartia l products thus obta ined, and the result

wil l be the entire product.

P roof .

Multip ly the multip l ier by the multip l icand. If the work is

correct, the two products wi l l be equa l .

58 MULTIP LIC'ATION .

92 . P erms of Exampl es in Mul tipl ication.

1 60 . Mul tipl y by 9, by 8, and by 7, and add

the products.

1 61 . Mul tipl y 78 by 1 0, by 3000, and by 600, and add

the products .

1 62 . Find the product of 368 by 59 .

1 63 . How many are 41 6 times 9851 64 . Two thousand four hundred ninety-nine x seven

hundred eight 9

1 65 . x 67 plus 1 900 x 800 what number1 66 . x 432 are how many1 67 . Take for a mul tipl icand and 1 59 for a.mul ti

pl ier, and find the product.1 68 . Amul tipl ier is and the mul tipl icand thirty

six thousand sixty-nine ; what is the product1 69 . One factor being and the other 6005, find

the product.1 70 . What product wil l be obtained by using 25 as a

factor tw ice three times1 7 1 . x 506 1 72 . Mul tiply by 6 .

NOTE .— When a zero occurs at the right of a decimal fraction , it

may be disregarded. In the answer to the 1 72d exampl e , 750

thousandths is the same as 75 hundredths.

1 7 3 . Mul tiply by 1 5 ; by 3 ; by 5 ;and add the products.

1 7 4 . What is the product of 1 8 thousandths mul tipl iedby 1 4 91 75 . Mul tipl y by 1 0, by 1 000, by 1 00 ; and add the

products.

1 7 6 . At apiece for tumbl ers, what is the cost of5 l ots of 1 00 tumbl ers each1 7 7 . At 30 cents an hour, how much can a person earn

in 1 5 days of 9 hours each1 78 . What is the cost of 1 6 yards of silk at 8 a

yard, and 25 yards of cambric at 8 a yard

WEIGHTS AND MEASURES. 59

WEIGHTS AND MEASURES IN COMMON USE.

93 . LIQUID MEASURE . 9 4. DRY MEASURE .

4 gil ls (gi .) 1 pint 2 pints 1 quart2 pints 1 quart 8 quarts 1 peck

4 quarts 1 gal l on 4 peeks 1 bushel

95 . NUMBERS . 96 . P APER .

1 2 ones 1 dozen 24 sheets l quire .

1 2 dozen 1 gross. 20 quires 1 ream.

1 2 gross 1 great gross. 2 reams 1 bundl e.

20 ones 1 score . 5 bundl es 1 halo.

9 7 . AVOIRDUPOIS WEIG RT.

Common Weights .

1 6 ounces (oz . ) 1 pound

2000 pounds 1 ton

G ross Weights.

1 1 2 pounds 1 hundred weight20 cwt . (2240 1h. ) 1 long ton .

NOTE .— In weighing some articl es, as iron and coal at the mines,

and goods on which duties are paid at the U nited States customhouses, the l ong ton of 2240 pounds is used ; when the kind of ton is

not designated, the common ton of 2000 pounds is understood.

Oral Exercises .

98 . a . Repeat the tabl e of l iquid measure ; of dry measure ; of numbers ; of paper ; of avoirdupois weight.b. How many things are there in 9 dozen in a gross

in a great grossc. 4 gal l ons of oil are how many quarts pints gil l s

d. 2 bushel s of corn are how many peeks quartse. A ream of paper is how many quires sheetsf . 6 gal l ons 3 quarts equal how many quartsSOLUTION .

— Since 1 gal l on equal s 4 quarts , 6 gal l ons wil l equal6 times 4 quarts, or 24 quarts, which with 3 quarts added equal 27


g. 6 dozen and 4 equal how manyh. How many sheets of l etter paper are 2 quires 4 sheets

What must be paid fori . 2 bushel s 1 peck of potatoes at 40 cents a peckj . 3 peaks of berries at 6 cents a quarth. 4 bushel s of wal nuts at 1 0 cents a quartl . 3 quarts 1 pint of cream at 20 cents a pintm. 1 pint 3 gil l s of col ogne at 25 cents agil l

Exampl es for W ritten W ork.

99 . I l lustrative Examp le. Howmany ounces are 35 lb. 7 oz.?

Observe that the 7 ounces are added withWRITTEN WORK ’

the partial products to make 567 .

35 l b‘ 7 O ! ‘

1 79 . Howmany pounds are there in 61 6

l oads, each contain ing 2 tons 750 pounds ?2 1 0 1 80 . A gardener had 1 8 pounds of35 seeds which he put into bags, using

567 oz

4 bags for every ounce . How manybags did he use

1 8 1 . How many papers of matches are 48 great gross and1 0 gross

1 82 . How many l ines has a person written who has

fil l ed a quire of l etter paper both sides, writing 24 l inesto each page1 83 . How many l etters can be written on a ream of

l etter paper, using one hal f sheet for each l etter1 84 . What is the difference in pounds between 250 l ong

tons and 280 short tons1 85 . How much wil l a deal er receive for 6 gross ofcl othespins at 1 0 cents a dozen, 2 gross of sl ates at 4 centsapiece, and a great gross of buttons at 1 1 cents a dozen1 86 . Find the total cost of 4 gross packages of tacks at

8 a gross, 2 gross at 1 gross at and 7

gross at 8

TIME MEASURE. 61

1 00 . TIME MEASURE .

60 seconds (sec. ) 1 minute (min ) .60 minutes 1 hour24 hours 1 day

7 days 1 week

365 days,

or 52 weeks 1 day

366 days 1 l eap year.

1 00 years 1 century .

1 common year

The year begins with the first of January , and is divided into fourseasons of three months each , as fol lowsThe w intermonths are December, January , and February .

The spring months are March, April , and May .

The summermonths are June,July , and August.

The autumn months are September, October, and November.

Thirty days have September,April , June, and November,Al l the rest have thirty-oneExcepting February alone,

To which we twenty-eight assign

Til l l eap year gives it twenty-nine.

NOTE .— For expl anation of l eap year, see Suppl ement, Art . 26 .

Oral Exercises .

1 01 . a . Howmanymonths are there in 1 y . in 3 y . 4 mo.

in 8 y. 4 mo. ? in 1 6 y . 8 mo. ?

b. Whi ch months have 30 (1 . each Which have 31 (1 .

eachAl l owing 30 days to a month, change

6 . 2 mo . 3 d . to days. e. 5 mo. 1 0 d. to days.

d. 3 mo. 3 d. to days. f . 4 mo. 1 7 d. to days.

9 . Change 4 wk. 6 d. to days ; 5 d. 1 0 h . to hours.

h. Change 3 h . 20 min . to minutes ; 24 h. to minutes ;1 h . to seconds .

i . What date is 30 days l ater than April 5 than May 5j . What date is 30 days earl ier than Apri l 5 thanMay 5


xHow many days are there

It. FromJune 1 to Jul y 1 m. FromApr. 1 toMay 1 0l . FromNov . 25 to Dec. 5 n . FromApr. 20 to June 1 0o. How many minutes are there between 6 o’cl ock 45

min . A.M. and 8 o’cl ock 55min . A. M. between 1 1 o’cl ock 20min . A.M. and 2 o’cl ock 52 min . P .M.

MEASURES OF LENG TH.

How many inches l ong is thisbook ? How many inches wide

LONG MEASURE.

1 2 inches (in .) 1 foot

3 feet 1 yard

5 and 1 hal f yards, 1 (1 rdor 1 6 and 1 hal f feet} ro

320 rods , or 5280 feet 1 mil e

1 03 . Exampl es for W ritten W ork .

What is the price of 1 8 yards of rubber hose at

1 6¢ a foot1 86 . To find the W idth of his house l ot

,Mr. Al l en

appl ied a yard stick 57 times, and 2 feet then remained.

How wide was the l ot in feet ?1 89 . What is the cost of netting for the two ends of a

tennis court, each measuring 1 8 yards, at 75¢ per runningfoot1 90 . How many rods does a person travel in going 1 2

mil es In going 300 mil es ?1 9 1 . If there are quarts of milk in a l iter, how

many quarts are there in 1 0 l iters in 1 00 l iters1 92 . Mt. St. El ias is 3 mil es and 2060 feet high . What

is the height in feet

MEASURES OF SURFACE. 63

MEASURES OF SURFACE.

1 04. Aflat surface, as the surface of a sl ate or the top

of a tabl e, is a pl ane surface. Such a surface bounded byl ines is cal l ed a pl ane figure.

1 05 . Each of the figures in the mar

gin is bounded by how many straightl ines How many square corners haseach A pl ane figure bounded by fourstraight l ines and having square corners An Oblong

is a rectangl e.

Which of the two rectangl es has al lits sides equal Such a rectangl e is a

1 06 . A square each of whose sides is1 inch l ong is a square inch.

What is a figure cal l ed each of whosesides is 1 foot l ong 1 yard l ong

The Areas of Rectangl es .

1 07 . If a rectangl e is 4 inches l ong and 1 inch wide, howmany square inches does it containThe cut represents a rectangl e 4 inches long

and 2 inches wide, scal e 1 fourth of an inch to

1 inch ; how many times 4 square inches doesthe rectangl e contain

How many times 4 square inches wil l a rectangl e contain that is

4 inches long and 3 inches wide 4 inches long and 4 inches wide

How many square feet wil l a rectangl e contain that is 4 feet l ong

and 3 feet wide 5 feet long and 4 feet wide

1 08. The area of a surface is its contents reckoned insquare units .

To find the area of a rectangl e, mul tip ly the number ofunits in the length by the number of l ike units in the width.

Theproduct is the number of square units in the area .


Oral Exercises .

1 09 . a . Howmany square inches are there in a rectangl e6 inches l ong and 4 inches wide 8 inches l ong and 5 incheswide 1 2 inches ( 1 foot) l ong and 1 2 inches (1 foot) wideThen how many square inches are there in a square footb. How many square feet are there in a rectangl e 3 feet

( 1 yard) l ong and 3 feet ( 1 yard) wide Then how manysquare feet are there in a square yard

1 1 0 . SQUARE MEAsU RE.

1 44 square inches (sq . in . ) 1 square foot (sq .

9 square feet 1 square yard (sq.

c. Measure the l ength and width of the top of your desk,and find how manysquare inches its surface contains.

d. How many square inches of writing surface are on

your sl ate On four l eaves of your writing book6 . Howmany square feet of l and has a strip 20 feet l ong

and 4 feet wide Has a court 20 feet square (20 feet l ongand 20 feet wide)f . If you know your steps are 2 feet l ong, how can you

find the number of square feet in a rectangul ar l ot of l and

g. P ace off a portion of the school yard, and find its area.

h. Find the number of square feet in a school yard 200feet l ong and 1 00 feet wide . In a park 200 feet square.

What is the difference between 4 square feet and 4 feetsquare Make diagrams to show the di fference.

1 1 1 . Exampl es for W ritten W ork.

1 93 . A sitting room is 1 5 feet l ong and 1 4 feet wide .

How many square feet does the floor contain1 9 4 . How many square yards of oil cl oth wil l be required

to cover a floor that is 4 yards l ong and 3 yards wide, and

how much wil l it cost at 62¢ per square yard1 95 . How manymore square feet has a grass pl ot 50 feet

square than a pl ot 1 00 feet l ong and 25 feet wide

MULTIFLI0ATION

MEASURES OF VOLUME.

1 1 2 . What two dimensions do surfaces have Name

some things that have anotherdimension . A figure that hasthreedimensions, l ength, breadth,and thickness, is a sol id. A sol idthat is bounded on al l sides byrectangl es is a rectangul ar sol id.

Name a rectangul ar sol id. Howmany rectangl es does it haveThe rectangl es are the faces of

the sol id and together make itssurface. The l ines formed by themeeting of the rectangl es are theedges of the sol id.

1 1 3 . A sol id that is boundedby six equal squares is a cube.

If the edges are 1 inch l ong the sol id is a cubic inch. Whatis the sol id cal l ed if each edge is 1 foot l ong 1 yard l ong

BECTANOU LAE Sou p s.

A Cube.

1 1 4 . The volume of a sol id is its contents reckoned incubic un its.

The Vo lume of Rectangul ar So l i ds .

1 1 5 . If a bl ock of marble is 4 feet l ong and 2 feet wide,how many square feet does its l ower surface contain ?

If the block is 4 feet long, 2 feet wide ,

and 1 foot thick, how many cubic feet

does it contain 2 How many times 8cubic feet wil l it contain if it is 2 feet

thick if it is 8 feet thick

The volume of any rectangul arsol id is found by mul tip lying the

number of units in the length by the number of l ike units in

MEASURES OF VOLUME . 67

the breadth, and this product by the number of l ike units in

the thickness . This is expressed, for brevity, as multip lyingtogether the length, breadth; and thickness.

Ora l Exercises .

1 1 6 . a . What is the number of cubic inches in a cube1 2 inches l ong, 1 2 inches w ide, and 1 2 inches thick, or in 1

cubic foot How many cubic inches are in a cubic footb. How many cubic feet are in a cube 3 feet l ong, 3 feetwide

, and 3 feet thick, or in 1 cubic yard

1 1 7 . CURIc MEAsU RE .

1 728 cubic inches 1 cubic foot (cu .

2 7 cubic feet 1 cubic yard (cu.

1 28 cubic feet 1 cord used in measuring wood.

6 . How many cubic feet has a rectangular bl ock of marbl ethat is 3 feet square at the end and 9 feet l ong 4 feetsquare by 8 feet l ongd. How many bl ocks, each of whose edges is 1 inch, can

be made from a cubic foot of wood, no al l owance to bemade for waste in sawing6 . When a number is used three times as a factor, the

product is cal l ed the cube of the number. Name the cubesof the numbers from 1 to 1 0 .

1 1 8 . Exampl es for W ritten W ork .

207 . Yel l owstone P ark is a rectangl e 65 mil es l ong and

55 mil es wide . What is its area in square mil es208. How many cubic feet of earth must be removed in

digging a cel l ar 25 feet l ong, 1 6 feet wide, and 1 1 feet deep209 . How many cubic inches would 1 000 bricks occupy,

each brick measuring 8 inches by 4 inches by 2 inches2 1 0 . How much must be paid for 1 2 drain til es at 25

cents each, 6 hours of masons’ work at 40 cents an hour,and 7 hours’ digging at 20 cents an hour


2 1 1 . It is 90 mil es fromN ew York to P hiladel phia. Howmany mil es wil l he travel ed by a conductor who makestwo trips daily for 31 3 days21 2 . In one season

,the heav iest peach train to Boston

contained 40 carl oads. At 540 baskets a l oad, and 8a basket, what was the value of the peaches21 3 . Mr. Gray sol d hops for a farmer to the amount

of 8 365, taking for sel l ing them 1 0 cents on each dol l ar hesol d them for. How much money did he take, and how

much was due the farmer21 4 . A invested 8 200 and B 8 590 in business. They l ost

45 cents on each dol l ar invested . How much did each l ose2 1 5 . A capital ist sol d for 8 78 each, 1 000 shares of stock

which cost him 8 1 4 a share . How much did he gain2 1 6 . Two steamers started at the same time from the

same pl ace, and sail ed in opposite directions, one at the

rate of 1 8 mil es an hour,and the other at 22 mil es an

hour. How far apart were they at the end of 6 hours21 7 . If the above steamers had been 680 mil es apart, and

sail ing towards each other, how far apart woul d they havebeen at the end of the same time2 1 8 . One horse-power can raise 550 pounds 1 foot in

a second ; how many pOImds can a 50 horse power engine

raise 1 foot in a second how many pounds can it raise1 0 feet in a second2 1 9 . A charity col l ection contained 2 fifty-dol l ar bil l s, 9

tens, 1 1 fives, 4 twos, 27 ones, 2 hal f-eagl es, 1 6 hal f-dol l ars,26 quarters, 46 dimes, and 1 9 nickel s . What was the

value of the whol e col l ection2 20 . What must be paid for 250 crowns, Austrian money,

at 8 per crown2 2 1 . A steel company in Harrisburg advertises to make

door mats of any size for 75 cents a square foot. Howmuch must I pay for 6 mats, each 3 feet l ong and 2 feetwide, and one dozen, each 5 feet l ong and 3 feet wide


22 2 . The tel egraphic pl ateau beween Newfoundl and and

Irel and is 1 640 miles long, with an average width of 400mil es. How many square mil es is its area223 . What is the cost of 50 pounds sterl ing at

per pound224 . A square mil e is 640 acres. How many acres has

the District of Columbia, which contains 70 square mil es225 . Grace has 8 Charl es has 5 times as much as

Grace, and Del ia has 8 times as much as Charl es and Gracetogether. How much money has Del ia22 6 . How many square rods are there in a rectangul

fiel d rods l ong and 1 3 rods W ide227 . Mr. Mead has acres of l and in one l ot, 4

times as much in another l ot, and acres in a thirdlot. How many acres has he in al l ?

228 . At 1 8 cents per pound, find the cost of butter in43 firkins of 58 pounds each.

229 . A man invested 8 1 50 in potatoes, and by sel l ing

them gained 25 cents on every dol l ar invested. What washis gain What were his total receipts230 . Abazaa r was attended by 5326 persons, 1 35 of whomwere admitted free ; the remainder paid 75 cents each fortickets. What were the receipts for admission231 . I buy each year 1 0 tons of stove coal and 8 tons of

furnace coal . Last year stove coal was 8 a ton,and

furnace coal 8 this year each kind costs 8 a ton .

How much more does my coal cost this year than l ast232 . A deal er has 225 cords of pine wood, 540 cords of

oak, and 1 64 cords of mapl e. He se l l s 1 38 cords of pine at

8 8 a cord, 1 25 cords of oak at 8 a cord, and 88 cordsof mapl e at 8 a cord. How many cords has he l eft ?How much money does he receive for what he sel l s

233 . A man receives 8 40 a month as night watchman,and works besides 5 hours a day for 24 days of the monthat 20 cents an hour . How much does he earn in a year

70 MULTIP LIC'ATION .

234 . Some oil fiel ds in Ohio are said to yie l d 5370 barrel sof oil daily. At this rate, how much wil l they yiel d in thel ast six months of the year2 35 . At the rate of 8 on a thousand dol lars

,what

wil l be the tax on 8 5000 worth of real estate, and 8of personal property236 . The owner of a wood l ot valued at 8 5850, paid each

year 3 cents on a dol l ar to insure himsel f against l oss by itstaking fire . What sum did he pay in 5 years23 7 . At 60 cents per square yard, find the cost of flagging

to l ay two wal ks, one 220 yards l ong, and the other 560yards, each 3 yards wide.

238 . How many pickets, 5 to a yard,wil l be required

to fence a rectangul ar fiel d 69 yards l ong and 65 yardswide2 39 . If in one yard of cl oth there are 580 fibers of warp

and 432 of fil l ing, and each fiber of warp contains 32 strands,and each of fil l ing 48, how many strands are there in the

yard240 . In 1 850, J . M. Hosmer’s ranch in Cal ifornia yiel ded

bushel s of potatoes at 8 5 per bushel , poundsof onions at per pound, pounds of pumpkinsat per pound . Find the total value of the yiel d.

241 . In a Vermont marbl e quarry were moved upon a

truck 3 bl ocks of marbl e, each 6 feet l ong and 4 feet squareat the end. If a cubic foot of marbl e weighs

pounds, what was the entire weight

What is Mul tip l ication What is the mul tipl icand P the mul ti

pl ier the product What are factors Mul tiply oral ly 45 by 6 .

P erform and explain an exampl e in which the mul tipl ier has at least

two terms . G ive the rul e ; the proof. How do you multipl y by 1 0,1 00, 1 000, etc . How do you proceed if there are zeros at the right

of the expression of the mul tipl icand or the mul tipl ier, or both

Tens x units what U nits x tens Thousands x tens Tens

x hundreds Ten-thousands x hundreds P

DRILL TABLE.

DRILL TABLE No . 8. (See Supplement, Art.

1 1 9 . For suppl ementary practice in mul tipl ication.

p q t

1 2 3 4 5 6 7 s 9 1 0 1 1 1 2 1 3 1 4 1 5 1 6 1 7 1 8 1 9 20

Exercises upon the Tabl e .

In each of the twel ve l ines a, b, 6 , etc. , mul tiply

242 — 2 53 . w by 6 . 350 —36 1 . m by n by 0 .

2 54-2 65 . x by 8. 362 —3 73 . t by u by v.

266 -2 7 7 . y by 9 . 3 74—385 . w by 34 .

ave-ass. 8p , q¢ by 7 . 3864 97 . by w.

290-301 . 8 r,s¢ by 1 1 . see—409 . by y.

302—3 1 3 . (m n) by 0 . 41 0 -4 2 1 . 8 by x.

31 4—32 5 . y by 1 2 . 42 2 —433 . x by w.

326 —337 . p by q. 434 —445 . y by x.

338—349 . r by s. 446 —4 57 . y by w.

7 1

SECTION V

DIVISION .

1 2 0 . I l lustrative Examp le. Separate 48 chi ldren intoequal groups of 1 2 chi ldren each. How many such groupswi l l there beSO LUTION.

— By mul tipl ication we know that four 1 2’s are 48, so 48

chil dren can be separated into four equal groups of 1 2 children each.

Separating anumber into equal parts is dividing .

1 2 2 . The number to be separated or divided is thedividend. The number to divide by is the divisor. The

resul t obtained by dividing is the quotient.

1 23 . Division is the process of separating, or dividinga number into equal parts.

1 24 . In the exampl e, Divide 50 by the quotient is4, with 2 l eft u ndivided. The number l eft after the product of the equal parts is subtracted is the remainder.

1 25 . The usual sign of div ision is The expression48 —1 1 2 4, means, and is read,

“ 48 divided by 1 2 equal s

or“ 1 2 in 48, 4 times,” or 1 2

’s in 48,

Division is al so indicated thus, 48 1 2 , and 44.

Oral Exercises in Division .

1 26 . I. How many Dividea . 2 ’s in 1 4 ? 1 8 ? 24 ? 22 ? g. 30 by 6 .

b. 3’s in 2 1 ? 1 8 ? 27 ? 36 ? h. 27 by 9 .

c. 4’s in 1 6 ? 24 ? 36 ? 28 ? i . 80 by 1 0.

d. 5’s in 25 ? 45 ? 60 ? 55 ? j . 49 by 7 .

e. 6’s in 24 ? 42 ? 54 ? 1 8 ? k. 64 by 8.

f. 7’s in 42 ? 84 ? 63 ? 56 ? z. 77 by 7 .

74 DIVISION .

Divide as fol l ows the numbers in the columns under

Exercise III.

, previous page.

m. a to c by 3 . p . b to f by 6 . s. b to i by 9 .

n . b to d by 4 . q. b to g by 7 . t. b to k by 1 1 .

o. b to e by 5. r. b to h by 8. u . b to l by 1 2 .

IV. a . How many 3’s are there in 240 in 2400

SOLUTION .— 240 equal s 24 tens. 3’s in 24, 8 ; in 24 tens, 8 tens,

or 80. Ans. 80 threes in 240 ; 800 threes in 2400 .

Tel l at sight, how many times

5. 2 in 80 ; in 1 20 ; 1 60 ; 1 40 . f . 4 in 800 ; 2400 , 3600 ; 2800.

c. 3 in 90 ; in 2 1 0 ; 270 ; 1 80 . g. 7 in 1 40 , 4200 ; 2 1 00 ; 3500 .

d. 6 in 1 20 ; in 480 ; 360 ; 540 . h. 5 in 450 ; 1 500 , 3500 ; 5500.

e. 8 in 240 ; in 1 60 ; 560 ; 320 . i . 9 in 720 ; 540 ; 6300 ; 81 00 .

1 27 . j . When car fares are 5 cents each, howmany ridescan be had for 35 cents for 42 cents

SOLUTION.— As many rides can be had for 35 cents as there are

5’s in 35, which is 7. Ans. For 35 cents, 7 rides ; for 42 cents,

8 rides, and 2 cents remain .

k. At 8 6 per square for tinning a roof, howmany squarescan be tinned for 8 42 for 8 75

l . The l oss on the sal e of some cattl e was 8 7 each . If

the total l oss was 8 84, how many were sol dm. Howmany steps, 9 inches high, are required to ascend

6 feet 1 2 feet

How many How manyn . P ints are 48 gi . ? 50 gi . ? r. Bushel s are 60 pk. ? 34 pk

o. Quarts are 50 pt. ? 25 pt. ? 8 . Feet are 36 in . ? 74 in . ?

p . Gal l ons are 44 qt. ? 50 qt. ? t. Yards are 30 ft. 21 0 ft.q. P eeks are 96 qt. ? 1 00 qt. ? u. Weeks are 60 d. 72 d.

v. How many whol es are 27 hal ves 36 fourthsw. How many whol es are 32 thirds 55 eighths

SHORT DIVISION . 75

SHORT DIVISION.

1 28 . Il lustrative Examp le. Divide 358 by 4.

WRITTEN WORK .Writing the dividend and divi .

D ividend sor as in the margin, and beginning at the l eft, as far as possibl e

D1 v1 sor4) 358— 2,Rema1 nder.

divide each term separately . As

89, Quotient. 4 is not contained in 3 hundred

any hundreds of times, begin with

35 tens , thus : 4’s in 35 tens, 8 tens, and 3 tens remain . Write 8

under the tens of the dividend, and unite the 3 tens remaining with

the 8 units, making 38 units.

4’s in 38 units, 9 units, and 2 remain ; write 9 under the units ofthe div idend, and the 2 remaining at the right of the dividend, asin the margin . Ans. 89 , Remainder 2 .

In dividi ng, simply say, “ 4’s in 35, 8, and 3 over ; 4’s in 38, 9 , and

2 over”

or abbreviating stil l more , “ 4’s in 35, 8 ; in 38, 9 , and 2

remain .

”

To prove the work, mul tip ly the divisor by the quoti ent, and add

the remainder if the work is right the resul twil l equal the dividend ;thus, 89 x 4 2 358.

Exampl es for W ri tten W ork .

1 29 . P erform and prove the fol l owing exampl es1 . 2 . 3 .

6 . 7 . 8 . 9 .

3) 1 505 3) 1 070

1 1 1 2 1 3 1 4

4)3224 4 ) 1 1 85 5) 1 1 35 5)31 40

Divide the numbers given bel ow as indicatedBy 6 By 7 By 8

1 6 . 2725 2 1 . 2208 26 .

1 7 . 2 1 84 22 . 3822 2 7 .

In 5007 23 . 28

1 9 . 999 1 2 4 . 2 9 .

20. 2 1 86 25 . 30 .

76 DIVISION .

36 . 8459 6 42 . 7 48 . 9

37 . 7 43 . 6 49 . 8

38 . 6 44 . 4 50 . 6

39 . 7 45 . 7 51 . 4

40 . 8 46 . 8 52 .—1 9

41 . 9 4 7 . 9 53 . 9

54 . 3865 pounds of tea were put up in 5-pound packages.

How many packages were there55 . To hold 264 quarts of milk, how many 6-quart cans

wil l be required How many 2-gal l on cans56 . Aman empl oyed an agent to purchase a l ot of cows,

paying for the serv ice 8 7 a head. If 8 1 743 was paid the

agent, how many cows were bought57 . At 1 1 tons a carl oad, how many cars are needed

to transport 4092 tons of coal ?58 . A deal er sol d a quantity of wood for 8 9 a cord, and

received 8 6858. How many cords did he sel l ?

59 . In 4824 sol diers, how many squads are there of 8each of 9 each60 . The distance across the Atl antic Ocean being 3000

mil es, in what time wi l l it be crossed by a ship sail ing at

the rate of 4 mil es an hour By a ship sail ing at the rateof 1 2 mil es an hour6 1 . In 2745 days how many weeks are there, and how

many days remain6 2 . How many yards are 482 feet 5374 feet63 . Change 6284 months to years. 5424 inches to feet.

PARTITIVE FORM O F DIVISIO N.

1 30 . I l lustrative Examp le. Separate 48 children into 4equal groups ; how many chi ldren are there in each groupAns. 1 2 chi ldren .

SOLUTION .— In this exampl e the number of the groups, four, is

given , and it is required to find the size of each group which is one

P ARTITIVE FORM. 77

fourth part of 48 children , or 1 2 children . This is the partitive formof division .

In the example in Art. 1 20, Separate 48 chil dren into equal groupsof 1 2 chil dren each ,

” the size of each group, twelve, is given , and it is

requi red to find the number of the groups. This is the measuringform of division.

Al l exampl es in div ision are either of the measuring or of

the partitive form. In the “measuring” form of division

,

the dividend and divisor are of the same denomination ;in the “

partitive” form, the div idend and quotient are of

the same denomination.

1 3 1 . What is one of the equal parts cal l ed, when a thingor number is div ided into 2 equal parts into 3 4 5 6

1 0 ? 20 1 00 1 000

These parts are written thus

one hal f, one fourth one sixth, 11,

one third, «at one fifth, i one tenth, 1etc.

The partitive form of division is denoted thus

i-of 26 ; 11; of 36 ; “In, of 480 ; “ In of 1 876 . These ex

pressions are read one ha l f of 26 ; one third of etc.

Exampl es for W ri tten W ork

1 32 . I l lustrativeExamp le. What is 1 twel fth of 1 61 mi l es1 twel fth of 1 6 1 miles is found by

dividing 1 6 1 by 1 2 .

mil es 1 twel fth of 1 6 tens is 1 ten ; 4 tens

1 3 157mil es, Ans.

remain . 1 twel fth of 4 1 un its is 3 units,

and 5 units remain . If 1 twel fth of each

of the 5 un its is taken, 5 twel fths wil l be taken . This may be expressedas in the margin . The entire quotient is then 1 332 . Ans. 1 35, mi les.

In doing the work, say simpl y, 1 twel fth of 1 6 , 1 of 4 1 , 3 and

WRITTEN WORK.

Find the quotients of the fol l owing64 . 4} of 3007 6 7 . i of 70 . 8 of65 . 8 of 68 . of 7 1 . {f of66 . 4 of 69 . of 7 2 . J, of

8 DIVISION .

1 of 8 ofI,of 3 of

g, of of4,of 4

,of

i of I,of

i of g, of

i of 8 of

4 of 8 of

Il lustrative Examp le. In 3 bins there were 27

bushel s of corn . How many bushel s was this to a bin ?

SO LUTION.— Since in 3 bins there were 27 bushel s, in 1 bin them

was i of 27 bushel s, which is 9 bushel s. Ans. 9 bushels .

a . Arent was 8 84 a year. Howmuch was this a monthb. How many yards of singl e track rail road can be l aid

with 1 20 yards of rail ? How many yards of doubl e trackcan be laid with 800 yards

c. 3 score equal s 5 dozen. How many units is 1 scored. The errors made by 4 boys in spel l ing were severall y

6, 7 , 5, and 1 0. Find the average number of errors to a boy.

NOTE .— The average of two or more quantities is their sum

divided by the number of quantities added. The whol e number of

errors is 28. i of 28 ia. 7 . Ans . 7 errors.

6 . A mountain party of 7 had expenses as fol l owsrefreshments 8 6, hotel 8 1 5, teams 8 1 4 . What was theexpense to each personf . Find the average earnings per week of 2 weavers, each

earning 8 1 3, and 5 carders, each earning 8 6 .

g. If 4 feet 6 inches of wire are cut into 9 equal pieces,what is the l ength of 1 pieceh. If a piece of work can be done by one man in 63 hours,

how l ong wil l it require 7 men to do the same worki . If 2 bunches of 1 00 1aths each wil l l ay 1 2 square yards,

how many l aths wil l be required to l ay 1 square yard

DIVISION OF DEUIMALS. 79

1 34 . Exampl es for W ri tten W ork.

89 . Six persons shared equal ly in the profits from the

freight of a vessel . The net profits for one year were8 1 435. How much was each one’s share90 . There are 2394 pupil s in the town of Grafton, and 7

school s, 9 cl asses to a school . What is the average numberof pupil s to a school ? to a cl ass9 1 . One third of an estate of 8 fal l s to a widow,

and the remainder is l eft to 7 chil dren. How much do al l

the chil dren receive How much does each receive92 . From November 1 to April 1 there were burned in a

furnace 6 l ong tons of coal ; in a cooking range 4 l ong tons.

What was the average number of pounds burned per monthin the furnace in the range

93 . When a grocer’s bil l for 6 months is 8 What

does it average a month94 . If 8 mil es of the U nion P acific rail road cost 8what was the average cost per mil e95 . John has 8 James has 1 fifth as much, and

Harry has 1 eighth as much . How much have James and

Harry together

96 . The number 9296 is 9 times what number 1 1 timeswhat number

DIVISION OF DECIMALS.

1 35. I l lustrative Examp les. ( 1 ) What is 1 eighth of

(2 ) 1 eighth of 947 (3) 1 seventh of

( 1 ) (2 ) <3)WRITTEN WORK. WRITTEN WORK. WRITTEN WORK.

8) 7

or

In dividing a decimal by an integral number, insert the decima l

point in the quotient when the decima l point in the dividend is reached.

80 DIVISION.

In example the remainder, 3 units, is changed to tenths and

divided, leaving a remainder which is changed to hundredths and

divided, and so on.

In exampl e after the division has been carried to thousandths,

there is stil l a remainder, which may be changed to ten-thousandths,

and the division be continued ; or, to Show that the division is incom

p lete, dots may be written in the quotient.

In the fol lowing exampl es, where the division is incompl ete, the

answersmay be carried to thousandths .

9 7 . Divide by 3 . 1 00 . Divide by 5.

98 . Divide by 4 . 1 01 . Divide by 1 1 .

9 9 . Div ide by 7 . 1 02 . Divide 8 by 6 .

1 03 . A quart dry measure contains cubic inches.

How many cubic inches are there in a pint in a gil l ?1 04 . A bushel contains cubic inches. How many

cubic inches are there in a peek

1 05 . When coal is 8 a ton, how much wil l hal f a

ton cost How much wil l 1 fourth of a ton cost1 06 . Shrimps at 8 per dozen cans are how much

per can

1 07 . Three partners share equal l y a profit of 8What is each one’s share1 06 . A house rent of 8 1 875 for 9 months is how much

per month1 09 . The P resident’s sal ary of 8 a year is how

much a month

LONG DIVISION.

1 36 . When the divisor is not l arger than 1 2 , as in the

exampl es thus far given, it is customary to div ide withoutexpressing the entire operation. The process is then cal l edshort division.

When the divisor is l arger than 1 2 , it is convenient individing to express the entire operation. The process isthen cal led long division.

82 DIVISION .

In the fol l owing exampl es, answers may be given in

either or in both forms shown on page 81 . If the secondform is used, continue the division to thousandths.

In each exampl e bel ow, what number wil l you take as a

trial div isor

1 1 0: Divide 3864 by 2 1 . 1 1 9 . Divide 5278.

by 93.

1 1 1 . Divide 6438 by 31 . Divide by 24 .


1 1 3 . Divide 4076 by 62 . Divide by 7 1 .


1 1 5 . Divide 1 357 by 52 . Divide by 53.

Divide 3579 by 63 . Divide 6438 by 32 .

Div ide 2543 by 74 . Div ide by 54 .

1 1 8 . Divide 3857 by 81 . 1 2 7 . Divide by 82 .

1 38 . I l lustrative Examp le. Divide 1 508 by 1 97 .

WRITTEN WORK As 1 97 is nearly 200 , 1 508 divided by 1 97

m, Ans.

wil l give about the same quotient as 1 500

1 379divided by 200 , or as 1 5 divided by 2 . There

1 29fore make 2 a trial divisor. Ans . 7m.

1 2 8 . Divide by 89 . 1 36 . Di’vide by 283.

1 2 9 . Divide by 78. 1 3 7 . Divide by 579 .

1 30 . Divide by 27 . 1 38 . Divide by 394.

1 31 . Divide by 1 92 . 1 39 . Divide by 886 .

1 32 . Divide by 281 . 1 40 . Divide by 1 78.

1 33 . Div ide by 698. 1 41 . Divide by 481 .

1 34 . Div ide by 56 . 1 42 . Divide by 1 845.

1 35 . Div ide by 29 . 1 4a. Divide by 3951 .

1 39 . From the preceding examples may be derived thefol l owing

Rul e for D iv ision.

1 . Write the dividend ; at the left draw a curved l ine ; and

at the left Of this l ine write the divisor.

LON G DIVISION . 88

2 . Divide the highest term or terms of the dividend by thedivisor.

3. I n long division write the result for the first term ofthe quotient at the right of the dividend ; in short division

write it beneath.

4. Mul tip ly the divisor by this term.

5. Subtract the p roduct thus obta ined from the part of thedividend used.

6 . Un ite the next term of the dividend with the rema inderfor a new partia l dividend ; divide, mul tiply, and subtract as

before ,"and so continue ti l l a l l the terms of the dividend are

usedfi“

7 . Express the division of thefinal rema inder, should therebe any, in thefractiona l form.

Or, change the rema inder to tenths, hundredths, thousandths,etc., and continue the d ivision as far as desirable.

P roof .

Find the p roduct of the quotient and divisor, and add to it

the rema inder, if there is one. The resul t should equal the

dividend.

1 40 . D iv ision by tens , hundreds , etc .

Il lustrative Examp le. Divide 5487 by 1 0 ; by 1 00 ; by1 000.

Moving the decimal point to the left has

5487 1 0 the effect of moving the figures the opposite

way (Art.

1338 24

42; Hence, to divide a number by 1 0, 1 00 ,87

1 000, etc . , move the decima l point as manyp laces to the left as the divisor has zeros.

Ameter contains inches . How many inchesare there in a decimeter, which is 1 tenth of a meter

If at any time the divi sor is not contained in a partial dividend ,

write a zero for the next figure of the quotient, and unite with the

partial dividend the next term of the given dividend.

84 DIVISION .

1 45 . In cents,how many dimes are there how

many dol l ars1 46 . mil l s are howmany cents ? howmany dol lars ?1 4 7 . If cents are to be put up in packages of 1 00

each, how many packages wi l l there be, and how manycents remaining1 48 . If a mil eage ticket for 1 000 mil es sel l s for $ 25,what is the cost per mi l e1 49 . When 1 00 shares of rail road stock sol d for $31 830,

what was the price per share1 50 . 5741 pounds of nai l s wi l l fil l how many 1 001pound

kegs ?1 51 . Howmany cental s (1 00 lb. ) ofgrain are pounds?1 52 . Divide by 1 0

,by 1 00, by 1 000, and add the

quotients.

1 53 . Div ide by 1 00, by 1 0, by and add

the quotients.

Il lustrative Examp les. (1 ) Divide by 4200.

(2) Div ide by 1 200 .

NOTE . When a divisor has zeros at the right, the process of dividing may be shortened, as in the written work below.

(1 ) WRITTEN WORK. (2) WRITTEN WORK.

42 OO)508 760 2 14

519,42

88 ( 1 ) Since 4200 42 x 1 00 , we first divide

84 by 1 00 , which is indicated by a vertical l ine

476 drawn so as to cut off two figures of the

dividend, and then div ide the result by 42

this gives for a quotient 1 2 , with a remainder of 476 . Ans. 1 2 151 665 .

(2 ) In the second exampl e, the division by 1 00 is indicated by the

placing of the decimal point (Art . This gives Divid

ing this by 1 2 , the answer is

1 54 . Divide 6842 by 200 . 1 57 . Divide by 7000.

1 55 . Divide 3872 by 1 53 . Divi de by 840 .

1 56 . Divide by 3200 . 1 59 . Div ide by 650.

DIVISION OF UNITED STATES MONEY. 85

1 60 . How many customers can be suppl ied by using a

ton of ice (2240 lb.) if each customer receives 60 lb. ? 80 lb. ?

1 6 1 . When 40 cubic feet of coal weigh 1 l ong ton, howmany pounds does 1 cubic foot weigh1 62 . Change seconds to minutes. To hours.

1 63 . At 2 1 0 pounds of sal t to a barrel , how many barrel sof sal t were produced in the sal t works of New York in the

year 1 892 , the product, as reported, beingpounds

1 42 . To div ide one sum of money by another.

Il lustrative Examp le. At each,how many chairs

can be bought for 32, and how much money wil l remain

To divide one sum of money by another(measuring form of division) both dividend

(25 and divisor must be expressed in the same de

WRITTEN WORK

250 nomination. Here the divisor being cents, the700 dividend must be changed to cents (Art.

625Dividing 3200 cents by 1 25 cents, the quotientis 25 , and there are 75 cents remaining.

Ans. 25 chairs, and 75 cents remain.

75

1 64. I have is5 with whi ch to purchase rail road ticketsat 2 1 cents each. How many can I buy and how muchmoney wil l remain1 65 . At 36 each, how many trunks can be bought for1 00

1 66 . At 85 each, how many Smyrna rugs can a deal er

buy for $ 50 for $ 75 ?1 67 . Amerchant has sent 8 75 to be spent for flour, at

per barrel . How many barrel s can he buy and how

much money w il l remain

1 68 . Divide s57 by s 1 7 1 . s 1 70 43¢1 69 . Divide 8 by 8 1 7 2 . 48 8

‘P

1 70 . 1 7 3 .

86 DIVISION .

1 74. The city of Holyoke was taxed for school purposeson property valued at 8 What was the

tax on a dol l ar1 75 . The city of Brockton was taxed 8 on

property valued at What was the tax on a

dol lar

1 7 6 . Divide by 1 1 .

1 7 7 . If a dividend is and the di v isor is 24, whatis the quotient1 78 . 7 6624 3456 what 1 44

1 7 9 . 8 85 mul tipl ied by what number gives a product of81 80 . What number mul tipl ied by 59 gives a product of

8 8673

5460 is 52 times what number 1 00 times what1 82 . 7 1 times what sum of money equal s 8 25561 83 . What is the quotient of 73

1 8 4. Divide by 1 0, by 1 000, by 1 00, and add the

quotients1 8 5 . How many are1 86 . The product of two factors is 8 7448, one of them

is 98. What is the other1 87 . The product of three fac tors is two of them

are 24 and 3 1 . What is the third1 88 . What is 1 eighth of 8 1 3

1 89 . Find 1 el eventh of 81 90 . i of what number1 9 1 . Divide 8 by 81 9 2 . 8 545 8 8 what 81 93 . A quotient is 1 6 men ; the dividend men .

What was the divisor1 9 4 . A div idend being mil es, the remainder 249

mil es, and the quotient 1 1 , what was the divisor


1 44 . Bxamfl es for W ritten W ork .

1 95 . How many sheets of paper are requl red for a bookof 768 pages, 24 pages to a sheet1 96 . How many square mil es are there in a township of

51 68 acres, 640 acres making 1 square mil e1 97 . Mr. P ratt put 1 980 bushe l s of appl es into barrel s

each hol di ng 2 bushel s 3 peeks. How many barrel s did heuse

1 98 . Mr . Granger received 8 2548 in payment of a debt,took 8 275 from the savings bank, and added to this 8 1 77of hi s earnings. With the money he’ bought 1 60 acres ofl and. What was the price per acre1 99 . Mr. Adams bought a house for 8 6200, paid 8 1 500

down,and the remainder in four equal payments. HOW

much was each payment200 . If an oil wel l yiel ds 6664 barrel s of oil in 8 weeks,what is its daily yield201 . Mr. P rince has acres of pasture l and

,1

seventh as much meadow l and, and 1 eighth as muchwood l and as pasture . How many acres has he in al l

202 . Charl es is to keep 1 twentieth of al l he receivesfrom the sal e of peaches. How much in dol l ars and centswil l he keep if he se l l s 8 35 worth on Friday and 8 47

worth on Saturday203 . A broker l et a store for 8 240, and received for his

services 8 How much did he get on a dol l ar204 . In 1 892 there were students in the different

col l eges in the U nited States, with 9326 teachers. Findthe average number of students to a teacher.

205 . The U nited States in 1 892 maintained school s forIndians, accommodating pupil s, at a total cost of

Find the cost for each pupil .206 . An operative can weave yards of sheeting

in 52 weeks. How much is that a week

88 DIVISION .

207 . What is the sum of 4 and 1 tenth of

208 . The read ings of a thermometer kept by a schoolboy in Lowel l for 4 successive days were as fol l ows

Monday 6 A.H. 56° 1 2 M 64° 4 m a. 70°

Tuesday 48° 69° 62°

Wednesday 45° 63° 59°

53° 70° 62°

Find the averages for 6 A.H ., 1 2 M., and 4 P .M. ; al so

the highest average for one day.

20 9 . Very heavy cannonading has been heard at a distanceof feet. What is the distance in mil es and feet2 1 0 . If the earth is mil es from the sun, and

the moon at its ful l is mil es farther on, and ifl ight travel s at the rate of mil es a second, howmany seconds is it in passing from the sun to the moonand back to the earth (Answer to tenths.)2 1 1 . The l imit of perpetual snow at Quito, on the equator,

is feet above the l evel of the sea . Find the heightin mil es

,carrying the answer to hundredths.

The National debt of the U nited States in 1 890 was8 and the number of inhabitantswasHow much was the debt to each inhabitant ?

What is Division 7 What is the dividend the divisor the

quotient the remainder P erform and expl ain an exampl e in short

division ; prove the work. P erform and explain an example in longdivision . G ive the rule ; the proof . How do you divide by 1 0 , 1 00 ,1 000 , etc. How do you divide when the expression of the divisor

contains zeros at the rightMake examples to il lustrate the measuring form of division ; the

partitive form.

When the dividend and quotient are given, how can you find the

divisor When the divisor and quotient are given, how can you find

the dividend When the mul tipl ier and product are given ,how can

you find the mul tipl icand When the mul tipl icand and product are

given, how can you find the mul tipl ier

SECTION VI .

MISCELLANEOU S AND REVIEW S .

SYMBOLS OF O PERATION.

1 46 . The signs indicating the different arithmetical processes are sometimes used in the same expression.

Thus 8 — 3 x 2 = 2 . 1 2 x

In every such case the mul tip lying and dividing indicated

should be done first, the adding and subtracting afterwards.

If, however, the terms to be operated upon are joined by a

parenthesis or v inculum the expression j oined istreated as if it denoted a single number.

To il lustrate : 8 3 x 2 means that 3 x 2 is to be taken from 8 ;

resul t 2 . But (8 3) x 2 , or 8 3 x 2 , means that 3 is to be takenfrom 8, and the remainder mul tipl ied by 2 resul t 1 0 .

Again , x but (8 + 3) x 2 , or 8 + 3 x 2 =22 .

So (8 + 2 )— 2 .

8) x 3 6 ] 5 means that the sum of 4 and 8 is to be multi

pl ied by 3, the product to be diminished by 6 , and the remainder

divided by 5 .

24+ 4 x 2=l 2 ; but 24+ (4 x 2 )=3.

1 47 . In performing a series of operations indicated bysigns,

First,operate on the numbers that are j oined by a paren

thesis Or vinculum as indicated by the signs. Next, multi

p ly and divide, using the signs x and in order as they

occur. Final ly, add and subtract as indicated by the signs

and

SYMBOLS OF OP ERATION . 9 1

1 48 . Exercises for Oral or W ritten W ork .

1 . (1 2 + 8) x 4 4 . 20 — 8

5 . 4 x 6 + 2 x 3= ?

3. (20 — 8) 6 .

1 7 — 59

2 3

8 . (3 + 8) x 4

9 .

1 5 x 4 — 6 x 3

7

1 1 .

In sol ving probl ems, the pupil wil l be aided by expressing the

required operations using the proper symbols, as shown in the fol

lowing exampl es .

1 0 . 28

1 2 . How many square yards are there in a roadway 200feet l ong and 40 feet wide 200

;40

Ans.

1 3 . From the freezing to the boil ing point in the Centi

grade thermometer is in Fahrenheit’s it is HOWmany degrees of Fahrenheit’s equal 40 of 1 80 x 40

Centigrade 2 1 00A”

1 4 . How many degrees Centigrade equal 1 00 x 36

36° Fahrenheit ? 1 80

1 5 . When 360 hours of the month of 360

Jul y have passed, how many days remain 24Ans

1 6 . A and B,two candidates for office, received a total

of 2584 votes,A receiv ing 430 l ess than 2534 430

B. How many votes did each receive 2A’

s'

A’s 430 B ’

s.

1 7 . If coins can be stamped by 8 presses in 1 hour,how many can be stamped by 1 press in 1 9200 x 40

40 minutes W1 8 . For how much must 8 tons of hay, which cost 8

a ton,sel l , to gain 8 per ton x 8 Ans.

Ans.

92 MISCELLANEO U S AND REVIEWS.

1 9 8 2500 per acre was paid for 1 0 acres of l and. For

how much must it be sol d to gain 8 200 per acre ?8 2500 8200 X 1 0 A1 1 8 .

20 . Amerchant sol d 40 yards of cl oth for 8 1 40, havingbought it for a yard. What was his gain per yard ?

M Ans .

40

2 1 . The charges of a student in a preparatory school lastyear were

,for tuition , 8 75, and for incidental expenses 8 9 .

This year in col l ege the charo'es were twice as much for tui9

tion, and 5 times as much for incidental s. What was the total

cost for the two years ? $ 75 + 8 9 + s75 x x 5 =Ans.

2 2 . I owe John Brent 8 600, due 6 months hence . If I

pay him 1 hal f now,he is wil l ing to take off 5 cents on a

dol lar of the other hal f of the debt. What wil l the bal ancebe ? — 80 .05 x 300 _—_Ans.

2 3 . Find the cost of storage of the fol l owing l ots of flourat 3 cents on a barrel per month.

200 barrel s, 2 months. 1 75 barrel s, 3 months.

1 50 barrel s, 1 month. 280 barrel s, 2 months .

280 ) x 2 1 50 + 1 75 x 3] x 3 Ans . in cents.

Mis ce l l aneous O ral Exampl es .

1 49 . I l lustra tive Examp le. a . If 8 yards of cotton cost72 cents, how much wil l 5 yards costSOLUTION . If 8 yards cost 72 cents, 1 yard wil l cost 1 eighth of 72

cents, or 9 cents, and 5 yards wil l cost 5 times 9 cents , or 45 cents.

Ans. 45 cents.

b. If is paid for 7 days’work

,how much wil l be

paid for 1 0 days’work

c. HOW many hours’ work 8 an hour w il l pay for3 l oaves of bread 8 a l oaf and 4 lb. of beef at 8a 1 h.

d. If 9 rows of corn yield 36 bushel s, how many bushel swi l l 20 simil ar rows yiel de. If an express train runs 1 1 4 mil es in 3 hours

,how far

can it go in 4 hours

MISCELLANEOU s ORAL EXAMP LES. 93

f . If 8 2 wil l buy 1 bushel 2 peeks of pease, how many

pecks w il l 8 1 2 buyg. If 6 boys can put the school room in order in 1 0 min

utes,how many boys can do the work in 1 2 minutes

h. A fiel d of rye was reaped by 1 2 men in 5 days. Whatl ength of time would be required for 1 5 men to reap the

same amounti . If a cistern can be emptied in 20 minutes by 3 pipes

of the same size, in what time can it be emptied when only2 of the pipes are openj . A sea breeze was fel t on the coast at hal f-past four.

If its vel ocity was 4 mil es an hour, at what time was it fel t9 mil es inl andk. The Israe l ites l eft Egypt in the year 1 49 1 B .C. In

what year did they enter Canaan 40 years l aterl .

,If it takes a dozen l emons tomake l emonade for 24 per

sons,how many l emons must be al l owed for 300 persons

m. When a floor is 1 1 feet wide, what must be its l engthto contain 1 32 square feetNOTE .

— Since the number of feet in l ength mul tipl ied by the number of feet in W idth equal s the area, to find the l ength, 1 32 must bedivided by the number of feet in the W idth (Art.

n . A grocer has 4 grades of coffee worth 29 cents,2 1

cents,40 cents, and 42 cents, which he mixes, using equal

quantities of each . At what price can he afford to sel l themixture0 . It took 1 2 yards of carpeting 3 quarters of a yard wideto carpet a room. How much woul d it take of carpetingthat is 4 quarters widep . A newsboy sel l s papers for an agent at 5 cents each,

and is al l owed 2 cents a copy for every paper sold. If his

al l owance is 24 cents, how many papers does he sel l , and

what does the agent receiveq. What is the profit in buying 1 0 shares of stock at 8 93

per share and sel l ing them at 8 1 1 5 a share

94 MISCELLANEOUS AND REVIEWS.

1 50 . Mi scel l aneous Exampl es for W ritten W ork .

2 4 . If 6 yards of pl ush cost what is the cost of1 3 yards25 . Five hundred passengers were carried by special

train for How much shoul d be paid for carryinga family of 7 persons at the same rate2 6 . A crew of 1 2 persons had provisions for 36 days,

when they took in 9 more persons froma wreck. Howmanydays’ prov ision had they for al l ?2 7 . How many strips of flooring 4 inches wide must be

used to l ay a floor which requires 1 44 strips 6 inches wide28 . Howmany l oaves of bread, 7 ounces of flour to a l oaf,

can be made from a barrel of flour ( 1 96 pounds)29 . At 1 7 cents a yard for paper bordering, What wil l be

the cost of bordering for a room 1 6 feet l ong and 1 4 feetwide30 . What must be the l ength of a lot of l and 80 feetwide to contain square feet31 . A cheese weighing 24 pounds cost For how

much a pound shou ld it be sol d to give a profit of 1 0 centsa pound32 . What shal l I receive for se l l ing 8 7800 worth of

goods if I receive 8 7 on each 8 1 0033 . When taxes are 8 1 6 on 8 1 000, how much is a person

taxed who has 8 worth of real estate and 8 ofpersonal property34 . The bel l of St. P eter

’s in Rome weighs pounds ;

that of the Kreml in in Moscow, the largest in the world,

weighs pounds. How many times as heavy as the

former is the l atter35 . 1 8 acres of l and were bought for 8 4050 and sold for

8 5000. What was the gain per acre36 . If the l and above had been sold for 8 3900, what

woul d have been the l oss per acre

MISCELLANEO U S WRITTEN EXAMP LES . 95

37 . What must be paid for insuring my house worth8 2500, at 8 on each 8 1 0038. It is estimated that in 1 890 there were

peopl e in North America, in South America,and in Europe . How many more peopl ewere there in Europe than in North and South Americatogether39 . Mr. Jaynes bought 4 1 acres of land at 8 60 an acre

,

and sold it for 8 3000. Did he gain or l ose,and how much

How much per acre40 . One person can make by hand 40 pairs of sl eeve buttons in a day, another with a machine can make 9000 pairsin a day. How many men working by hand would be re

quired to do a day’s work of one man with a machineDora deposited in the sav ings bank in May,

twice the amount in June, 8 5 in July, in August ;she took out 8 8 in September, deposited 8 in October,

in November, and took out in December.

How much remained in the bank42 . Wh at is your ice bil l for the year if you commenceto take ice in May and pay 8 9 from May through September

, and 30 cents a hundred for the remainder of the year,using 260 pounds in October, 1 55 in November, and 85 inDecember43 . A deal er sel l s a bushel of berries in quart boxes for

which he receives a box . If he pays 8 3 for his

berries and 2 cents for every box he uses, what are his

profits44 . Four chil dren amused themsel ves by counting watermel on seeds on their pl ates. They counted 25

, 70, 1 43, and

21 2 . Suppose these were 1 third of the seeds in the whol emel on, how many seeds did the mel on contain45 . In 1 880 Boston had inhabitants in 1 890

there were inhabitants. What was the increase in1 0 years, and the average increase per year

96 MISCELLANEO U S AND REVIEWS.

46 . By the U S . census of 1 890, the ages of seven very

old persons, were 1 28, 1 25, 1 23, 1 22 , 1 1 8, 1 1 5, and 1 1 3 yearsrespective ly. What was their average age

4 7 . Mary is 6 years 2 months ol d, Charl es is 1 1 years 3

months, Ruth 1 1 years 1 month, and Horace 1 3 years 2months. What is the sum of their ages in months Whatis their average age in months In years and months48 . On five successive mornings the barometer stood as

fol l ows : inches, inches,

inches, inches,and inches. What was the average height49 . Agrocer had 2 pounds of tea worth 30 cents a pound,

4 pounds worth 40 cents a pound, 1 pound worth 74 cents,and 1 0 pounds worth 75 cents a pound, what was the value

of the entire l ot If he shou ld mix the teas, for how much

per pound could he afford to sel l the mixture50 . A grocer mixed together 3 pounds of coffee worth

34 cents a pound, 4 pounds worth 29 cents a pound, 6

pounds worth 1 8 cents a pound, and 2 pounds worth 32cents a pound. What was the value of 1 pound of themixture

5 1 . In the Linco ln school there were three classes as foll ows : first cl ass, 27 pupil s, average age 1 4 years ; secondcl ass, 65 pupil s, average age 1 0 years ; third c lass, 1 06

pupi l s, average age 7 years. Find the average age of the

pupi l s in the whol e school . (Answer to tenths.)52 . A man had a stock of goods worth 8 1 0 000 ; these

he expected to sel l at a gain of 8 1 5 on every 8 1 00 worth.

How much did he expect to receive for them53 . The above goods became damaged by fire, and Were

sol d at a l oss of 8 40 on every 8 1 00 . How much did the

deal er receive for them54 . A man se l l s books for a publ isher at each.

For sel l ing he is to receive 8 on every book sold.

If he receives 8 24, how many books are sol d How muchis due the publ isher

8 MISCELLANEOU S AN D REVIEWS.

How many inhabitants to a square mil e were there66 . In Maine 69 . In Mas sachusetts6 7 . In New Hampshire 70 . In Rhode Isl and68 . In Vermont 7 1 . In Connecticut72 . It is estimated that there are peopl e in

the world, and square mil es of l and. What isthe average number of inhabitants to a square mil e

The area of the surface of the system of great l akesof North America is as fol l ows

Lake Superior, sq . m. Lake St. Cl air, 41 0 sq . 1 1 1 .

Lake Michigan , sq . m. Lake Erie , sq . 1 1 1 .

Lake Huron , sq . m. Lake Ontario, sq . m.

73 . What is the sum of the areas of these l akes74 . How

‘

much larger is Lake Ontario than Rhode Isl andand Connecticut together75 . How much smal l er is Massachusetts than Lake Erie76 . Which is larger, al l the great l akes together or New

Engl and, and how much7 7 . The number of inhabitants in the U nited States in

1 880 was in 1 890 it was Whatwas the average increase per year78 . The area of the U nited States being square

mil es, what was the number of inhabitants to a square mil ein 1 880 9 ’ In 1 890

7 9 . October 1 , a gas meter registered feet con

sumed. January 1 , it registered feet. What was thecost of the gas for the quarter at 8 per thousand feet80 . Tel egraphic rates to Canton being 8 per word of

ten l etters or l ess, with doubl e rate for each word exceedingten l etters, what is the charge for the fol l owing tel egram“Bronson, Canton River and harbor appropriation bil l

passed senate Wednesday by bare majority.

”

‘l'Answer to hundredths.

ACCOUN TS AND BILLS. 99

ACCO U NTS AND BILLS.

1 51 . Bel ow is a record by a painter and gl azier ofarticl es sold and serv ices rendered by him to Mr. Edward

Crane .

NEW YORK, Dec . 1,1 894.

To H . H . HU N T , P a in ter an d G l az i er ,Dr.

8 sf/i sttas 2 5 4 nod e’

s w e s t 4 5 ¢

2 8 Ci t are, effici ent ext, 4 tee .

died/n il 57 7!

36601449 956 1414»

4 km w’

wo ve, 80 15

3 6742251229W roof6 1151249 flowSu l fa /1:6 3 for, fist/ 1.40 4.

1 52 . Such a record as the above is cal l ed an account .

1 53 . The person who owes a debt is a debtor The

person to whom a debt is owed is a creditor (Cr. Awrittenstatement of sums due on account rendered, by the creditorto the debtor, is a bil l . (See exampl es 81 , 82 and

Who is debtor in the account above Who is creditor Whichof the items are charges for work done Which are charges for material furnished

1 54 . When the bil l is paid, the creditor, or some one

authorized by him,signs the bil l , after the words “ Received

payment ” or“ P aid.

” The bil l is thus receipted. (Seeexammes 82 and

1 00 MISCELLANEOUS AND REVIEWS.


1 55 . Find the cost of each articl e in the fol l owing bil l s,and their several amounts

NEW YORK, May 1 , 1 895.

777C W téé flx 05122To W O OD , BARKER CO . ,

Dr.

1 8 9 5 .

7 $7

0 2 8 fl. Wei/6nd ,

pt. son s ,

8 8

hou w’ MW Q,

Received payment,

CHICAGO , MAY 1 9 , 1 896 .

777v . 777. 168vBought Of L . D . COBB <9. CO .

W ee ,

f em/moan d ew,

é oeo

fitm t fa becé J aw/b,

c

a

fe/mas thead ,

Received payment,

L. D . COBB CO .,

P er J . L. C.

1 02 MISCELLANEO U S AND REVIEWS.

85. Nov . 3d, 1 894, Mrs. J . H . Dol e bought of Smith and

Fel ton, New York, 5 yards of vel vet at 8 yards of

cashmere at 87 4 yards of Sil esia at Nov . 8, she

bought 7 yards of flanne l at 43 yards of sheeting at

and 4 yards of cambric at 37 Date the bi l l the firstof the fol l owing January.

86 . Joseph Sears bought of Mr. S. H . Al l en,June 4

,1 895,

2 pounds of steak at squash l ettuce 1 553 June 7,1 0 pounds of veal at 1 7 June 1 1 , 2 pounds of steak at

l ettuce June 1 8, 7 pounds of l amb at beets

pease 25¢ June 28,8 pounds fow l at 1 7 pease

87 . April 1 , Mr. M. H . Humphrey repaired for Mr.

David French,a sweeper for 50¢ and a hose for using

2 clamps at 1 0¢ each ; June 3, he sold him a kettl e for 90¢and a pail for June 5, 4 bol ts at 8 pounds l ead

pipe at 7 and charged for 3 hours’ l abor at

88 . Bonney and Rogers shipped to Thomas Durgan Co.

from Cincinnati,2 oxen, 1 246 lh.

, 1 31 8 lh.

,at

2 steers, 922 1b.

, 893 lh. , at

3 hogs, 31 9 l h., 366 lh.

, 384 lb., at

89 . Martha Jones made a dress forMrs. Mary Fitch and

furnished 6 yards cambric at 7 braid silk and twist3 bones at 2 yards Si l esia at buttons

5 yards ribbon at the work was val ued at 8 1 2 .

90 . P urdy White cl eaned carpets for Mr. Seth Grangeras fol l ows : April 24, 23 yards at 3¢ and 82 yards at

April 25, 64 yards at 4¢ and a stair carpet 8 April 29 ,28 yards at May 2 , 30 yards at9 1 . Sel l four different articl es from a grocery at current

prices and make out the bil l , giving credit for 4 dozen eggs

at the current price .

92 . Sel l five articl es from a meat and vegetabl e marketand make out the bil l with credit for 2 bushel s of pease at8 a bushe l .

TEST EXERCISES. 1 03

TEST EXERC ISES.

1 56 . In each column bel ow commence at the bottom and

add or subtract as indicated.

a b c d

+8 6 6 5

— 7 9 7 9

+ 9 7 4 5

6 9 8 9

+8 5 7 4

+8 6 4 3

+5 + 1 7 + 25 + 32

W ritten W ork .

1 57 . Add the fol l owing across the page, Without writingthe numbers in col umns93 .

94.

95 .

96 . s 3 1 7406,

1 58 . Subtract the fol l owing without writing the num

bers, but preserve the remainders. Add al l the minuends

and then al l the subtrahends, and find the difference of theirsums. Compare this difference with the sum of the remainders :

8 8

8 2

9

8 2

2

7

8 8

h

8

4

9

2

9

5

%

9

9

8

7

9

5

8

w

f2

4

9

3

5

8

M

04 MISCELLANEOU S AND REVIEWS.

1 59 . Supply the missing numbers in pl ace of x in thefol l owing exampl es

1 05 . 98765 x 1 36542 . 1 09 . x 1 93826 248701 .

1 06 . 26784 x 9789 . 1 1 0 . x 82394 340786.

1 07 . 5934 1 x x= 7 1 20 92 . x x 2681 1 1 5283.

1 08 . 434 1 6 + x 1 809 . 1 1 2 . x + 681 2 9283.

1 60 . Copy the fol l owing tabl e and fi l l out with the sum,

difference, product, and quotient of each pair of numbers,A. B.

, exampl es 1 1 3 to 1 1 7 , and find their total s as indicated

Sum. Difference . P roduct. Quotient. Total .

1 1 8 . At 54 cents per square yard, what wil l be the costof cementing the bottom of a cel l ar, the dimensions ofwhich are shown in the fol l owing diagram

89 feet.

2 1 feet.

1 1 9 . What wil l be the cost of a trench wal l to surroundthe above cel l ar at 8 per running foot, an al l owance of

1 6 feet being added for corners ?

1 06 FACTORS.

1 65 . When the same number is used several times as a

factor of a number, the number of times it is so used is

indicated by a smal l figure cal l ed an exponent .

Thus,52 5 x 5 . 52 is read “ Five to the second power

”

2 x 2 x 2 x 2 x 2 . 25 is read “ Two to the fifth power.

”

DIVISIBILITY OF NUMBERS.

1 66 . Numbers divisibl e by 2 are even numbers ; al l othernumbers are odd numbers . Al l prime numbers except 2are odd.

The factors of a number may be found by division. The divisibil ity

of numbers by some factors can be determined by the fol lowing tests :1 . A number is divisibl e by 2 if the units are divisible by 2 .

2 . A number is divisibl e by 3 if the sum of its digits* is divisible

by 3 . Thus, 285 is divisibl e by 3 , for 2 8 5 1 5 is divisibl e by 3.

3 . A number is divisible by 4 if its tens and units together aredivisible by 4 . Thus, 6724 is divisibl e by 4 , whil e 6731 is not.

4 . A number is divisible by 5 if the units’ figure is either 0 or 5.

5. A number is divisibl e by 6 if it is divisibl e by 2 and by 3.

6 . A number is divisible by 8 if its hundreds, tens, and units

together are divisibl e by 8 . Thus, 6728 is divisible by 8, while 6724

is not.

7 . A number is divisibl e by 9 if the sum of its digits is divisibl e

by 9 .

8. A number is divisibl e by 1 1 if the sums of its al ternate digits

are equal , or if their difference is divisible by 1 1 . Thus , 1 782 and

1 859 are divisibl e by 1 1 , While 4987 is not.

9 . A number is divisible by a composite number, if it is divisibl e byeach of the factors of the composite number. Thus , 3825 is divisi

bl e by 1 5, for it is divisible by 3 and by 5.

NOTE . Reasons for these tests are given in the Supplement, Art. 6 .

1 67 . Exercis es for W ritten W ork .

1 . Turn to page 89 and, using the tests above, write thenumbers expressed in w,

l ines a,b, 6 , etc.,

that contain as

a factor,2 ; 3 ; 4 .

A digit here means the number denoted by a figure independentlyof its place or order.

P RIME FACTORS. 1 07

2 . Write the numbers e xpressed in x,l ines a

,b,c,etc. ,

that contain as a factor, 5 ; 6 ; 8 .

3 . W rite the numbers expressed in l ines a , b, 6 , etc .,

that contain as a factor 9 ; 1 0 ; 1 1 .

Find by inspection, or by trial when necessary, al l thefactors not greater than 1 2 that are contained in the num

bers given bel ow :

4 .

5.

6 .

1 68 . To find the prime factors of a number.

I l lustrative Examp les. ( 1 ) What are the prime factorsof 735 (2 ) What are the prime factors of 409 ?

WRITTEN WORK .

(1 ) 3 735 (2 )38 23

2 9 1 79

1 9 1 6 1

1 0 1 8

( 1 ) Applying the tests (Art . 1 66 ) to the given number, we findthat 2 is not a factor of 735, but that 3 is, and by dividing, we see

that 735 3 x 245.

Continuing this process, we find that x 49 , and that 49 7 x 7 .

Therefore 735 3 x 5 x 7 x 7 , and the prime factors are 3, 5, 7 , and 7 .

(2) Applying the tests (Art. 1 66) to the second exampl e , we find

that 409 is not divisible by 2 , 3, or 5. We then try to divide by the

other prime numbers in order until we reach 23, when we see that thequotient is l ess than the divisor. There can then be no prime factorin 409 greater than 23, for if there were , there would be another factor (the quotient) l ess than 23, which we shoul d have found before

reaching 23 . The number 409 is, therefore, prime .

As 735 is found to equal the product of al l its prime factors, so wil lit always be found that a composite number eq

'

ua l s the product of a l l

its prime factors.

08 FACTORS.

1 69 . From the preceding exampl es may be derived thefol l owing

Rul e .

To separate a number into its prime factors1 . D ivide the given number by one Of its prime factors.

2 . D iv ide the quotient thus Obta ined by one Of i ts prime

factors ; and so continue div iding unti l a quotient is Obta ined

that is a prime number.

3 . This quotient and the severa l divisors are the prime

factors sought.

P roof .

Multip ly together the prime factors thus found. T he prod

uct should equa l the given number.

NOTE . If no prime factor is readily found by which to divide, trythe several prime numbers in order. If no prime factor is foundbefore the quotient becomes l ess than the trial divisor, the given num

ber is prime .


1 70 . Separate into prime factors the fol l owing numbers

1 6 . 1 1 6 1 9 . 204 22 . 342 2 5 . 567

1 7 . 1 64 20 . 252 2 3 . 364 26 . 644

1 8 . 1 76 2 1 . 270 24 . 363 2 7 . 684.

Sel ect the prime numbers and find the prime factors ofthe composite numbers among the fol l owing23 357 32 . 490 36 . 627 40 . 908

29 . 372 33 . 497 37 . 7 1 1 4 1 . 972

so. 377 34 . 499 38 . 754 42 . 1 728

3 1 . 367 35 . 570 39 . 760 43 . 1 830

What is a factor of a number 7 What is a composite number ?a prime number a prime factorWhat is an even number an odd number ? What numbers are

divisible by 2 ? 3 ? 4 ? 5 ? 6? 8 ? 9 ? 1 0 ? 1 1 ?

How can you find the prime factors of a number A compositenumber equal s what product ?

1 1 0 FACTORS.

58 . If a bar of iron 8 feet l ong weighs 36 pounds, whatwil l a bar of iron of the same thickness, and 1 00 feet l ong,weigh59 . If a tree 69 feet hi gh casts a shadow 90 feet, what

l ength of shadow wil l a tree 92 feet high cast at the same

time of day60 . If the work of 6 men is equal to the work of 9 boys

how many men’s work wil l equal the work of 2 1 boys6 1 . A yachting party has provisions for 1 2 persons 60

days. How l ong wil l it l ast 2 1 persons62 . A merchant exchanged 1 26 pounds of sugar at 6

cents a pound for eggs at 28 cents a dozen . How manydozen did he receive63 . How many yards of cl oth 24 inches wide will be

required to l ine six yards of cl oth 28 inches Wide64 . A rectangul ar pl at of ground 63 feet by 48 feet is

turfed with sods 6 feet l ong and 1 foot wide . How manysods does it take65 . 1 00 rol l s of dimes, 20 to a rol l

,were given in ex

change for a rol l of 8 5 bil ls . How many bill s were therein the rol l6 6 . How many shares of stock worth 8 1 75 per share

wil l pay for 750 shares at 8 77 per share6 7 . The freight on pounds of coal was

What is the freight on 1 00 l ong tons at the same rate68 . How many years wil l be required to pay a debt of

at the rate of 8 650 in 5 years69 . I sol d 20 barrel s of appl es at a barrel and

spent the money thus obtained for cl oth at a yard.

This I afterwards sol d at 60 cents a yard and bought a

horse with the proceeds . How much did I pay for the

horse70 . On a certain rail road, a passenger counted 25 tel e

graph pol es passed in 3 minutes. If the pol es were set 88

yards apart, how many mil es per hour was the train going

SECTION VIII .

COMMON FRACTIONS .

1 74 . When a unit is divided into two equal parts, whatis each part cal l ed ? Whatis each part cal l ed when the

W

unit is div ided into threeequal parts into four ? five

J‘ l— W — H W

six ten twenty one hun fl—W — U T — H — fi

dred

1 75 . One of the equal partsof a un it is a fractional unit . A col l ection of fractionalunits is a fractional number. By common usage fractionalunits and fractional numbers are al l cal l ed fractions .

1 76 . A fraction is one or more of the equal parts of a

unit.

1 77 . The unit which is divided to make a fraction is

cal l ed the unit of the fraction.

1 78. The number of equal parts into which the unit of

the fraction is divided,is the denominator (namer) of the

fraction . The number of equal parts taken is the numerator

(numberer) of the fraction . The numerator and denominator are cal l ed the terms of the fraction .

In two th irds of an inch what is the fractional unit 7 What is the

unit of the fraction 7 The denominator 7 The numerator 7

1 79 . Tenths, hundredths, thousandths, etc.

,are decimal

fractions (Art . Al l other fractions are cal l ed commonfractions .

1 1 2 COMMON FRACTIONS.

1 80 . To w ri te common fractions .

The terms of a fraction are written, the numerator aboveand the denominator be l ow a l ine . Thus,

Numerator, 2two thirds Of an inch 1 6 wri tten as 1 1 1 the Denominator, 5margin .

The form of writing fractions as shown above , is the same as one

of the forms used to indicate division (Art .

The expression 8may mean'two thirds Of one or one third of two.

By the fol l owing il lustration , the two quantities are shown to be

equal .

1 inch 1 inch

W Ml — 4 —

1— I — I

g of 1 equal s 11, of 2 .

1 81 . a . Name a fraction that has for its denominatorfour ; eight ; el even . Name a fraction that has for its

numerator six ; twel ve ; nine .

b. Which is the greater part of a thing, e} or 1} or116

c. What is meant by the expression 152of a mil e

Ans. It means 5 of the 1 2 equal parts into which the tinit 1 mile isdivided, or it means 1 twel fth of 5 mil es.

d . What ismeant by the expression 4of a foot 110 of 1

8 . Show by a diagram that of 1 equal s 4 of 3.

1 82 . To change a fraction to l arger or smal l er terms .

By the fol l ow ing il l ustration it wil l be seen that thefractions 4g, g, and of the same unit are equal .

In changing to 4, both terms of the fraction i are made twice

as large ; in changing g to 3, both terms of 4 are made 3 times as

l arge .

So, on the other hand, in changing

{gto 3, both terms of {8 are made 1 34 l l l l

hal f as l arge, and in changing g to -i— i -i

8 both terms of s are made 1 thirdg

1 1 4 COMMON FRACTION S. .

1 87 . To change fractions to l arger terms .

Il lustrative Examp le. Change 3 to 1 2ths.

To change 4ths to 1 2ths the denominator mustWRITTEN WORK be mul tipl ied by 3 . To preserve the value of the3 X 3 9 fraction the numerator also must be mul tipl ied

4 x 3 1 2 by 3 (Art . Ans. 192

And, general ly , to change a fraction to l argerterms, mul tip ly both terms by such a number as wi l l change the denom

inator to the required denominator.

Non . If the number to mul tipl y by is not readily seen, it can be

found by dividing the required denominator by the denominator of

the given fraction .

Oral Exercises .

1 88 . a . Change 55 to equival ent fractions having for

denominatms

b. Change%to equival ent fractions hav ing for denominators

c. Change%to equival ent fractions hav ing for denominators

Changed . To 1 2ths :

35

, g,2

. Is. To 48ths : g,H.

e. To 1 6ths : 7 , g , 13, 1 , l . To 56ths : 9, 1215

.

f . To 24ths : 2 , 117 , {U Q. m. To 72ds : g, 1 1 7 ,

9 . To 30ths : {0 g, g, T5,” 125

. n . To 80ths : 5, g, 1 1 5 .

h. To 36ths : g, 1 52 , 1 99 , 0 . To 1 00ths :395 , 12

25.

i . To 40ths : g, 1 1 6 5 g. 1 5. To 1 20ths : fr , T1125 2

11

.

j . To 42ds : 13, g, 111

. q. To 1 50ths : 1225, £56 .

1 89 . A proper fraction is a fraction that is l ess than an

integral unit. A fractional number that equal s or exceedsan integral unit, as 5g, is cal l ed an improper fraction.

1 90 . A mixed number is a number consisting of an inte

ger and a fraction, as 3g.

RED U CTION . 1 1 5

1 9 1 . To change improper fractions to integers or to mixed

numbers.

I l lustrative Examp le. Change 1 91 1 and 1 352 to integers orto mixed numbers.

Since 7 sevenths equal a unit,RITT N ORK.W E Win i gfl there are as many umts

7 ) 750 as there are 7 ’s in 560 , or 80 .

80 1 07 1 11"

Ans.fi g-0 80 ;

1 39.

1 9 2 . From this il lustration may be derived the fol l owingRul e

To change an improper fraction to an integer or to a

mixed number : D ivide the numerator by the denominator.

Oral Exercises .

1 93 . Change to integers or to mixed numbersa 4

15555 ; 515 ; 575 ; 595 ; 1 1551 ; 1 1 515 5 1 151 5 ; 1 8 1 5 1 35 5 935

b ”755915535551 —8 -1-85 5 1 51

9 55 4

55H5

1— t5 5 5 85


1 94. Change the fol l owing to integers or mixed numbers,

and express the resul ting fractions in their smal l est terms1 7 . 2 2 . 2 7 . 32 .

285 3 7 .

1 8 11742 23 5

5505 28 5

5135 33 38 1

9595

1 9 . 24 .L352 2 9 .

£ 531 34 .81916 39 .

20 .L331 as.

aw 30 . 35 .i fs—

552 40 .

1 95 . To change an integer or a mixed number to an im

proper fraction .

I l lustrative Examp le. Change 483r to fourths .

WRITTEN WORK .

Since in 1 there are 4 fourths, in 48 there are1485 —5.

48 times 4 fourths, which with 3 fourths added4

are 1 95 fourths. Ans.1 34 1 .

1 95

1 1 6 COMMON FRACTION S.

1 9 6 . From the foregoing il lustrations may be derivedthe fol l owing

Rul e

To change an integer or a mixed number to an improperfraction : Mul tip ly the integer by the denominator of the f rac

tion,and to the p roduct add the numerator ; the result wil l

be the numerator of the required f raction.

Oral Exercises .

1 9 7 . Change the fol l owing to 7ths ; 8ths ; 9ths ; 1 2ths .

8 ; 3 ; 5 ; 2 ; 4 ; 7 ; 6 ; 9 ; 1 1 ; 1 0.

Change to improper fractions

a 4i s 3 11 ; 9i s 755 ; 38; 4 1 15 5 l 2i 5 653; 1 1

1553

b 31 ; 5555 ; 7i 5 45; 48 ; 3555 451515 ; l otf0 852 ; fit ; 65; 85; 78; 8535; 71 55 1 2

15

d 455 91 75 ; 855 5 8355 5 95; Hit ; 1 2

155


1 9 8 . Change the fol l owing to improper fractions

42 . s1 i 46 . so. 969,

54 . 3335, se ss4g,43 . 4 7 . 684; 51 . 55 . 59 . 5671

1,

44 . sag 48 . 52 . 87g . 56 . 1 665, so. 264255

45: 3751; 49 . 893. 53 . 99; 57 . 4533 61 .

What is a fractional unit? A fractional number ‘

2 What name

is appl ied to both Name and define the terms of a fraction . Ex

plain the expression 3. How do you change fractions to smal lerterms To larger terms When is a fraction expressed in its

smal l est terms How do you change improper fractions to integersor mixed numbers How do you change integers or mixed numbers

to fractions

First change the fractional part to smal l est terms.

COMMON FRACTION B.

202 . Exampl es for Oral or W ritten W ork .

63 . 66 . 6 9 . 7 2 .

64 . 6 7 . 70 .

— 9 7 3 . g -3

203 . In adding fractions, auv common denominator maybe used

,but the least common denomina tor is to be preferred.

This is always the least multip le common to a l l the denomi

na tora and is cal l ed the l east common mul tipl e (L. C.

A common mul tipl e of the denominators must contain al l the de

nominators, and hence al l their prime factors ; the l east common

mul tipl e must contain only these factors, each occurring as many times

as it occurs in any one of the numbers. When these factors cannot

be readily seen, the method at the l eft of the written work be lowmaybe used for finding them, and from them obtaining the l east common

denominator. For another explanation, see Supplement, Art. 8.

204. I l lustrative Examp le. Add 3, and 191

.

WRITTEN WORK .

5 5 x 7 x 3 £5 ,

8 1 68 1 68’

8 8 x 2 x 4 64.

2 1 1 68 1 68’

1 4 2 1

r—5

L. c.

9 9 x 4 x 3 1 08;

1 4 1 68 1 68

Here by repeated divisions the277

factors which are common to two 1 68 1 68

ormore of the denominators, and — 1 11-8-g Ans .

which , therefore, shoul d enter into the common denominator but once,are taken out. The product of these factors with those that are not

common must be the l east common denominator, which is 1 68.

To change 3 to 1 68ths, the denominator 8 is mul tipl ied by 7 x 3,so the numerator 5 must be mul tipl ied by 7 x 3 .

In a similar way , i f is found to equal and 151 to equal

Adding these, the sum is i t} Ans.

8

ADDITION . 1 1 9

205. From the preceding exampl es may be derived thefol l owing

Rul es.

I. To change fractions to equivalent fractions having theleast common denominator1 . For the common denominator, find the least common

multip le of the given denominators.

2 . For the new numerators, mul tip ly the numerator ofeach f raction by the number by which its denominator must

bemul tip l ied to produce the common denominator.

II. To add fractions1 . If they have a common denominator, add their

numerators for the numerator of the answer.

2 . If they have not a common denominator, change themto equiva lent fractions that have a common denominator,and then add their numerators.

206 . Add the fol lowing in l ines and in col umns

78. 7 9 . a) . 81 . 86 . 87 . 88 . 89 . 90 .

74' 82

75 ‘

lll' f +ti5+ t“ 83

155

°

77 ° 35 “Hz? 85

Nora . In performing the fol l owing exampl es, add the integers

and fractions separately. (See Suppl ement, Art .

1 02 . 1 03 . 1 04 .

91 . 25 ; as +843,

92 . 4ag+ 1 6g 9 9 . 4033+

as. 946+ 6 195

. 1 00 . 1 2 ,1r+ 25ti +9 1H

94. 1 0 1 . 6534+ 84fi +55fi

1 20 COMMON FRACTIONS.

1 05 . Out of a barrel of vinegar were drawn gal l ons, 33gal l ons, and 2 } gal l ons, after which 1 2 } gal l ons remained.

Howmany gal l ons did the barrel contain at first1 06 . HOW much silk is there in 4 remnants measuring

1 2gyards, 1 51 456 yards, 1 1 75 yards, and 9 } yards ?1 07 . How much i ron rail ing is needed to fence a court,

whose sides are 4 1 1r rods , 3 i,5f rods, 2 ; rods, and 72 rods ?1 08 . John weighs 92 17 pounds, James 651-5 , Charl es 85g,

andRichard 783pounds. What 1 s the sumO f these weights ?1 09 . From a piece Of 1 ibbon measuring 1 0 yards 33. yards,

2 1} yards, and 3gyards were cut. How much remained ?

SU BTRACTION OF FRACT IONS.

O ral Exercises .

207 . I l lustrative Examp le. From Of a yard of vel vet

g. Of a yard was taken . What part remained Ans. atFind the remainders in the fol l owing exampl es :

a . c. 55? — 171 " 6 . 35

.- l g. g. 8

b. fay— 136

. d.

11546—15095

. j : 1 § h. 2 §

t . 5g g m. 3; 23. q.

n . 2735 1

115

. r. 6257 511+

k. o. 7 1415 31

35

. 8 . 1 5376

l . 8% p . 5T5§— 2 -

1t. 82

35— 33

55

.

When the minuend and subtrahend are l ike fractions, how do you

subtract In mixed numbers when the fraction in the subtrahend

is larger than the fraction in the minuend, how do you proceed

208 . I l lustrative Examp le. If Mary had g of a yard Of

satin and has used of a yard, how much has she l eft

That the subtraction may be performed,WRITTEN WORK

these fractions must be changed to equiva

3 2 9 8 1 l ent frac tions having a common denominator.

4 3 1 2The least common denominator is 1 2 .

aud i t192 Ans . r

‘

r yard.

'

1 22 COMMON FRACTION S.

1 2 6 . A pol e is standing so that g of it is in the‘water,

in the mud, and the rest in air. What part is in the air1 2 7 . The average age of the pupil s in Arthur

’s school is

1 2mgyears ; Arthur is 1 01 1 , years old. How much is hisage bel ow the average1 2 8 . If 69} yards of cl oth shrank 1 } yards in bl eaching,what was the l ength of the cl oth after bl eaching1 2 9 . If you l ose 3 } a bushel by sel l ing wheat at $ 1 } a

bushel,how much did your wheat cost you a bushel ?

1 30 . Mr. Gaue bought flour at $ 51 11,T a barrel and gainedin sel l ing it $ 1 55 a barrel . For how much a barrel did hesel l it1 31 . Al bert has 851 1 1 6 . How much more must he get to

buy a cap worth 3 2} and a book worth 3 g1 32 . Aman wil l ed 1 17 Of his property to his mother, to

an orphan asylum, 116to the publ ic l ibrary, and the remain

der to a col l ege. What part of his money did he wil l tothe col l ege .

‘7

1 33 . From what number must you take 1 94 to l eave‘7

1 34 . What number must you take from 50} to l eave‘7

1 35 . What number must you add to 1 915,to equal

To equal1 36 . From 35} take 5} 2

1’s 4}

1 37 . 5} pounds of powder were made by mixing sal tpeterwith $5} of a pound of charcoal and an equal quantity ofsul phur. How many pounds of sal tpeter were used

1 33 . 1 43 . 1 6

1 39 . 1 44 . 1 6 — 4}n o s+r— a— s )

— 9 1 4s. 1 6 — (e1 41 fi — t 1 46 . i fs 9

1 42 . 1 4 7 . 54

What are l ike fractions How do you prepare unl ike fractions

for adding and subtracting . Sel ect three fractions of different denominators ; add and explain . G ive a rul e for addition of fractions . How

do you add mixed numbers How do you subtract fractions

MULTIP LICATION . 1 23

MULTIPLlCAT lON OF FRACTIONS.

2 1 2 . To mul tipl y a fraction by an integer.

I l lustrative Examples. (1 ) How many are 5 times 11,

(2 ) 6 times ‘

izf ( l ) 5 times 1

7, are 1 3

Ans . 23 . Here the numerator,WRITTEN WORK“

7 , is mul tipl ied by 5.

7 X 5 35 (2 ) In mul tiplying r’r by 6 ,

1 2 1 2 the mul tipl ication is expressedas in the previous case , and the

7 X 6 7 c is cancel ed. This has the

1 2 2 effect of dividing the denomi2 nator by 6 . Ans .

Hence to mul tiply a fraction by an integer : Multip ly the

numerator or divide the denominator by the integer.

In mul tiplying a fraction by an integer, when shoul d you divide

the denominator Why‘

1’

Oral Exercises .

2 1 3 . Expressing the resul ts in their smal l est terms,a . Mul tiply } by 2 ; 5 ; 8 ; 9 .

b. Mul tiply by 3 ; 6 ; 7 ; 8 ; 1 1 .

0 . Mul tiply 35} by 2 ; 3 ; 4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 1 0 ; 1 1 ; 1 2 .

d. How many pounds of wool are there in 9 yards ofcl oth if there are of a pound to a yard ?Find the cost of :e. 8 dozen eggs at a dozen . At 85 At 35

j : 6 bushel s of potatoes at 3 1 56 a bushel . At 35g.

g. 6 hal f-barrel s of flour at 2} apiece.

Norm. Mul tipl y the integer and the fraction separatel y.

h. A tail or uses 5} yards of cl oth for a suit of cl othes .

How much does he require for a dozen simil ar suitsi . If a quart of kerosene wil l supply 5 l amps 2 31 , hours,how l ong wil l it supply 1 l ampj . If 1 wil l pay for 7Q yards of ribbon

,for how many

yards wil l 358 pay For how many wil l 351 1 pay



21 4 . I l lustrative Example. What is the width of 7 housel ots, each 932 feet wide

WRITTEN WORK.

932 Mul tiplying the fraction and the integer sepae7

rately, and adding the resul ts, their sum is

5} Ans. 666 } feet.651

656}

1 48 . If 1 3 men can do a piece of work in 8} days, howl ong wil l it take 1 man to do it ?1 49 . When an el ectric car goes at the rate of a mil e in

71290minutes, in how many minutes wil l it go 9 mi l es 20

mil es1 50 . By using 3 teams, a contractor can cart a l ot of

gravel in 1 4} days . How l ong wil l it take him if he usesbut 1 team1 51 . I own acres of l and which is 1 fifth of an

undivided l ot. How many acres are there in the l ot1 52 . There are 57fie cubic inches in a quart, l iquid meas

ure. How many cubic inches are there in a gal l onIf 1 7 men can shear a l ot of sheep in 9 1 26 days, in

what time can 1 man shear the lot1 54 . How many square inches are there in the surface of

a desk 28} inches l ong and 1 5 inches wide In a board425. inches l ong and 1 7 inches wide1 55 . What must be paid for 2 shares of Canadian P acific

stock at 635. a share, and 5 shares Fitchburg rail roadstock at 3 69} a share1 56 . Mul tiply 9 1 7} by 8. 1 60 . x 1 31 57 . Mul tiply by 9 . 1 61 . 951

5} x 1 8

1 58 . Mul tiply 1 9 1 27 by 1 1 . 1 62 . 1 1 8151 x 1 5

1 59 . Mul tiply by 5 . 1 63 . 324 15; x 24


21 7 . From the previous il l ustrations may be derived thefol l owing

Rul es .

1 . To mul tiply an integer by a fraction, or a fraction byan integer : Multip ly together the integer and the numerator

Of the fraction for the numerator Of the product, and take

the denominator Of the fraction for the denomina tor Of the

product.

2 . To mul tiply a fraction by a fraction : Multip ly the

numerators together for the numerator Of the product, and

multip ly the denominators together for the denominator Of the

p roduct.

2 1 8 . I l lustrative Examp le. Mul tipl y 87 byWRITTEN WORK .

87

23 Mul tiplying by the fraction and integer sepa

7)522 rately , and adding the products as in the mar

74 } gin, the resul t is Ans.

1 74

248;Exampl es for W ritten W ork .

1 64 . 79 1

.

x 1 66 . 487 x 2g . me. 901 2 x1 65 . 938 x 1 6 7 . 804 x 8

112

. 1 69 . 3456 x

I l lustrative Examp le. Find 3 of

WRITTEN WORK. In mul tiplying together fractions and

7 x 3 mixed numbers, first change the mixedi Ii» . Ans .

3 x 4 numbers to improper fracti ons .

Mul tiply the fol l owing in l ines and in col umns as indicated

1 7 2 . 1 7 3 . 1 7 6 . 1 7 7 . 1 81 .

1 70 . } of 1 7 4 .

153;of 1 78 . gof 1 82 . 2} x

1 7 1 . } of 1 1 }. 1 7 5 . g of 1 79 . .5r of x

MULTIP LICATION . 1 27

1 9 2 . 1 93 . 1 96 . 1 9 7 .

1 8 6 . g of 1 90 . 26} x 1 9 4 . 87} x 9 .

1 8 7 . 1 5} x 1 9 1 . x 1 5. 1 95 . x

2 20 . I l lustrative Examp le. What part of an estate isof of g of itW RITTEN WORK.

NOTE . Such express1ons as of {gof g, gofetc . , are cal l ed compound fractions, but they

merely indicate the mul tipl ication of fractions,

3x 5 XA or the finding of a part of a part.

1 98 . 200 .

1 9 9 . } of } of 3} months = ? 201 .

Mul tiply the fol l owing in l ines and in columns2 1 1 . 2 1 2 . 2 1 3 . 2 1 7 . 21 8 . 2 1 9 .

202 . 208 . } x } x 2 1 4 .

155 x 8x 1 8}.

Oral Exercises .

22 1 . a . If 1 4 yards of cl oth wil l make a dress,and as

much a wrapper, how much wil l make the wrapper ?b. How many acres remain of a 7-acre lot of l and afterof it is sol dc. Of a j ourney of 56 mil es, how many mil es remain

after 3» of it is travel ed.

d. A boy bought a flute for 80 cents, and sol d it for ofits cost. How much did he l osee. He bought a boat for 24, and sold it for Of its cost.How much did he gain ?f . Mining stock which cost 1 00 a share has decl ined to

17} of its cost. What is it now worth

g. A book, the retail price of which was 8 1 , was soldat whol esal e for of the retail price with 1 13 off from that,for cash. For how much was it sold

COMMON FRACTION8 .

222 . Exampl es for W ritten W ork .

2 20 . A house costing $5260 was mortgaged for of itscost. For how much was it mortgaged22 1 . If

Th of $475 is paid for its use for a year,how

much is paid2 22 . At a mark-down sal e

, a suit formerly markedwas sol d for of this price . How much did it bring22 3 . Of 1 50 acres of l and, was sold to one man, and

1} to another. How many acres were sol d224 . Find the cost of 9} yards of cl oth at 29 cents a yard .

22 5 . A quotient is 1 517} and a div isor 1 1 7 . What is thedividend2 2 6 . Find the cost of 1 8} feet of boards at 23 cents a foot22 7 . How many 1 -0 unce bags wil l be required to hol d 32}

pounds of flower seeds2 28 . A store worth $9550 was insured for of its value.

How much was insured ? What was the cost of insuringit at of a cent on every dol l ar insured22 9 . A tax of of the value of a l ot of gl assware

imported from France, and valued at $250, was paid at the

CustomHouse . How much was paid2 30 . There are 272} square feet in a square rod. How

many square feet are there in 9 1 5} square rods231 . What is the value of 12; of a square rod of land at 8}cents a square foot232 . The wheel of a wagon, 1 2 1 5} feet around, revol ves

28} times in going to the rail road station. How manyfeet is it to the station2 33 . Apiece of ground 1 6} yards l ong and 4} yards wide

is to be sodded. What wil l the sodding cost at 1 } centsa square foot2 34 . How many square feet are there in a lot 87 feet

l ong and 43} feet wide, and what is it worth at $180a

square foot

1 30 COMMON FRACTION S .

f . Aw alk of 8} mil es in 3 hours is an average of howmany mil es per hour

When the dividend is a mixed number andless than the divisor, as in example e.

, th e

WRITTEN WORK .

dividend is changed to an improper fractionbefore dividing .

132 When the dividend is greater than the divisor ,2 ‘H as in exampl e f . , the integer is first divided, and

the remainder on l y is changed to an improperfraction . Ans . 2h} .

g. Aweek’s work for 1 0} is how much per day ?

h. A tank can be emptied by 1 pipe in 1 0} hours ; inwhat time can it be emptied by 3 such pipesi . When ice forms 1 1 } inches thick in 4 successive

nights, what is the average thickness formed per night

j . If a boy working al one can pick a lot of pease o in 6}days, how l ong wil l it take with 3 boys to hel p himk. Ral ph travel s on his bicycl e 75} mil es in a day of

9 hours ; what is his average rate per hour

225 . Examp l es for W ritten W ork .

235 . If an engine goes mil es in five minutes, whatis its average rate per minute2 36 . At the rate of 51 } mil es per hour, how far does an

express go in 1 minute237 . When a product is 27g and the mul tipl ier 8, what is

the mul tipl icand238 . When a product is 54} and the mul tipl icand 9, what

is the mul tipl ier239 . A rope 1 48} feet l ong was cut into 1 2 equal l engths.

How many feet were there in each l ength240 . What is the price of 1 yard of fur trimming, the

cost of 1 1 yards being $ 1 7241 . A seamstress put 1 4} yards of ribbon into 46 equal

Ioops. How much was put into each l oop

DIVISION . 1 31

2 2 6 . To div ide an integer or a fraction by a fraction .

a . How many thirds are there in 2 how many fourths ?eighths tenthsb. 3 are how many times

T117 ? 2

16 ?

I l lustrative Example. How many yards of cl oth at g ofa do l l ar a yard can be bought for $6

As many yards can be bought asWRITTEN WORK there are times 3 in 6 . 6 changed

5 48 5 48 to 8ths There are as many

8 8 g g935 times 51 in as there are 5’s in 48,

which is 93. Ans . 9g.6 X 8 5 6 X 8

9gIn the last form of written work,

8 8 5 we have the dividend, 6 , mul tipl ied

by g. But g is the inverted expres

sion of the divisor g.

or, simpl y, 6 x g

Oral Exercises .

2 27 . c. How many hours’ work at13}of a dol lar an hour

can be obtained for $3 For $6 For $9

d. How many steps of a foot high wil l be required toascend 8 feet 1 2 feete. How many pounds of tea at of a dol l ar a pound can

be bought for $5f . A heaped bushel is of an even bushel . How manybushel s of appl es, heaped measure, can be put into barrel sthat hold just 7 bushel s of shel l ed beans, even measure

g. A dea l er bought $9 worth of hats at 19}of a dol l ar

each. How many hats did he buyh. If the deal er had bought his hats at $ 1 } each, how

many could he have boughti . 6 are how many times g ?

j . How many are 8 6 4 9

Dividing by is equivalent to mul tiplying by whatTo divide an integer by a fraction , how must the integer be

changed What change must be made in a mixed number used as a

divisor


75° Divide fr by 226 by 236 5“ by 3's ; fi by 12

7;

How is the division of a fraction by a fraction performed whenboth fractions have a common denominator

m. How many pil l ow sl ips of a yard l ong can be madefrom 1 2 yards of cl oth From 8} yards

n . How many paces, each 2} feet, are there in a. rod

O . When coffee cups hold of a pint each, how manycups can be fil l ed with 5} pints of coffee With a gal l on

228 . I l lustrative Example. How many yards of cl othat of a dol lar a yard can be bought for of a dol l ar

WRITTEN WORK .The fractions and g changed

2 5 1 6 1 5 1 6to fractions havmg a commondenominator equal and H» .

D ividing the new numerator ( 1 6)3 8 24 13

or, of the dividend by the new nu

2 X 8 3 x 5 2 x 81 1

merator ( 1 5) of the di visor, the24 24 3 X 5

T5 ’ resul t is 1 1 1 5 . Ans . 1 115 yards .

Or, using the factors by which

2 8the new numerators are obtained,

xin changing to a common denom

3 X 5 inator, we have the original divi

dend (g) , mul tipl ied by g. But g is the inverted expression of the

divisor g!

22 9 . From the above il lustrations may be derived thefol l owing

or, simply,

Rul es .

1 . To div ide a fraction by an integer : Divide the numerator Or mul tip ly the denominator by the integer.

2 . To divide an integer or a fraction by a fractionChange the div idend and divisor to fractions having a com

mon denominator, and then divide the numerator Of the

dividend by the numerator Of the divisor. Or,

Invert the exp ression Of the divisor and proceed as in mul

tip l ication Of f ractions.

2 For other explanations of division of fractions, see Suppl ement,Art. 1 0 .


280 . An agent sel l s flour,receiv ing 45 cents on a dol l ar

of the amount of the sal es. If he receives 35 what isthe amount of the sal es

281 . If tel egraph pol es are set one to every 5rlrrods,

how many are set in 3 1} mil es282 . How many rail road ties wil l be required for 300

feet of a singl e track rai l road which is supported at the

ends and throughout by ties 25 feet apart

231 . I l l ustrative Examp le. What is the quotient of g—g?WRITTEN WORK .

20denotes that 20 is to be divided

5}20 20 x 2 40 by (Arts Mul tipl ying both

1 1 fi312? terms by 2 , we have 40 di vided by 1 1 ,

which equals Ans. 3fi .

P erform the operations indicated bel ow8 2 l 1 7h286 . 29 2 .

81} 4 6} 1 0

33 1} I l it 3” 4

s 2 i 1 0 4a

fit 29 1 . [fig 294 . fit.ir t

Expressions l ike those above are cal l ed compl ex fractions .

How do you mul tip ly a fraction by an integer ? a mixed numberby an integer Expl ain , by an exampl e, the method of mul tiplyingan integer by a fraction. Mul tipl y a fraction by a fraction ; explain

and give the rule . How do you mul tipl y a mixed number by a.

mixed number or by a fraction How can you simpl ify compound

fractions

How do you divide a fraction by an integer a mixed number,by

an integer ? an integer by a fraction Dividing by Q is equivalentto mul tipl ying by what Mul tipl ying by 4 is equival ent to dividingby what Exp lain by an exampl e the method of dividing a fraction

by a fraction , and give the rul e . How can you simpl ify the expres

sions cal l ed comp l ex fractions .

SECTION IX .

MISCELLANEOU S AND REVIEW S .

232 . To find w hat part one number is of another.

a . What part of 5 is l ? 2 ? 3 ? 4 ? Of’

l is 4 ? 5 ?

b. One is what part of 3 ? 5 ? 1 0 ? 20 ? 1 00 ?

In comparing 1 with any number to see what part it is of that number, what do you take as the numerator as the denominator

c. What part of 30 days is 1 day of 32 quarts is 1 quartd. What part of 1 dozen is 6 ? 3 ? 4 ? 9 ? 8 ? 2 ?

.

1 0 ?

e. What part of 1 score is 1 0 ? 4 ? 5 ? 1 5 ? 1 6 ? 8 ? 1 2 ?

f . If a cistern can be drained in 1 2 minutes, what part ofit can be drained in 5 minutes

g. A firm fail ing can pay but 30 cents on a dol lar of whatthey owe . What part of their debts can they payh. Goods which cost 50 cents per yard sold for 75 cents.

What part of the cost was the gaini . What part of the cost is the l oss, when brooms whichcost 30 cents sel l for 24 cents

j . Fred and Frank buy a quire of paper together ; F red

uses 8 sheets, Frank uses 1 6 . What part of the cost shouldeach pay

233 . From the foregoing il lustrations may be derived

the fol l owingRul e .

To find what part one number is of another : Make

number which is the part the numerator Of a fraction, and

number wi th which it is compared the denominator.

1 35

1 36 MISCELLANEOU S AND REVIEWS.


234 . What part of1 . 8§ is 5 ? 6 . 1 00 i3 1 2 1} ? 1 1 .

296 3 3

16?

2 . 1 05 is 3 ? 7 . 1 00 is 1 2 . g is

3 . is 7 ? 8. 1 00 is 874 ? 1 3 . 4} is

4 . 9 . fi is 1 4 . 1 6§ is Gt ?

5 . 25 is 1 0 . fi is jflg ? 1 5 . 83 1} is 1 00 ?

1 6 . What part of 2724 square feet is 9 square feet1 7 . The material for a cl oak cost $ 1 24, the making

$ 1 1 5. What part of the entire cost was the cost of making

1 8. If a man can do a piece of work in 25 days, whatpart of it can he do in 6%days1 9 . A can l ay a hal l floor in 6 days, B can do it in 8

days. What part of the work can each do in one day

What part can both together do in one day ?

20 . Three men working at the same rate per day did a

piece of work for $ 90. A worked 5 days, B 65 days, and

C Si days. How much did each receive per day ? Howmuch did each receive for his work2 1 . In a rectangul ar l ot 1 00 ft. l ong and 80 ft. wide there

is an asparagus bed, al so rectangul ar, 50 ft. l ong and 1 2 ft.w ide . What part of the l ot does the bed occupy2 2 . What part of his earnings does aman save who works

6 days a week, receiving per day, and paying for

his board $ 4 a week, and for al l other expenses a

quarter ( 1 3 weeks) ?

235 . To find the w hol e w hen a part is giv en .

a . In 3; of a reamof paper there are 8 quires. Howmany quires are there in of a ream In a whol e ream

b. 8 is gof what numberSOLUTION .

— Since 8 is 3, of a number,1 hal f of 8, or 4, is i of the

number, and gof the number is 5 times 4 or 20.

MISCELLANEOU S AND REVIEWS.

2 3 . 4 l b. of what 2 6 . 6” of what24 .

16,ft . of what 2 7 . 33 gal . 1

26of what

25 . fl A.

131of what 28. 13.

-Of what

29 . If g of a barrel of flour weighs 731} pounds, what isthe weight of a barrel30 . A farmer received in one year 3575 as a royal ty for

O il produced on his farm. If this was 3. of the val ue pro

duced, what is the value producedWhat is the cost per pound31 . Of veal when 5§ pounds can be bought for 92 cents ?32 . Of butter when 4,1I pounds can be bought for33 . Of sugar when pounds can be bought foradol lar ?34 . Of tea when a box of 1 3 13 pounds costs $ 1 0 ?35 . What 1 3 the cost per gross of pencil s when 5} gross

can be bought for $ 74 9

36 . When 74 gal l ons of mol asses costs what is the

price per gal l on ?37 . If 4 :

1f pounds of hal ibut costs what 1 8 the cost

per pound ?38 . After l osing 4 of his property a man had $ 2400 l eft.

How much money had he at first39 . A sold 9, of his farm and had 41 2 acres l eft . How

many acres had he at first40 . Mr.White owning of a yacht sold 4of his share for

$ 550 . What was the va lue of the yacht at the same rate4 1 . What number is that which increased by i of itsel f

equal s 29 1 1}

42 . What number is that which diminished by i Of itsel fequal s

43 . Water in changing from the temperature of ice to thatof boil ing, 1 ncreases its bulk about 1 cubic foot 1 n every 24 .

At this rate of increase,how many cubic feet of boil ing

water wil l equal the we ight of 1 000 cubic feet of we water ?

ALIQU OT P ARTS. 1 39

44 . From a hogshead ofmol asses 31 4 gal l ons were taken,after which 4 of the original quantity remained. How

many gal l ons did the hogshead contain at first45 . When 4 of 4 of a vesse l is worth 980, how much is

the whol e vessel worth46 . In a col l ege of 342 students there were 4 as many

women as men. What was the number of each4 7 . A deal er sol d an umbrel l a for which was 1 4times what he paid for it. How much did he pay for it,and what was his gain48. A man fail ing in business paid onl y 4 of the value

of his bil l s. If Mr. Taft l ost 2050 by him,what was the

amount of Mr. Taft’s bil lFour chil dren picked cranberries which they sol d forOne picked 5 quarts, another 7 quarts, another 6

quarts, another 8 quarts. How much did they

“

receive perquart How much shoul d each receive for al l he had

ALIQ UOT PARTS OF NUMBERS.

237 . What is one of the four equal parts of 8 of 9 ?One of the equal parts of a number is an al iquot part of

the number. Thus,24 is an al iquot part of 9 .

Oral Exercises .

a . Of the number l O find 4 ; 4 ; 4 ; 4 ; 4; 4.

b. Of the number 30 find 4 ; 4; 4 ; 4 ; 4, 41

4; 4;

0 . Of the number 50 find4; 4 ; 4 ;

d. Of the number 60 find 14 ; 4

1

; 4 ; 4 ;e. Of the number 1 00 find 4 ; 4; 4

°

f . Of the number 1 00 find 4 ; 3

4; 4 ; 4;

g. Of the number 1 44 hud 4 ; 1

4; 4 ; 4 ;

h. Of the number 200 find 4 , 4 ; 4°

4; 4 ; 4 ; 416 ; 414 ; 4.

By using the al iquot parts of numbers,especial ly of 1 0,

1 00, and 1 000, the work of mul tiplying and dividing can

often be material ly shortened.

1 40 MISCELLANEOU S AND REVIEWS.

238 . I l lustrative Examp le. What is the cost ofshingl es at $24 per thousand

shingl es at $1 a thousand woul d cbstWRITTEN WORK

shingles at $1 0 per thousand

4) woul d cost ten times asmuch, or at 824

per thousand, they woul d cost 4 of or

Ans.

i . Give the shortest way of mul tiplying by 25 ; 1 24; 374 ;624; 874; 34; 334; 664.

j . Give the shortest way of dividing by 25 ; 1 24; 34;334, etc.


239 . What is the amount of the fol l owing purchases50 . 2 tubs of butter, each containing 35 pounds at 25fi,

and 1 tub of 45 pounds at 3741551 . 2 boxes of cheese, each containing 75 pounds at 1 2441,

and 1 box of 82 pounds at 1 6451 ?52 . 1 quarter of beef weighing 1 90 pounds at and

1 quarter weighing 2 1 0 pounds at 84¢53 . 60 yards of cashmere at 62445; 54 yards at 7 36

yards at 8754 . 1 9 yards of vel vet at $ 1 3 34; 25 yards at

22 yards at55 . 34 yards of crash at 1 3¢ 74 yards at 1 1 55; 5 yards

of broadcl oth at

NOTE .— First find the cost of 1 0 yards .

56 . 1 24 yards of silk at 7 334yards at 374 yardsat

57 . How many pounds of cocoa at 334¢ per pound, canbe bought for 1 For $ 25

se 428 x 1 64— 9 -1

59 . 262 x 624= ? 61 . 482 x 374= ? ea.

42 MISCELLANEOU S AND REVIEWS.

n . A can walk of a mil e in 1 5 minutes. If he starts at2 o’cl ock to walk 2; mil es, at what time can he finish ‘ hiswd k ?

24 1 . Miscel l aneous W ri tten Exampl es.

64 . What is the cost of 1 33. pounds of Formosa tea at

7515 per pound and 1 84 pounds of bacon at 1 0¢ per pound65 . A woman bought remnants of silk as fol l ows : «5yd.

,

yd.,1 7} yd.

, and 1} yd. How much did she buy in al l ,

and how much did she pay, at 80¢ a yard66 . From a piece of duck containing 42%yd. were cut

35yd.,2 1} yd.

, 5§ yd.

,and 3; yd. How much remained ?

6 7 . From a piece of cotton containing 38 yd. , worth 1 8¢per yard, were made as many sheets, 123. yd. l ong, as possibl e. How many sheets were made, and What was the costof cl oth for each ‘7

68 . At 50¢ a cord for each time the sticks are sawed,howmuch must be paid for sawing 8%cords of wood, eachstick to be sawed into three pieces69 . Suppose each stick of the above to be sawed into

four pieces, what shoul d be paid ?70 . How many bushel s of wheat at per bushel can

be bought for 20

7 1 . If l and is worth per square foot,'

what is the

val ue of a l ot which measures 1 00 feet by feet ?7 2 . When 3} of a mil l is worth $ 2700, what is g of the

mil l worth7 3 . Hav ing travel ed 564 mil es, I find I have gone just

3 of my journey. How many mil es have I stil l to go ?74 . A’

s share in the profits of a specul ation was

If this wasT1555 of the whol e profit, what was the whol e

profit ?7 5 . My strawberry patch yiel ded 41 40 boxes of straw

berries in two years, yiel ding as much the second year as

the first. What was the yiel d each year ?

MISCELLANEOU S WRITTEN EXAMP LES. 1 43

76 . Amixed school of 800 pupil s had as many boys as

girl s. What was the number of each ?7 7 . A farm was sold for 6580, which was 3 1} times whatit cost. How much did it cost78 . 9 men can do a piece of work in a certain time,

working 8%hours a day ; how many men woul d be requiredto do the work in the same time, working 671, hours a day79 . When wood was $37 1} a cord, I gave cords for 1 ;

tons of coal . What was the coal worth a ton

80 . Owning of a yacht, I sol d of my share for 1 520 .

What was the value of the whol e yacht at the same rate

81 . How l ong wi l l a barrel of flour ( 1 96 l b. ) l ast 1 2 per

sons at the rate of of a pound a day to each person82 . What wil l 425 quires of paper weigh, at 1 2 pounds to

the ream

83 . I shipped to a man in P aris $ 3 1 8 worth of cotton,

and in return received 1 200 francs. If the exchange val ue

of a franc is 1 9 1 80“ cents, how many francs are stil l due me84 . Two ships, l 7511,r mil es apart, sai l ed towards eachother. After one had sail ed 26g mil es and the other 37mil es, how far apart were they85 . Two persons start at the same pl ace at 6 o’cl ock

5 min . A.M. , and travel in opposite directions, one at the

rate of 7 mil es an hour, the other at the rate of 4 ; mil esan hour. How far apart wi l l they be at 2 o’cl ock 1 5

min. P .M.

86 . If ‘

a body fal l s 1 61 1 f feet the first second, 3 times

as far the next second, and 5 times as far the third second,how far does it fal l in 3 seconds87 . If the circumference of a wheel is 9 } feet, how many

times wil l it turn in going a mil e88 . A shil l ing of the ol d N ew Engl and currency was

worth cents, and 1 2 pence made a shil l ing . Whatwas the val ue in cents of the coin cal l ed ninepenceHow much was fourpence ha

’

penny worth

1 44 MISCELLANEO U S AND REVIEWS.

89 . A New York shil l ing was worth cents. Howmany New York shil l ings were equal to 6 shil l ings N . E.

currency90 . What part of a shil l ing N . E. currency was a New

York shil l ing What was the difference of their values9 1 . When the exchange value of a franc is 1 9 }I what

is the val ue of a 5-franc piece How many francs can bebought for and how many cents wil l remain

'

9 2 . Find the cost of storing, at 4} cents a month perbarrel , the fol l owing goods

2000 barrel s appl es, 1 3}mo. 1 24 barrel s squashes, g78 parsnips, l fi 66 onions, 1 1 36

953. A deal er purchased 1 6 acres of woodland whichaveraged 40 cords to the acre ; he sol d 1 75; cords of oak,49 ; cords of beech

, and 362%cords of pine . What woul dthe bal ance come to at the rate of a cord9 4 . If the average yie ld of sap per mapl e tree is 1 6}

gal l ons, and 30 gal l ons of sap make 7 pounds of sugar, whatweight of sugar wil l be made from 500 trees9 5 . In a certain school of the pupil s bel ong to the

fourth grade, 4 to the third, 125;to the second, and the re

mainder, which is 96 , bel ong to the upper grade. Howmany pupil s are there in the whol e school ?96 . A man l ost l , and g of his money, and then had

$ 26 1 3 l eft. What sum had he original ly9 7 . A farmer had 2364 sheep, and sold hal f “of them

to one man , and to another. How many had he l eft98 . Aman owning 26 1 § acres of l and, sold

161

" of it toone man and 7; of the remainder to another. How muchdid he sel l to both men9 9 . A. man owning a meadow of 1 82 acres

, sol d 27;acres to one man and s of the remainder to another. At

$32 an acre, what was the value of the l and he kept, andhow much did he receive for the l and he sol d

1 46 MISCELLANEO U S AND REVIEWs.

Dam . TABLE No . 5. (See Suppl ement, Art.

For suppl ementary practice in fractions.

2 3 4 5 6 7 8 9 1 0 1 1 1 3l1 4 1 5

i f l f t i’r t f t i f wi l l

l t l l l i l f t t i t ?

it ti l t in?fi ft fi i i fi ft f

tfi if C

4& 7 l 4 l 6fi 3l 2 l l t 3 l 4l fit 4 4 3; n

In each of the columns, 1 , 2, 3, etc.,

Add A and B . 24 7-2 61 . Mul tiply A by B.

Add B and 0 . 2 62 - 2 76 . Mul tiply B by 0 .

Add 0 and D. 2 77 — 29 1 . Mul tiply D by 0 .

1 574 71 . Add A,B, and C. Mul tiply D by

Add B, C, and D . 307—3 2 1 . Div ide C by B.

1 87—201 . AddA,B, C, and D. 32 2 —336 . Div ide D by 3.

202-2 1 6 . Take C from D. 337 — 351 . Divide A by B.

2 1 7 - 2 31 . Take B from D. 352—366 . Divide B by 0 .

232—246 . Take B C fromD. 36 7 — 381 . Div ide D by A.

382-396 . Mul tiply C by A by B .

39 7—4 1 1 . Find Aof B x C of D.

41 2 —42 6 . Divide C D by B.

42 7-4 4 1 . Div ide (D A) by (A44 2 -4 56 . C g of what ? 47 2 -486 . A x C g of what ?

D : g of what ? 48 7—50 1 . 4&xD is 1 1 , of what ?

How do you find what part one number is of another

What effect does mul tiplying the numerator of a fraction have

upon the fraction Why In what other way could you produce

the same efiect, and why What efiect does dividing the numeratorhave upon a fraction Why In what other way could you produce

the same efiect, and why

SECTION X .

DECIMAL FRACTIONS .

243 . A decimal fraction is one ormore of the decimal partsof a un it, as 1 tenth, 2 hundredths, 25 thousandths, etc.

NOTE . Some el ementary appl ications of Decimal s were treated

with integral numbers. (Art. 27—30 , 44 , 60 , 89 , In this sec

tion are given appl ications which require a knowledge of principles

taught in common fractions.

244 . To read and virrite decimal s .

The manner of reading and writing decimal s is taught inArt . 29 and 30. The fol l owing tabl e shows the correspondence between the names of decimal s and integers

TABLE.

d

2 53°

b oo g w’

F ow l "

:$ 0 3

2 0 O za d '.

z moo < m ¢ m < o w zP LACE NAMES. o m x w m t oJ

o s a a s fg a n : a:

Jz z o z z _ 2 2 0 z _,

— =u 2 9 m 2 w a r m :

E r r- b z i- a l- I l- t- z i

j i f i . .

c o l“ l ‘ ti ci lt s s z m w fl “ a m t -0 5

Integers. Decimals.

Name the orders of decimals from tenths to mil l ionths ; frommil l ionths to tenths .

Il lustrate the manner of reading and writing decimal s.

How are integral numbers and decimal s read when written together1 47

1 48 DECIMAL FRACTIONS.

Exercises in Reading and W riting Decimal s .

245 . Read or write in words the fol l owing

1 . 7 . 1 3 .

2 . 8. 1 4 . 1 6 4076,

a 9 . 1 5 .

4 . 1 0 . 1 6 .

5 . 1 1 . 1 7 .

5 . 1 2 . 1 8 .

Write in figures the fol l owing25 . 208 ten-thousandths. 29 . 648 mil l ionths.

26 . 1 , and 45 thousandths. 30 . 6 thousand 48 mil l ionths.

2 7 . 1 8 ten-thousandths. 31 . 4 1 hundred-thousandths.

28 . 582 hundred-thousandths. 32 . 8, and 4003 mil l ionths .33 . 45

, and 673 mil l ionths.

34 . 5 thousand, and 51 thousandths.

35 . 6 hundred, and 48 mil l ionths.

36 . 1 0 mil l ion, and 75 ten-mil l ionths.

Write in decimal form37 mim ‘i

rtt ; are” ; n’u’n ; rottin

38 . Write fl , and883

. Ans.

1 00

333;39 Write a ;1 0

, l tt s ittt ;1 00

Write the fol l owing in decimal form, each exampl e as one

number :

40 °

18mT81 71 and Th

elm 43 "

170" fi

lm,"

11 731 70 , and Th41 °

110"

and T8" ? 44 ° 1 3 T31 1 7 W 's—(Hr, and T'

GQO'

U ‘

42 '

1“n “

fin " and 1

20“ 45 ° 50

: T6171 “

1 1 1 31 1 0 , and 1 1 1 036 1 7

(I . What is the denominator of the decimal

b. What are the numerators of the above decimal s

1 50 DECIMAL FRACTION S.

249 . From the preceding exampl e may be derived the

fol l owing

To change a common fraction to a decimal fractionAnnex zeros to the numerator and divide it by the denominator.

ADDITION AND SU BTRACTION OF DECIMALS.

For addition and subtraction of decimal s, see Arts. 44

and 60 .

250 . I l lustrative Example. Change 9 and 226 to decimal sand add them.

WRITTEN,WORK.

0 375Changed to decimals, g and

“

290

°

45the sumof which is Ans .

Change the fol l owing to decimal s, and add in l ines and

in columns :

65 . 66 . 6 7 . 71 . 7 2 . 73 . 7 7 . 78.

l _

64 7+7+a =7vo r+ee+l

s7 = 7 76 7 + a +s = 7

9

251 I l lustrative Examp le. Change 6} and 49 to decimal s and add them.

WRITTEN WORK .

fil =

ExactAns. Approximate Ans.

Neither nor 9, can be compl etely expressed in decimal form ;

however far the division is carried, there wil l stil l be a remainder.Whenever the denominator contains factors other than 2 and 5, the

division is interminable. Further explanations of such fractions wil l

be found in the Supplement, Arts. 1 1 - 1 4.

ADDITION AND SUBTRACTION . 1 51

The degree of exactness required in any particul ar exampl e wil ldetermine the exte nt to which the division shal l go . It is customaryto express the answer to the nearest tenth, hundredth, thousandth,

etc. This is done by carrying the quotient one pl ace farther than isrequired in the answer, and adding one to the tenths, hundredths,

thousandths, etc. , whenever the next term of the quotient equal s or

exceeds 5.

In the exampl es which fol low , the pupil shoul d give the decimalvalues to the nearest thousandth, unl ess otherwise directed.

Change to decimal s and add in l ines and in col umns

89 . 90 . 9 1 . 9 5 . se

sa g 92.

5g 87 . 1

775-7-T

51} 9 3 .

4 +fi +fl as. 9 4 . 1 83123 4 40 51

1,

Change to decimal s and subtract in l ines and in columns

1 03 . 1 04 .

97 . 39 24 1 01 . 02 47 1 05 .

98. 4 1 02 . 1 06 .

1 1 1 . 1 1 2 . 1 1 5 . 1 1 6 .

1 09 . 1 4 y, 1 1 3 . 1 4g 4 1 1 7 . 1 31 1 0 . {if 1 1 8.

T3;

1 2 1 . Frommy home to church is of a mi le ; thenceto my neighbor Crook

’s is mi l es ; and to Mr. Knight’s

is mil es stil l further. What is the distance from myhome to Mr. Knight

’s

1 2 2 . If water covers of the surface of the earth,what part of the surface is l and1 23 . When of a journey is compl eted, what partof the journey remains1 24 . If from a meter, which is inches, are cut

inches, 7

—1; inches, 5%inches, and inches, how much wil lbe left


1 25 . If a bal l oon rises L} mil es, then sinks of a

mil e, and again rises {r of a mil e, how high is it1 2 6 . If the entire surface of the earth is taken at 1 000

parts, the Torrid ! one contains parts, and the

Temperate ! ones, parts. How many parts are in

the Frigid ! ones1 2 7 . The four sides of a park measure, respectively, 1 409

rods, 1 28grods, 98 1 36 rods, and rods. After riding a

mil e, how much farther has one to go to ride around the

park

MULTIPLIOATION OF DECIMALS.

For mul tipl ication of decimal s by integers, see Art. 89 .

In mul tiplying a decimal by an integer, how many decimal placesmust there be in the product 2

2 52 . I l lustrative Examp les. ( 1 ) Mul tipl y 28 by (2 )Mul tiply 28 by (3) Mul tiply by

( 1 ) Tomul tiply 28 by is to take 1 tenth

of 28, which is expressed by placing the deci

( 1 ) 28 x 0-1 2 8 mal point one place from the right. Ans .

(2 ) (3) (2 ) To mul tiply 23 by is to take 7

28 2 8

WRITTEN WORK .

tenths of 28. One tenth of 28 is (28 tenths) ,

0 7and 7 tenths is 7 times or Ans .

(3) To mul tiply by is to take 7

tenths of One tenth of is and

7 tenths of is 7 times or Ans.

253 . From the written work above may be derived thefol l owing

To mul tiply decimal s : Multip ly as in integers, and point

of as many places for decima ls in the product as there are

decima l p l aces in the mul tip l icand and the multip l ier counted

together .

Norm.— If there are not figures enough in the product, prefix zeros.


At a profit of 8 on the dol lar howmuch money wouldI make1 63 . On a carriage costing 8 3681 64 . On a l ot of appl es costing 8 2461 65 . On a house costing 81 66 . On a piece of property

'

costing 8 635

1 6 7 . On a gol d watch costing 8 3751 68. On a quantity of flour costing 8 35251 69 . At the rate of “

8 on 8 1 000 for taxes, what taxshould be paid on 85000 on 8 4800 on 8

1 70 . At the rate of 8 on one dol l ar, howmuch shouldbe paid for the use of 8 3475 for one year

DIVISION OF DECIMALS.

For division of decimal s by integers, see Art. 1 35.

In dividing a decimal by an integer, how many decimal places must

be al l owed in the quotient

255 . I l lustrative Examp les . (1 ) Divide by 1 2 .

(2) Div ide by 1 2 . (3) Divide byWRITTEN WORK .

( 1 ) (2 ) (3)1 2 1 220 9552

In Art . 1 35, we l earned that in dividing a decimal by an integer,

theo

decima l point must be i nserted in the quotient when the point in

the dividend is reached. ( 1 ) Ans . (2 ) Ans.

(3) This exampl e differs from the other two inasmuch as the

divisor contains a decimal as wel l as the dividend . Since mul tiplyingthe dividend and divisor by the same number does not al ter thequotient (Art. we may mul tiply the divisor by 1 00 , thus

making it an integer, 1 2 , provided we also mul tiply the dividend by1 00, making it This increase in the dividend may be ex

pressed by a mark, as a vertical l ine, put as many places at the

right of the decimal point as the divisor has decimal pl aces. The

exampl e wil l then be read , divide by 1 2,and the work wi l l

be done as in exampl e 1 . Ans.

DIVISION . 1 55

256 . From the preceding il lustrations may be derivedthe fol l owing

To divide decimal s : D ivide as in integers. If the div isor

is an integer, point of as many p laces for decima ls in the

quotient as there are decima l p laces in the d ividend.

If the div isor is not an integer, p lace a mark in the div i

dend as many p laces to the right of the decima l p oint as there

are decimal p laces in the divisor, and p lace a decima l point in

the quotient when the terms Of the di vidend are used as far as

NorE . When there is a remainder after al l the terms of the dividend have been used, the division may be continued as in Art . 1 35.

En n ipl es for W ritten W ork .

Div ide the fol l owing in l ines and in columns

1 71 . 7 1 74 . 3 .628 .5 1 78 .

1 72 . 5 1 75 . 81 1 79 .

1 88 . 1 89 .

Divide 1 9 2 . 42 4 1 .765 1 86 . 1 8.76 4 0 .004 7 4 0 .0009

by 1 33 4 1 37 . 4 1 9 1 . 600 4

Norm. In Ex . 1 9 1 first mul tiply both dividend and divisor by 3.

1 96 . 1 97 .

1 9 4. 1 93 406 5 4 80 99 202 . 8 1 85 4

1 95 . 33 344 9 34 1 99 . 56 .87 4 203 . 8 982 4

206 . How many pounds Engl ish money, at 8 each,

can be paid for with 8 50, and how many cents remain207 . How many rubl es of Russia, at 8 each, can bebought for 8 1 00, and howmany cents remain


258 . Mi sce l l aneous Exampl es .

208 . I have 5 yards of binding for a square bed Spreadyards on each side . How much more binding do I need

209 . A cubic foot of water weighs pounds ; graniteis times as heavy as water. What is the weight of acubic foot of granite ? of cubic feet2 1 0 . How many meters of silk, yards to a meter,

shoul d I buy to make a dress requiring 1 8 yards2 1 1 . How many yards are there in a piece of dress goods

measuring 1 5 meters, 5 inches to a meter2 1 2 . Howmany inches are there in meters In

meters2 1 3 . What woul d be received for the use of 8 480 for a

certain time, at 8 on each do l l ar2 1 4 . An architect received 8 for planning a house .

If this sum was of the cost of the house, what wasthe cost2 1 5 . Edinburgh is mil es fromLondon . With cl ose

connections a travel er can go from London to Edinburghand return in 1 8 hours, andmake stops equal to 24 minutes.

What is the average speed of the cars per hour whil e inmotion (Answer to the nearest hundredth. )2 1 6 . If pounds of grain l oses ounces of its weight

in thoroughly drying, what part of the original weight ofthe grain was l iquid (Answer to the nearest thousandth .)2 1 7 . If

,by exposure to dampness, a certain articl e weigh

ing pounds absorbs ounces of water, what part of theful l Weight is the weight of the water which is absorbed2 1 8. What woul d be received on 8 1 20 of an account that

is worth cents on a dol l ar2 1 9 . Aman owning of a store bought hal f of the

remainder. What part of the store did he then own If

his share was then worth 8 1 700, what was the value ofthe whol e store


232 . What is the cost of 475mel ons at 8 1 2 per hundred ?

Norm. 475 equal s (41 7050 ) hundreds . Since 1 hundred melonscost 8 1 2 , hundreds wi l l cost times 8 1 2 .

Find the cost of2 33 . 6 casks of l ime, each 240 pounds, at 8 5 per C.

‘

234 . 1 260 posts at 8 20 per C.

235 . 2842 pounds bituminous coal at 43¢ per C.

236 . 220 bunches of onions at per C.

237 . 397 heads of l ettuce at per C.

238 . What wil l be the cost of printing 6760 ems at 30¢

per thousand ems

239 . Make out a lumberman’s bil l for the fol l owing items1 230 ft. White Wood at 8 1 8 per Mfl

‘

284 ft. Spruce at 8 1 7 per M.

93 ft. P ine at 8 48 per M.

240 . My water meters registered gal l ons for myhouse and gal l ons for my store . Find the amount ofthe bil l for water at 35¢ per thousand gal l ons.

241 . My gas meter registered April l st feet ; thebil l for January 1 st showed feet . What is the bil lfor gas for the 3 months at 8 per thousand feet, ifof the bil l is deducted for prompt paymentWhat are Decima l Fractions 2 How are their written expressions

distinguished from those of integral numbers What indicates the

denomination of the decimal ? How do you read a decimal expression How do you write decimals What is the effect of annexing

ciphers to a decimal expression 2How do you change decimal s to common fractions Common frac

tions to decimals How do you add and subtract decimal s P er

form an exampl e in mul tipl ication by a decimal , expl ain and give the

rul e . P erform an exampl e in division by a decimal , expl ain and give

the rule. How do you express the mul tipl ication of a decimal by 1 0 ;1 00 1 000 ; How do you express the division of a

decimal by 1 0 ; 1 00 ; 1 000 ;

7 C is used for “ hundred ”and M for thousand.

”

DRILL TABLE. 1 59

DRILL TABLE No . 6 . (See Suppl ement, Art.

259 . For suppl ementary practice in decimal s.

Exam

Two, and 206 thousandths.

936 ten-thousandths.

54, and 54 thousandths.

806 , and 1 047 mil l ionths .

5 hundred, and 26 hundredths.

One thousand mil l ionths.

Twenty-nine mil l ionths.

84629 1 hundred-thousandths.

Five hundred el even thousandths

4271 , and 427 1 ten-mi l l ionths.

68 thousand, and 44 tenths.

One hundred 22 thousandths.

Eight, and hundredths.

Five hundred, and 4 tenths .


242—2 55 . Read the numbers expressed in E and in G.

256 — 2 69 . Write in figures the numbers expressed in F.

270 -283 . Change E to equi val ent common fractions inlowest terms.

284- 2 9 7 . In the same way change G.

298 —31 1 . Change H to equival ent decimal s (4 places) .31 2 —3 25 . Add E

,F, and G.

Find the difference between326 -339 . E and F. 340—353 . E and G. 354-36 7 . F and G.

368—381 . Mul tiply E by F. 41 0 —42 3 . Div ide E by F.

382—39 5 . Mul tiply E by G. 424 —4 3 7 . Divide G by E.

396-4 09 . Mul tiply F by G. 438—451 . Divide G by F.

4524 6 5. Add E x 1 0, F 1 00 and G.

SECTION XI .

COMP OU ND DENOMINATE NU MBERS .

260 . I l lustrative Examp le. How many pints are therein 2 quarts 1 pint ? Ans. 5 pints.

The number, 5 pints, is made up of units of one de

nomination only. Such a number is a simple number, and

because the kind of unit is named, it is a denominate number ;hence, it may be cal l ed a simpl e denominate number.

26 1 . The number, 2 quarts 1 pint, is made up of units oftwo different denominations. A number made up of unitsof two or more different denominations is a compound de

nominate number.

262 . Denominate numbers are al so cal led concrete num

bers , and numbers considered apart from any object are

cal l ed abstract numbers .

Name a simpl e denominate number ; a compound denominate

number ; a concrete number ; an abstract number.Nora . The tables ofmeasures in common use required to sol ve the

exampl es in this section wil l be found in the Supplement, Arts. 20—46 .

MEASURES OF EXTENSION.

Exampl es for Oral and W ri tten W ork .

263 . a . Repeat or write frommemory the tabl es of LongMeasure ; of Square Measure ; of Cubic Measure.

b. What is the standard unit of l engthc. What div isions are there in the yard besides those

given in the tabl e

1 62 COMP OUND DEN OMINATE“

N UMBERS.

MEASU RES O F WEIG HT.

266 . h. Repeat or write frommemory the table of Avoirdupois Weight ; of Troy Weight.

7 . Copy and fil l out the fol l owmg

1 ton pounds ounces.

1 long ton cwt. pounds.

1 pound Troy oz . pwt . gr.

Non . Apothecaries use the Troy pound and ounce, but formix

ing medicines they divide the ounce into 8 drams of 3 scruples of

24 grains each.

7 . What weight 1 8 used in weighing silver, gol d, preciousstones, etc.

j . What is the standard unit of weightIt . How many ounces are there in 3 pounds of sil ver in

3 pounds of coal ?l . At 80 cents a Troy pound for camphor, what is the cost

of an ouncem. How many more pounds are there in a l ong ton than

in a common ton8. What is the value of a go ld chain weighing 24 ounces

at 90 cents a pwt .

267 . Comparison of W eights .

1 75 lb. Troy 1 44 l b. Avoirdupois .

1 75 oz. 1 92 oz.

7000 gr. 1 lb.

n . Which is heav ier, a pound of gol d or a pound of leadan ounce of gold or an ounce of l ead9 . From two Austral ian mines, ounces of gol d

were produced during one week in 1 894 . What was the

product in pounds Troy in pounds Avoirdupois1 0 . Change 1 2 pounds Avoirdupois to pounds Troy.

CIRCU LAR AND AN G ULAR MEASURES. 1 63

C IRCU LAR AND ANG U LAR MEASU RES.

268 . A circl e is a plane surface bounded by a l ine, everypoint of which is equal ly distant from a

point within ca l l ed the center.

269 . The bounding l ine of a circl e isthe circumference. Any part of the cir

cumference is an arc.

270 . A straight l ine drawn from the

center to the circumference of a circl eis its radius . A straight l ine drawn through the center

,

having its opposite ends in the circumference of the circl e,

is its diameter.

A Circle .

I

27 1 . The circumference of a circl e is div ided into360 equa l arcs, cal l ed degrees eachdegree into 60 minutes and each minute into 60 seconds

27 2 . When two l ines, as ab and bc,

meet each other,they form an angl e.

The l ines are the sides of the angl e, and the point wherethey meet is the vertex . The size of the angl e is the

amount by which one side is turned away from the other.

An Ang le.

273 . If the center of a circl e is pl aced at the vertex of

an angl e, the are included

between the sides is the

measure of the angl e. Thus,if the are ac contains 30

degrees, the angl e abc is an

angle of 30 degrees. Again,if the arc ad contains 50 degrees, the angl e abd is an angl e

of 50 degrees.

1 64 COMP OUND DEN OMINATE N UMBERS.

274 . An angl e of 90 degrees is a right angl e ; an angl e

of l ess than 90 degrees is an acute angl e ; an angl e of morethan 90 degrees is an obtuse angl e.

275 . Lines which together form a right angl e are perpendicul ar to each other.

Nora . As the circumference of every circle has 360 degrees, the

length of 1 degree of a circumference differs in difierent circles .

27 6 . The l ength of a degree of the circumference of theearth at the equator is about common mil es.

277 . The l ength of a minute of the circumference of theearth at the equator is a geographical or nautical mil e, and

is about common mil es . When measuring the speed ofvessel s this is cal l ed a knot .

Exercises .

278 . a . Repeat or write from memory the table of Cironl ar and Angular Measure.

Copy and fil l out the fol l owing1 circumference 0 I

b. How many degrees are there in a semi-circumferencein a quadrant, or 4 of a circumference in a sextant, or 4of a circumference

c. How many degrees are there in a right angl ed. Through how many degrees does the hour hand of a

cl ock move in 3 hours in 1 hour ? in 2 hours3 . Through how many degrees does the minute hand of

a cl ock move in hal f an hour ? in a quarter of an hour ?in 5 minutes in 1 minutef The hour andminute hands of a cl ock forman angl e ofhowmany degrees at 2 o

’cl ock at 4 o’cl ock at 5 o’cl ockat 7 o’cl ock at 1 0 o’cl ock

g. The torrid zone is 47 degrees wide. How coul d youfind its width in nautical mil es

1 66 COMP OUND DENOMINATE NUMBERS.


282 . Change to dol lars : 8 1 00 equal s

1 00 (pounds) . 1 9 . How many pounds1 00 francs . 2 0 . How many marks

1 7 . 1 00 marks. 2 1 . How many roubl es1 8. 1 00 roubles . 22 . How many fran cs23 . The gold coin of Great Britain of the value of a

pound is cal l ed a sovereign . How many shil l ings are therein 1 guinea, 2 crowns, 3 sovereigns, 1 hal f-sovereign, and 3hal f-crowns

REDUCTION.

I l lustrative Examp le. Change 3 yd. 2 ft. 6 in . to inches.

WRITTEN WORK.

3 yd 2 ft 6 in.

Since there are 3 ft. in 1 yd. ,in 3 yd. there

are 3 times 3 ft. or 9 ft. 9 ft . and 2 ft . 1 1 ft.3

Since there are 1 2 in. in 1 ft. , in 1 1 ft . there are1 1 ft 1 1 times 1 2 in. or 1 32 in. ; 1 32 in . and 6 in.

1 2 1 38 in . Ans. 1 38 in.

1 38 in.

284 . From the foregoing i l lustration may be derived thefol l owing

Rul e .

To change a compound denominate number to uni ts of al ower denomination : Mul tip ly the number of the highest

denomination by the number Of units it takes Of the next lower

denomination tO make one of that higher, and to the p roduct

add the given number Of the next lower denomination. Mul ti

p ly this sum in l ike manner, and so p roceed ti l l the given

number is changed to units 0f the required denomination.

RED U CTION . 1 67


285 . How many

24 . Inches are 3 yd. 1 ft . 4 in . 29 . Feet are 3 rd. 4 yd.

25 . Minutes are 5 h . 40 min . 30 . Feet are 2 mi . 3 rd.

26 . P ounds are 2 T . 1 80 lb. 31 . Sq . ft. are 4 A. 9 sq . yd.

27 . Ounces are 6 1b. 9 oz. Av . 32 . Cd. ft. are 2 _cd. 5 cd. ft.28 . P ence are 5 68 . 33 . Minutes are 4°

34 . What is the value of 2 l b. 8 oz. of gold at 8 20 an

ounce35 . At 1 1 15a quart for nuts and 4515 a peek for appl es,

find the val ue of 44 bushel s of nuts and 1 2 bushel s of

appl es.

36 . At 7$5per quart for milk, and 30¢ per pint for cream,

how much w il l a deal er receive for 24 gal . 2 qt. of milk and

3 qt. 1 pt. of cream37 . What w il l it cost to fence both sides of a road 36 rd.

6 ft . l ong, at 2815 a foot38. The tropic of Cancer is 23

°30'north of the equator.

What is the di stance in geographical mil es What is thedistance in common mil es

286 . I l lustrative Examp les. ( 1 ) Change 4 of a rod toyards and feet. (2 ) Change of a rod to yards and feet .

WRITTEN WORK .

( 1 )

4 rd. = § of 1 4l yd = 3§ yd

i . = -Of 3 ft .

— 2 ft

Ans. 3 yd. 2 ft

(I) Since 54Yd . z 1 rd ., 3 Of a rd . 4 Of 1 21

yd. or 34yd. Since 3 ft . 1 yd . , 40 1 a yd. 3.

of 3 ft. 2 ft. Ans . 3 yd. 2 ft.

(2 ) rd . of 54yd. yd.

yd. of 3 ft. ft. Ans. 3 yd. ft.

(2 )rd.

£ 1733

330

yd.

3

ft.Ans. 3yd. ft.

1 68 COMP OUND DENOMINATE N UMBERS.

287 . From the foregoing il lustration may be derivéd the

fol l owingRul e .

To change a fraction of one denomination to integers ofl ower denominations : Change the fraction, as far as possible,to an integer of the next lower denomination . If a fraction

occurs in the result, proceed with it as with the first fraction,

and so continue as far as required.

Change39 . 4 rd. to ft. 43 . lb. Troy to pwt.40 . 4A. to sq . ft. 44 . 0 39

° to41 . 4mi . to rd. 45 . y . to days.

42 . 4 oz. to gr. 46 . 4mi . to in .

47 . At 20¢ a foot, what is the cost of 3; of an acre of land

Change to units of l ower denominations

48 . 4 T . 53 .

471mi . gal .

49 . 54 . g h. sq . rd.

50 .

115cu. yd. 55 . 4 of 4A. rd.

51 . 4. 56 . bu. 5 lb. Avoir.

52 . 5.624 . 57 .

494l b. Troy .

288 . To change a simp l e denominate number to a com

pound number of higher denominations .

I l lustrative Examp le . Change 424 feet to rods, yards, etc.

WRITTEN WORK .

54) 1 4 1 yd. 1 ft. rem.

2

25 rd. 4 yd. rem.

Ans. 25 rd. 34yd. 1 ft . ,

or 25 rd. 3 yd. 2 ft. 6 in .

Since there are 3 ft. in 1 yd. ,in

424 ft. there are 424 3 1 4 1 yd.

and 1 ft. remaining. Since there are

54yd. in 1 rd. , in 1 41 yd. there are

1 4 1 121 25 rd . and 34yd. remain

ing ; 4yd . _ 1 ft . 6 in . l i t. 1 ft.

2 ft. Ans. 25 rd. 3 yd. 2 ft. 6 in.

1 70 COMP OUND DEN OMINATE N UMBERS.

29 2 . From the preceding il lustration may be derived thefol l owing

Rul e .

1 . To change a compound number to a fraction of a.

higher denomination : Change the given compound number

to its lowest denomination, and make it the numerator of a

fracti on ; a l so change to the same denomination a unit of the

required denomination and make it the denominator of the

fraction .

2 . To change a compound number to a decimal of a

higher denomination : P roceed as above and then change

the common fractio n to a decima l .

7 7 . 1 pk. 2 qt . is what part of a bushel ?78 . 4 yd. 0 ft . 9 in . is what part of a rod ?

75 sq . rd. 904 sq . ft. is what part of an acreChange 6 oz. 5 pwt. to the decimal of a pound.

Change 87 rd. 1 0 ft. to the decimal of a mil e.

Change 4s . 6d. to the decimal of a. 53.

Change 1 1 8 . l 0d. to the decimal of a £3.Find the cost of 3 pk. 2 qt. of meal at $ 0.60 a. bushel .

. What wi l l be the cost of 2 qt. 1 pt. of kerosene at

a gal l on ?

86 . How much wil l a man earn in 5 days 7 hours at therate of $51 5 per week of 6 days of 1 0 hours each87 . At 8 60 an acre, what must be paid for 2 A. 1 1 34 sq.

rd. of l and88 . At 8 200 a lb.

, what is the value of 1 0 oz. 1 0 pwt. 1 0

gr. of gold89 . At per what is the value of £ 9 1 2s. 6d.

Engl ish money90 . Change 5 francs 50 centimes French money to

‘

U . 8 .

money. (Art.

9 1 . Change 1 0 marks 20 pfennigs German money to U . S.

money. (Art.

3

3

8

8

53

58

3

ADDITION AND SUBTRACTION . 1 71

ADDITION AND SU BTRACTIO N.

ass Il lustrative Examp les. (1 ) Add 2 mi . 1 72 rd. 6 ft.,

1 mi. 31 0 rd. 1 4 ft. , and 95 rd. 1 1 ft. (2 ) From 5 rd. 3 yd.

1 ft. take 1 rd. 4 yd. 2 ft.

WRITTEN WORK.

( 1 ) ADDITION. (2 ) SU BTRACTION.

2 mi . 1 72 rd. 6 ft. 5 rd. 3 yd. 1 ft.

1 31 0 1 4 1 4 2

95 1 1 3 rd . 3 4 yd. 2 ft .

4 m. 258 rd. 1 4 4 ft.21 in .

or,4 m. 258 rd. 1 4 ft. 6 in. 3 rd. 4 yd. 0 ft . 6 in .

( 1 ) The numbers are written so that units of the same order areexpressed in the same column, and the adding begins with the units

of the lowest denomination , as in simple addition .

The sum of the feet is 3 1 feet 1 rod 1 44 feet. Writing the 1 4}feet under the l ine in the place of feet, and adding 1 rod with the rodsof the given numbers, the sum is 578 rods, equal to 1 mil e 258 rods.

The sum of the miles is 4. Ans . 4 mi . 258 rd. 1 4 ft. 6 in .

(2) The subtracting begins with the units of the l owest denominx

tion, as in subtraction of simpl e numbers. As there is but 1 foot in

the minuend, 1 of the 3 yards is changed to 3 feet, making with the

1 foot, 4 feet ; 4 feet l ess 2 feet is 2 feet. As there are but 2 yardsleft in the minuend, 1 of the 5 rods is changed to 5} yards , makingwith the 2 yards, 74yards ; 74yards l ess 4 yards is 34yards . 4 rodsless 1 rod is 3 rods . Changing the 4 yard to feet we have 1 foot

6 inches, which added to 2 feet makes 3 feet 6 inches or 1 yard6 inches. Ans. 3 rd . 4 yd . 0 ft . 6 in.

Norm.— The operations upon compound numbers are similar to

those upon simple numbers, the difference being that in Operationsupon compound numbers varying sca l es are used instead of the uni

form scal e oi tens. Therefore, no special rul es are necessary for

addition , subtraction , mul tipl ication , and division .


92 . How much time is there in 2 h. 40min ., 3 h . 1 7 min .,

and 4 h. 45 min.

93. Add together 1 0°

4°9' and 1 5

°45'1 s" .

1 72 COMP O UND DEN OMINATE N UMBERS.

94 . Add together 5 1 7s . 1 1 d., 8 1 0s. 4d .

,1 88 . 6d.

, and

36 3 78 . 6d.

9 5 . Howmuchwood is there in 3 pil es containing, respectivel y, 5 cords 3 cord feet

,8 cords 4 cord feet

,and 1 0 cords

7 cord feet.96 . How much wheat is there in 1 bu. 3 pk. 7 qt.

, 2 bu.

1 pk. 5 qt.,1 bu. 6 qt.

,and 3 bu. 2 pk. 6 qt.

9 7 . How much l and is there in four pastures, the firstcontaining 7 A. 64 sq . rd.

, the second 1 5 A. 82 sq . rd. 260

sq . ft., the third 5 A. 1 7 sq . rd. 1 60 sq . ft.

, and the fourth1 9 A. 89 sq . rd. 36 sq . ft.

98 . Add 320 cu . yd . 20 cu . ft. 1 000 cu. in . , 29 cu . yd. 24

cu. ft. 968 cu. in .,500 cu. yd. 728 cu. in.

9 9 . Add 4 bu. and 5 qt .

NOTE.— Change 4bushel s to quarts .

1 00 . Add4mi . to 78 rd. 26 ft. 1 03 .

1 0 1 . Add 4 T . to 1 76 l b. 1 04 .

1 02 . Add “ y to 97 d. 1 05 .

417sq . mi .+44A.

1 06 . From a firkin of butter containing 44 lb. were sold1 8 l b. 1 0 oz. How much remained ?

1 0 7 . From a cask of Oi l containing 54 gal . were drawn 5gal . 2 qt. How much remained1 08 . A person who had 7 bu. 3 pk. 2 qt. of grapes sol d

3 pk. 4 qt. to one person and 2 pk. 5 qt . to another. Whatquantity ha dhe l eft 91 09 . How much remains of a journey of 54 mi . after 7

mi . 854 rd. have been travel ed1 1 0 . The l ongest day in Boston is 1 5 h . 1 6 min . What

is the l ength of the shortest n ight1 1 1 . Hav ing 1 0 sovereigns in my purse, I spent in Liver

poo l £ 1 1 08 . 6d.,

2 5s. 1 0d., and 1 1 8 . 6d. How much

money had I l eft1 1 2 . The Hoosac Tunnel is ft. l ong ; Mt. Cenistunnel is 74mi . l ong. What is the difference in feet

1 74 COMP OUND DENOMINATE N UMBERS.

297 . To find the difference of time betw een tw o date s.

Il lustrative Examp le. What is the time in years,months,and days fromMay 1 1 , 1 897, to August 7 , 1 899

SOLUTION .— From May 1 1 , 1 897 , to May 1 1 , 1 899 , is 2 years ; to

July 1 1 is 2 months more . FromJul y 1 1 to July 3 1 is 20 days, andto August 7 is 7 days more . Ans. 2 yr. 2 mo. 27 da.

298 . From the preceding il lustration may be derived thefol l owing

Rul e .

To find the number of years, months, and days betweentwo dates : F irstfind the number Of entire years between the

two dates,then the number Of ful l months remain ing, and

lastly, the rema in ing days, including the day Of the later date.

Nora — The method of finding difference of time by compound subtraction wil l be found ln the Suppl ement, Art . 1 5.

Oral Exercises .

299 . a . What time e l apsed between the birth of Benjamin Frankl in

,Jan . 1 7 , 1 706 , and that of GeorgeWashington,

Feb. 22,1 732

b. America was discovered by Columbus Oct. 1 2 , 1 492 ;the settl ement at Jamestown took pl ace May 1 3, 1 607 .

What time el apsed between these two eventsc. Lafayette was born Sept. 6 , 1 757 , and died May 1 9,

1 834 . What was his age at the time of his deathd. What was Lafayette’s age at the time of the Decl ara

tion of Independence, July 4, 1 776e. How ol d was Washington at the time of his death,

whi ch occurred Dec: 1 4, 1 799f How ol d was Lafayette when Washington died

g. What time el apsed between the l anding of the P il grims,Dec. 22 , 1 620, and the Decl aration of Independence

MULTIP LICATION AND DIVISION .

h. How many “years have el apsed since the Decl arationof Independencei . If a man was born March 5, 1 856, how old wil l he beif al ive at the cl ose of the nineteenth century

300 . I l lustrative Examp le. Find the exact number ofdays from J an . 7 , 1 896, to April 1 0, 1 896 .

SOLUTION .— There are 24 days remain ing in January , 29 days in

February ( leap year) , 31 days in March , and 1 0 days in Apri l .24 29 31 1 0 94 . Ans . 94 days .

Norm.— Any year is a l eap year when the number denoting the

year is divisibl e by 4 and not by 1 00, and when it is divisible by 400 .

(See Supplement, Art.

Find the exact number of days .

j . FromApril 1 0 to Jul y 5. l . From Oct. 4 to J an . 1 .

It. From Sept. 3 to Nov . 1 2 . m. From Dec. 2 to Feb. 1 .

n . From Dec. 6, 1 895, to March 8,1 896 .

0 . FromOct. 24, 1 895, to May 1 2 , 1 896 .

MU LTIPLICATION AND DIVISION.

301 . I l lustrative Examp les. (1 ) Mul tiply 5 oz . 7 pwt.

by 4. (2 ) Divide 5 A. 8 sq . rd. by 3.

WRITTEN WORK .

(1 ) 5 oz. 7 pwt. (2 ) 3 25 A. 8 sq . rd.

1 A. 1 094 sq . rd.

1 lb. 9 oz. 8 pwt.

( 1 ) 4 times 7 pwt. equal s 28 pwt . , which equals 1 oz. 8 pwt . Write8 pwt. , and keep the 1 oz. to add to the product of ounces. 4 times

5 oz. equals 20 oz . , which , with the 1 oz. kept, equal s 2 1 oz. , or 1 1h.

9 oz. Ans. 1 1h. 9 oz . 8 pwt.

(2) 5 acres divided by 3 gives 1 acre , with 2 acres remaining. We

write the 1 acre and change the 2 acres to square rods, which equal s320 sq . rd . This, with 8 sq . rd. , equal s 328 sq . rd . 328 sq . rd. divided

by 8 gives 1 09} sq . rd. Ans. 1 A. 1 094sq . rd.

1 76 COMP O UND DEN OMINATE N UMBERS.

302 . “ pl ea for W ritten W ork .

1 2 6 . How much wood is there in 5 l oads, each containing1 cd. 5 cd. ft .

1 2 7 . How much coal is there in 4 l oads, each containing1 ton 1 400 lb. ?

1 28 . If it takes 3 l engths of a string 36 ft . 9 in . l ong and

4 ft. 2 in . more to measure the depth of a l ot, what is itsdepth1 29 . Find the contents of 250 preserve jars if each jar

contains 1 pt. 3 gi .

1 30 . What quantity of peaches is brought into NewYorkin one car l oad of 450 baskets, if esfeh basket contains 1 pk.

6 qt .

1 31 . How much earth is removed in 1 5 l oads, each con

taining 1 cu . yd. 204 cu. ft.

1 32 . How much gl ass is there in 24 windows of 1 2 paneseach, if each pane contains 2 sq . ft . 8 sq . in .

1 33 . If 4 A. 8 sq . rd. is divided into 6 equal l ots, whatis the size of each1 34 . A farmer brought 5 bu . 3 pk. of corn to mil l . If

the mil l er took414 part as tol l , how much did he take

1 35 . When an express train travel s 1 00 mil es in 2 h.

30 min.

,what is its average time per mil e

1 36 . If a horse is fed 1 2 quarts of oats a day, how manydays w il l 6 bu . 4 pk. l ast himNorm.

— Change 6 bu . 4 pk. to quarts before dividing .

1 37 . How many barrel s, each hol ding 2 bu. 3 pk.

,wil l it

take to hold 1 00 bushel s of oysters, and how many peeksremain1 38 . A deal er has 3 hogsheads of mineral water, each con

taining 63 gal l ons, which he w ishes to put into bottl es holding 1 gal . 3 qt. 1 pt. each . Howmany bottl es can he fil l1 39 . Al l ow ing for a horseshoe 1 4 ounces of

.

iron,how

many horseshoes can be made from 60 pounds 7

1 78 COMP OUND DEN OMINATE N UMBER.

Standard Time .

304. By the system of standard time, four meridians 1 5°

apart are establ ished as central meridians ; reckoning fromGreenwich, these are the 75th, the 90th, the l 05th, and the

1 20th. The local time Of each Of these meridians is taken as

the standard time for a l l p laces within 74 degrees on either

side. Thus there are four bel ts, in each of which the timeby the cl ock is one hour ahead of that of the bel t next westof it, and one hour behind that of the bel t next east of it.These time bel ts are cal l ed the Eas tern, the Central , theMountain

,and the P acific bel ts.

a . Trace on yourmap the 75th meridian ; the 9othmeridian . Near what large city does each passb. In the same way trace the l o5th meridian .

c. The 1 20th meridian forms the western boundary ofwhat stated. Trace the div iding l ine between the Eastern and the

Central bel ts, and tel l what states l ie mostly in‘ the Eastern

bel t .e. What states l ie mostly in the Central bel t The

Mountain bel t The P acific be l tj I When it is 1 0 o

’cl ock by standard time in the Easternbel t, what is the time in the Central bel t The Mountainbel t The P acific bel t

Longitude and So lar Time .

305 . FromArt. 303 is derived the fol l owing

Rul e .

To find the difference of l ongitude between any two

pl aces when the difference of sol ar time is known : Mul tip ly

the difi'erence Of time between the two p laces , expressed in

hours,minutes, and seconds , by 1 5 . The product wi l l express

the difl'

erence Of longitude in degrees, minutes, and seconds .

LON GITUDE AND TLME. 1 79


306 . What is the difference in l ongitude between two

places, the difference in their solar time being1 47 . 4 h . 1 7 m. ? 1 49 . 6 h. 1 3 m. 1 0 s. ?

1 48 . 2 h. 9 m. ? 1 50 . 1 h. 5 m. 25 s. ?

In what l ongitude from Greenwich is a pl ace whose timecompared w ith that of Greenwich is1 51 . 3 hours earl ier 1 53 . 1 hour 1 2 minutes l ater1 52 . 5 minutes l ater 1 54 . 4 hours 8 minutes earl ier1 55 . A and B sai led together from San Francisco . A

kept his watch by San Francisco time, and B set his by thesun every day. After 1 0 days, A’

s watch was 4 hours 39minutes faster than B’

s. In what l ongitude were they then,the l ongitude of San Francisco being 1 22

°26'1 5" west

307 . FromArt . 303 is al so derived the fol l owing

Rul e .

To find the difference in time between any two pl aceswhen the difference in l ongitude is known : D ivide the dig

'

er

ence i n longitude, expressed in degrees, minutes, and seconds,

by 1 5 . The quotient wi l l express the dig'

erence Of solar timein hours , minutes, and seconds.


308 . U sing the l ongitude given in Articl e 296, find thedi fference in time between1 56 . Boston and P aris. 1 59 . London and San Francisco.

1 5 7 . P aris and Canton . 1 60 . Canton and San Francisco .

1 58 . New York and London . 1 6 1 . New York and Canton.

The l ongitude of Washington is 77°2'48" W. When it

is 1 2 o’cl ock by sol ar time in Washington, what is the time

1 62 . In P aris 1 64 . In Canton1 63 . In London 1 65 . In San Francisco

1 80 COMP O UN D DEN OMINATE N UMBERS.

What are the kinds of measure of extension Show how from l ongmeasure the measures of surface arise . Show how the units of cubic

measure arise .

What is the standard unit of l ength‘

2 Of weight Of l iquidmeasure Of dry measureDefine a circl e circumference arc diameter ; radius angle.

Name the diiferent kinds of angl es. How are anglesmeasured How

are compound numbers added, subtracted, mul tipl ied, and divided ?What is the diiference between these operations upon compound num

bers and upon simpl e numbers How is time between two dates

found What years are leap years How is difference of l atitude

found Of longitude

Describe Standard Time. In which of the four great time bel ts do

you l ive What is the difierence of longitude between the central

meridian of this bel t and the city or town in which you l ive When

it is 1 2 o’cl ock by the sun where you reside, what is the standard time

Rev iew Exercises .

of x in the fol l owing, supply its val ue1

y. 1 pk.

5bu .

b. 1 in . k. 1 qt .

_1_

x

0 1 S‘l ft : sq . yd.

l . 1 oz. Avoir.

d. 1 sq .

m. 1 oz. Troy

6 . 1 (31 1 . ft.

n . 1 pwt. Troy

cu . ft'0 . 1 gr. Troy

p . 1 cu. in .

h. 1 pt

q. 1 gr. Troy l b. Avoi r.

t 1 gi .Circumference diam. x x.

SECTION XII.

MENSU RATION OF SU RFACES AND SOLIDS .

Oral Exercises .

31 1 . a . How is the area of any rectangul ar surface found

(Art . Il lustrate by an example and diagram.

b. If the l ength is given in rods and the width in feet,what must first be done

c. If a board is 1 ft . 6 in. l ong and 4 in. wide, what is itsarea in square inchesd . When the area and one dimension of a rectangl e are

g iven, how can the other dimension be founde. What must be the l ength of a board 8 in. wide to

contain 1 00 sq . in .

f . What must be the width of a floor 1 2 ft. l ong to con

tain 1 32 sq . ft .g. Chinese matting is 36 in . wide. How many square

yards are there in a rol l 72 ft . l ongh. Brussel s carpeting is 4 yd. wide. How l ong must a

rol l be to contain 60 sq. yd.

3 1 2 . i . How is the volume of any rectangul ar sol idfound (Art. Show this by an exampl e.

j . When the contents and two dimensions of a rectangul arsol id are given, how do you find the other dimensionIt . What must be the depth of a cistern 4 ft. l ong and 3

ft. wide to contain 60 cu. ft . of waterI. What must be the height of a room20 ft. l ong and 1 5

ft .

'

wide to contain 3000 cu. ft. of air1 82

SQUARES AND OTHER RECTAN GLES. 1 83

SQ UARES AND OTHER RECTANG LES.

1 . How many bricks 8 in . l ong and 4 in . w ide, l aid flatwise, must be used to buil d a wal k 4 ft. wide and 1 00 ft . l ong

2 . What must be paid for a concrete wal k 5 rd. l ong and

5 ft. wide, at 90¢ per square yard3. What is the price of a building l ot in Brooklyn 25 ft .

wide and 80 ft . deep, at 8 a square foot What is theval ue of the land per front foot

4 . How much money wil l a man be worth who owns a

quarter of an acre of l and worth 20¢ a foot5 . My neighbor

’s garden is 1 00 ft. square ; mine con

tains 1 00 sq . ft . What is the difference in size6 . A and B have each a garden containing sq . ft.

A’s is 200 ft. by 50 ft .

, and B’s is 1 00 ft. square . C is to

fence both at 28¢ per running foot . How much shoul d A

pay How much shoul d B pay (Make diagrams of both

gardens . )7 . A buil di ng l ot contains 4 of an acre

,is rectangul ar,

and measures on the street 45 ft . What is its depth8 . A rectangul ar park contains acres and is

yd. l ong . What is its breadth9 . What is the area of the upper surface and sides of a

marbl e slab 3 ft. 34 in . by 2 ft. in. and in . thick1 0 . Amap drawn to a scal e of one inch to 34 mil es, is 5ft. 2 in . by 3 ft. 7 in . What area in square mil es does itrepresent1 1 . How many acres were covered by Machinery Hal l at

theWorl d’s Fair in Chicago, 1 893, the main building measuring on the floor 846 ft. by 492 ft. , and the annex 550 ft.

by 49 ft.1 2 . Find the cost for flooring the main buil ding above, at

3 1 2 per thousand sq . ft. for boards and 25¢ per hundredforwork, al lowing 1 tenth for waste of lumber.

1 84 SURFACES AND SOLIDS.

G OVERNMENT LAND8 .

3 1 4 . The U nited States publ ic l ands, before being offeredfor sal e, are divided by paral l el s and meridians into rectan

gul ar tracts, cal l ed townships, each being as nearly as prac ticabl e 6 mil es square, or 23040 acres in extent.

The l ines bounding a township extend due north and south, and

east and west, and a l ine on a paral lel of latitude is always establ ishedas a base. A l ine of townships extending north and south is cal l ed a

range . The ranges are designated by their numbers east or west ofthe principal meridian, and the townships in each range by their num

ber north or south of the base l ine .

SECTIONS OF ATOWNSHIP DIVISIONS OF ASECTION

Townships are subdiv ided into sections 1 mil e square,or

640 acres, and sections into hal f-sections, quarter-sections,hal f-quarter sections, and quarter-quarter sections or l ots.

Lots which for any reason are irregular in form are designated as

Lot 1 , 2 , 3 , 4 , etc., of a particul ar section . City and vil lage plots are

subdiv ided into blocks, and these again into smal ler l ots.

1 township 36 sq i . 23040 acres

1 section 1 640

1 hal f-section 4 320

l quarter-section 1 60

1 hal f-quarter section 80

1 quarter-quarter section 414

40

a . What must be paid for the NE . quarter of section 1 1of a South Dakota township at $53 per acre ?

1 86 SURFACES AND sou p s .

31 7 . Exampl ea for W ri tten W ork .

1 6 . Find the area of triangl es having the fol l owingdimensions . Give the answers in square feet.

Base 30 ft., height 1 5 ft.Base 1 24 rd.

,height 1 44 ft.

Base 1 2 yd. 2 ft.,height 5 yd 1 ft.

1 7 . What must be the height of a triangl e that contains2250 sq . ft.

, and whose base is 1 80 ft . in l ength1 8 . How many square feet are there in a triangul ar flower

bed whose base is 1 8 ft. and the height 8 ft. 6 in .

1 9 . Mary is to crochet a hal f square breakfast shawl ,each of its two equal sides being 1 4 yd. l ong . How manysquare feet wi l l she crochet20 . Find the surface of an octagonal spire, each of whose

eight faces is 1 94 ft. at the base and 51 gL ft. high.

CIRCLES.

31 8. A circl e may be considered as made up of triangl esthe sum of whose bases is thecircumference of the circl e and

Whose height is the radius .

Hence the area of a circl eequal s the product of the circum

ference by ha lf the radius .

What is the area of a circl e2 1 . Whose circumference is ft. and radius 6 ft.22 . Whose circumference is ft. and diameter 74 ft .

31 9 . In every circl e the circumference very nearly equal sthe diameter multip l ied by How

,then, when one

dimension of a circl e is given can the other be foundNora . For many purposes 34or instead of is accurate

enough . Work the examples in this article with the decimal Value.

RECTAN G ULAR sou p s . 1 87

What is the l ength of the circumference of a circl e23 . When the diameter is 2 ft. 4 ft. 3 in .

24 . When the radius is 5 in . 1 1 yd.

Nor-n . Draw diagrams to il lustrate the fol l owing examples

25 . What is the distance around a circul ar pond that is1 00 ft. across26 . What is the diameter of a circl e, the circumference

being 1 00 ft. 50 rd.

2 7 . What is the radius of a circl e whose circumference is500 ft ? 1 2 ft. 9 in . ?

Find the area of a circl e when28 . The diameter= 4 rd. 30 . The circumference=560 ft.29 . The radius 5 ft. 4 in. 31 . The circumference=480 ft.32 . Over how many square feet can a cow feed whentethered so that her head reaches 20 ft. from the stake 933 . The trunk of the largest tree now standing in Cal i

fornia has a circumference of 1 06 ft. What is the distancethrough and what is the area of a cross section34 . How many feet of surface has a circular mirror in

side a frame which has a breadth of 4 in ., and whose outside

edge is the circumference of a circl e whose diameter is4 ft. 5 in.

35 . How many square feet are there in the frame of theabove mirror

320 . RECTANG U LAR somos.

36 . How many cubic inches are there in a bl ock of marbl e1 ft. l ong, 8 in . wide, and 5 in . thick

, and what is its weightat of a pound to the cubic inch37 . What is the weight of a bl ock of granite 8 ft. l ong,

1 4 ft. wide, and 1 4 ft. thick, at 1 65 l b. to the cubic foot38 . What must be the l ength of a beam 1 6 in . by 22 in .

to contain 1 1 54 cu . ft.39 . How many cubic yards of earth must be removed in

digging a cel lar 20 ft. l ong, 1 8 ft. w ide , and 1 0 ft. deep ?

1 88 s URFACES AND sou ps

WOOD MEASURE.

32 1 . Wood is usual ly cut for marketing into 4 feet

sticks. Two l engths of such wood pil ed 4 ft. high and 4

w ide contain how many cubic feet1 28 cu. ft. is a cord used principal ly in measuring wood .

1 6 cu. ft. , or 4 of a cord, is a. cord foot (cd.

The fol l owing il l ustration represents 1 cd. and 1 cd. ft.

1 CORD FOOT 1 com)

Exammes for Oral and W ri tten W ork .

322 . a . How many cords of seaweed are in a pil e ft.high, 4 ft. wide, and 1 6 ft. l ong 20 ft. l ong 40 ft . l ongb. At $ 6 a cord for wood, what is the cost of 4 cd. ft . 9

of 1 2 ed. ft. of 6 cd. ft.At $4 a cord for wood, find the cost of a l oad measuring0 . 4 ft. by 4 ft. by 4 ft. e. 8 ft . by 4 ft. by 1 ft.d. 4 ft . by 4 ft. by 2 ft. j ? 9 ft. by 4 ft. by 4 ft.40 . What is the cost at per cord of a l oad of wood

8 ft. by 4 ft. by 3 ft. 1 2 ft. by 6 ft. by 4 ft.4 1 . If wood is cut in l engths of 44 ft. and pil ed 4 ft.

high, how l ong must the pil e be to contain a cord42 . If wood cut 3 ft. 9 in . l ong was sol d for 4 ft. wood,

at a cord,what deduction should be made in the bil l

for what was sol d for 20 ed.

1 90 SURFACES AND sou p s.

LATHS, SHING LES. AND CLAPBOARDS.

324 . Lathe are 4 ft. l ong, 1 4 in . wide, and are usual l y laid

4 in. apart at the sides, and cl ose together at the ends.

60 . How many square yards wil l be l aid by a bundl e of1 00 l aths61 . How many bundl es, 1 00 laths each, wil l be required

to lath a ceil ing 1 6 ft. by 1 5 ft.

325 . Shingl es are 1 6 in. l ong and of various widths, but

in reckoning, each width of 4 in. is considered one shingl e ;a bundl e contains 250 shingl es . Shingl es are ordinarilylaid from 4 to 4 to the weather.

Norm. A shingle 1 6 in . long, l aid 4 to the weather is 54 in. to the

weather, and therefore covers 54 in . x 4 in. of surface. A shingl e

1 6 in. long laid 4 to the weather covers 4 in . x 4 in . of surface .

62 . When shingl es are l aid 4 in . to the weather, howmany shingl es wil l cover the two sides of a roof each 40 ft .

l ong and 25 ft. wide63 . How many square feet of surface wil l 1 000 shingl es

cover when l aid 4 to the weather How many square feetwil l they cover if l aid 4 to the weather64 . When 1 bundl e of shingl es is al l owed for 30 sq. ft.,

how many bundl es wil l be required for a roof 20 ft. l ongand 1 8 ft. wide, and What wil l they cost at a thousand

326 . Cl apboards are sometimes cut 4 ft. l ong and 6 in.

wide, and sold in bundl es of 25 each.

6 5. When cl apboards are l aid 34 inches to the weather,how many square feet wil l 6 bundl es cover ?66 . How many cl apboards laid 34 in . to the weather wil l

be required to cover 1 40 sq. ft.6 7 . How many wil l be required to cover the side of a

house 40 ft. l ong and 20 ft. high

CARP ETIN G . 1 91

PAPERING .

327 . Wal l papers are usual ly 4 yd. wide and sol d in rol l seach containing 8 yd. The paper is l aid up and down thewall s .

Nora 1 .— In reckoning the amount required to cover the wal l s,

without al l owing for windows and doors

Find the distance in yards around the room, and reckon 2 stripsfor every yard . Divide this number of strips by the number thatcan be made from a rol l . The quotient wil l be the number of rol l s

Nora 2 .— In estimMing the amount of paper necessary to cover a

certain area, a whol e rol l is al l owed for any part of a rol l requiredNora 3 . Bordering is sol d by the yard.

How many rol l s of paper wil l be required to paper aroom 1 8 ft. l ong by 1 64 ft. , 8 ft . from the baseboard to theceil ing, making no al l owance for windows and doors69 . How many rol l s woul d be required for the ceil ing of

the above room70 . How many rol l s wil l be required to paper a hal l 1 64

ft. by 6 ft. , 7 ft. between baseboard and ceil ing, deductingfor a door and a window, each 34 ft. wide

Measure and estimate the number of rol l s of paperthat woul d be required to paper one side of your schoolroom, deducting the width of windows and doors .

CARPET ING .

328. Il lustrative Examp le. How many yards of carpeting, l aid l engthwise of the floor

,wil l be required to carpet

a room 1 8 ft. by 1 4 ft .

,no al l owance being made for match

ing ; the width of the carpeting being 1 yd. How manyyards 4 of a yard wide wil l be required

Sonnrron . 1 ) A floor 1 8 ft. long wil l require 6 yd . of yard wide

carpeting to each breadth ; the room being 1 4 ft . wide, it wil l require4] breadths, practical ly 5 breadths . Hence, to cover the floor withyard wide carpeting wil l require 5 times 6 , or 30 yards .


(2 ) Carpeting being 4 of a yard wide, it wil l require as manybreadths to carpet the above floor, l aid lengthwise, as there are times

4yd. in 1 4 ft. , which is 64times practical ly it wi l l require 7 breadths .

Hence, to cover the floor with carpeting 4yd . wide w il l require 7 times6 , or 42 yards.

Norm1 .— Since 4 of a breadth is l ess than a hal f , the breadth can

be spl it, in which case 64breadths wil l suffice. There is usual ly some

waste in matching the patterns of carpets, which must be al lowed for.

Nora 2 .— In each of the fol l owing examples, draw a plan of the

floor showing how the carpeting should be laid down .

7 2 . Howmany yards of carpet 1 yd. w ide wil l be requiredto carpet the above room if it is l aid crosswise7 3 . How many yards of carpet 4 yd. wide, l aid cross

wise, would be required to carpet the room74 . Which way is it best to l ay a carpet 1 yd. wide on a

floor 2 1 ft. by 1 9 ft. so that no breadth may be spl it or

turned under,and how many yards wil l be required, no

al l owance being made for matching figures7 5. If the above carpet is l aid the otherway and a breadth

Spl it, how many yards wil l be required7 6 . If the floor in Exampl e 74 is carpeted with W il ton 4

of a yard wide, and l aid l engthwise, how many yards wil lbe required, one breadth being spl it7 7 . Measure the floor of your school room

, and estimatethe number of yards that woul d be required to carpet itw ith carpeting 1 yd. wide . Al so find the number of yardsrequired to carpet it with carpeting 4 of a yard wide.

STONE AND BRICKWALLS.

329 . Bricks of common size are 8 in . l ong, 4 in . wide, and2 in . thick .

In laying wal l s the mortar occupies about 4 of the space, so that 22bricks to a cubic foot is a fair estimate .

Stone work is estimated by the perch, which is a rod ( 1 64ft.) long,1 4 ft. wide , and 1 ft. thick, and which contains 244cu . ft . In ordinary

cal culations the perch is considered to be 25 cu. ft.


CAPACITY OF BINS, CISTERNS, ETC .

33 1 . In estimating roughl y the capacity and volume of

bins, cisterns, etc .,there are in common use certain units

of approximate value in capac ity, volume, and weight.

1 T . of coal (2000 lb. ) about 35 cu. ft.

1 bbl . (31 4gal .) about 44cu. ft.

1 bu. cu. in. ) about 1 4cu. ft.

1 bu . heaped measure about 1 4 cu. ft .

1 gal . of water (231 cu. in . ) about 13, cu. ft . 84 lb . weight.

1 pt . of water about 1 l b.

1 cu . ft. about 4 of a bushel about 74gal .1 cu. ft . of water 1 000 oz . 624 l b.

In measuring bulky fruits and vegetabl es, as appl es and potatoes,

the measures are heaped. In measuring smal l frui ts, grain, etc. , the

measures are evened, or stricken with a. straight edge.


332 . Roughl y estimate the fol l owing86 . How many bushel s of grain can be put in a bin 4 ft.

by 3 ft. by 5 ft. ?

87 . How many bushel s of appl es can be put in the

above bin88. How many tons of coal can be put in a bin 7 ft . by

5 ft. by 2 ft.89 . How many cubic feet are there in a cistern hol ding

1 000 gal .

90 . How many pounds do 1 0 cu. ft. of water weigh9 1 . What must be the depth of a box 1 5 in . by 20 in . , to

hol d 3 bu. 3 pk. of rye92 . A can of l inseed oil 1 6 in. by 1 4 in. by 9 in. containshow many gal lons9 3 . What is the weight of the above oil , it being as

heavy as water9 4 . A jar weighs 8 lb. when empty and 60 l b. when

fil l ed with water. How many gal l ons does it hol d 7

MISCELLANEOU S EXAMP LES. 1 95

95 . What wil l 6400 sheets of paper cost at a.

ream?

9 6 . How many rods of fencing are required to incl ose asquare l ot, each side of which is 2344 feet l ong

9 7 . How many furrows, each 20 inches wide, wil l bemade in pl owing l engthwise a lot of l and 8 rd. 1 0 ft . wide

98 . What is the width of a rectangular fiel d whoselength is 2 1 74 feet, and the area is square feet

99 . A quantity of sil ver weighed 4 lb. 1 0 oz. 3 pwt .

before refining, and 3 l b. 1 1 oz. 2 pwt . 9 gr. afterwards.What weight was l ost in the process1 00 . If I burn 1 20 pounds of coal in a day, and buy mycoal by the l ong ton at per ton, what is the costfor December and January1 01 . John Adams was born Oct. 30, 1 735, and died at the

age of 90 y. 8 mo . 4 d. What was the date of his death ?1 02 . Dating the commencement of the Civ il War in the

U nited States at April 1 2 , 1 86 1 , and its cl ose at May 26,1 865, how l ong did it continue1 03 . Make a bil l against Wi l l iam Rice, and receipt it, for

24 gross of penci l s at $ 24 a gross, 74 quires of paper at

a ream, and 300 envel opes at per thousand.

1 04 . How many gal l ons of water wil l be contained in a

tank 3 feet square, if the water is 5 ft. 4 in . deep

1 05 . Change 1 5 l b. 8 oz. Avoirdupois to Troy weight.1 06 . Add 4 of the month of February, 1 893, to 4 of the

days fromMarch 2 1 to June 1 9 .

1 07 . If a factory can make 1 200 yards of cl oth per hour,how many yards coul d be made by working 1 0 hours a dayfrom July 7th to January 4th, al l owing for 26 Sundays1m. How many perches of masonry (25 cu. ft. ) are there

in a wal l 1 00 ft. l ong, 1 0 ft. high, 1 4 ft. thick How manybricks

,22 to a cubic foot, wil l construct this wal l

1 96 S URFACES AND SOLIDS.

Here is the floor pl an of somerooms in a cottage . The roomsare 9 feet high.

Rooms. a , Study ; b, Sittingroom ; 0 , Hal l .

Doors. FD, the front door,5 ft . by 8 ft. ; D, D, 6 ft. by7 ft. ; (1 , d, 34 ft. by 7 ft.

Windows. W, W, 34 ft. by74 ft. The others, 3 ft . by6 ft .

Baseboards. Al l the baseboards 9 inches high.

”ft. x'

1 5f f.

Norm— P lastering , painting, and kal somining are usual ly com

puted by the square yard.

1 09 . Find the number of square feet in the baseboard ofthe study, passing under al l the windows.

1 1 0 . Find the number of square feet in the W indows anddoors of the study.

How many square feet are in the wal l s and ceiling of the study, deducting for baseboards, doors, and windows ?1 1 2 . .At 40¢ per sq . yd. find the cost of pl astering the study

wal l s, deducting hal f the area of the doors and windows.

1 1 3 . What is the cost of kal somining the ceil ing of thehal l and other rooms at 9¢ per square yard1 1 4 . How many square feet of gl azing in the above pl an,W,W hav ing each 4 panes 42 inches by 1 8 inches, and theothers hav ing each 4 panes 3 1 inches by 1 5 inches1 1 5 . How many yards of carpet, 1 yard wide, wil l be

required to carpet the study ? How many yards 4 of a

yard wide w il l be required, no al l owance being made formatching figures

1 98 SURFACES AND sou p s.

1 2 5 . At 50 cents each, find the cost of planting trees, 25feet apart in the boundary of the l ot, beginning at a corner1 26 . A pitched roof

,each side of which is 25 ft. wide and

32 ft. l ong, is to be covered with sl ates 1 6 in . by 1 2 in.

How many sl ates wil l it take, al l owing them to l ap one

hal f1 2 7 . What weight of sl ates wil l the above roof have to

support, the average thickness of the sl ate being in .

,and

the weight being times as heavy as water (Art.1 28 . How many cords of wood can be put into a shedwhose inside measure is 1 0 ft. by 1 4 ft. 8 in. and 9 ft.

high, a space of 6 ft. square from the floor upward be ingreserved

1 2 9 . From a pil e of wood 58 ft. l ong, 4 ft. hi gh, and

4 ft. wide, were sold at one time 2g cords, at anothercords. How much is the remainder worth at $ 5 a cord1 30 . A man purchased 75 cords of wood for $ 360 ; he

sold the fol l owing l ots, 1 0%cords, 1 5 cords, and 1 1 g cords,al l at 5 per cord. Howmuch did he gain on what he sol d ?1 31 . At 20 cents per cord for each sawing, what woul d be

the cost of sawing the remainder of the 75 cords, so thatevery stick should make four pieces1 32 . If the first Atlantic cabl e weighed 1 9 cwt. a mi l e,

what was the weight of the entire cabl e in l ong tons, thedistance fromVal entia to Newfoundl and being 2435 mil es1 33 . A farmer divided one hal f of hi s l and of 540 acres

1 6 square rods equal ly between his two daughters, and thebal ance, after setting off 1 6%acres, equal ly between his twosons. What was the share of a daughter of a son

1 34. What must I pay for a dozen sil ver spoons, eachweighing 2 oz. 9 pwt.

, at per ounce1 35 . How many bushel s of buckwheat wil l a bin containthat is 8 ft. l ong, 5 ft . wide, and 3 ft. deep (Art .

1 36 . To what height wi1 1 62 bu. 2 pk. of grain fil l a bin1 0 ft. l ong and 8 ft. wide

MISCELLANEOU S EXAMP LES. 1 99

1 37 . In grading my lot which is 1 00 ft. square, I used thematerial dug from a cel lar 60 ft. by 24 ft .

, and 8 ft . deep.

What was the depth of the grading of the l ot1 38 . How many square feet does the surface of a bl ock

contain which is 5 ft. 6 in . l ong, 3 ft. wide, and 2 ft. 4 in .

thick1 39 . A yacht race took pl ace in Engl ish waters Aug. 4

,

1 894, which resul ted as fol l ows. Yachts started at 1 0 o ’cl ock30 min . 5 sec . The Vigil ant compl eted the course, whichwas 48 mil es, at 2 o’cl ock 36 min . 1 5 sec.

, the Britannia at

2 o’cl ock 41 min . 43 sec . By how much did the Vigil antbeat her rival ? What was her average rate per hour1 40 . How much carpeting i yd. wide wil l cover the top

and sides of a bl ock 3 ft. l ong, 9 in . wide, and 6 in . high

No a l l owance for waste .

1 4 1 . At 1 5¢ per pound, what is the cost of l ead , 5 1h.

to a square foot, to l ine a tank 6 ft . by 4 ft.

, and 5 ft. deep ?1 42 . The earth turns upon its axis once in 24 hours. How

many degrees does it turn in 1 6%hours1 43 . It is said that bricks were put into the

Art P alace at the Worl d’s Fair in 1 893 . How many cubic

feet of brick were there (See Art.

1 44 . How many common bricks 22 per cubic foot wil l ittake to build a wal l 38 ft. l ong, 8 ft. high, and 1 7} ft. thick,al l owing 1 fifth of the wal l for mortar1 4 5 . The circul ar dome of the Horticul tural buil ding at

the Worl d’s Fair was 1 80 ft. in diameter. What was itscircumference ? (Art.

1 46 . What must be the diameter in yards of a trottingpark that the circumference may be 1 mil e around

1 4 7 . What wil l be the area of the above park in acres1 48 . At $ 36 per thousand, what is the cost of chestnut

plank 2 71g in . thick to cover a circul ar cesspool , the cover tobe 9 ft. in diameter, and 1

1;of the material purchased being

al l owed for waste

200 SURFACES AND SOLIDS.

1 49 . How many paving stones 8 in. square wil l be re

quired to pave 1 0 rd. of a street that is 40 ft. wide1 50 . Estimate the cost of feeding a. yoke of oxen during

the winter of 1 895 and 1 896, hay being per ton, one

ox weighing 1 772 lh. ,the other 1 431 lb.

, al l owing 316of

their weight in hay per day1 51 . \Vhat was gained on ft . of lumber by buying

at per thousand and sel l ing at per foot1 52 . How many feet of boards wil l be required to incl ose

and separate, with a tight board fence 4 ft. high, 3 rec

tangul ar l ots lying side by side, each 5 rd. wide in frontand 1 0 rd. deep, the fronts being in the same straight l ine1 53 . From a barrel of 42 gal l ons of kerosene costing 8¢

per gal l on, there were sold at 1 0¢ per gal l on 1 5 gal . 2 qt.

,

1 8 gal . 3 qt . 1 pt. , and 4 gal . 3 qt. 1 pt. The rest wastedby evaporation and l eakage . What was the gain or l oss onthe barrel1 54 . How much wil l it cost to tin a roof 9 ft. by 1 7 ft. ,

al l owing 1 2 in . al l around additional for gutters at $ 8 persquare of 1 00 ft.1 55 . A wal k al ong a bl ock of 50 house l ots averaging 4

rd. in front,was l aid with 550 flagstones 4 ft. square .

What was the width of the wal k1 56 . In a school room 28 ft. 6 in . by 30 ft. 8 in. and 1 1 ft.

1 0 in . high, there are 48 pupil s. How many cubic feet ofair are al l owed for each pupi l ?

A school room shoul d have from4 to 4 as much l ightingsurface as floor surfac e . Cal culate these surfaces in the

fol l owing rooms stil l in existence, and find the fractionalpart which the l ighting surface is of the floor surface.

1 57 . First room,40 ft. x 28 ft. ; 6 windows, each 1 2

l ights 1 0 in . by 1 2 in . of gl ass.

1 58 . Second room,24 ft . x 28 ft. ; 4 windows, each 1 2

l ights 1 0 in . by 1 5 in . of gl ass.

SECTION XIII .

P ERCENTAG E.

334 . I l lustrative Examp les. (1 ) A nursery man set out

young trees in his nursery, but l ost 1 5 out of every 1 00 ofthem from a drought. How many hundredths of his treesdid he l ose

(2 ) Another nurseryman l ost 9 out of every 50 of histrees. What part of his trees did he l ose How manyhundredths

(3) Another l ost 4 trees out of every 25. How manyhundredths of his trees did he l ose Compare his l osswith that of each of the others.

In these questions the number 1 00 trees is used as a standard of

comparison . It is usual in such questions to empl oy the phrase per

cent, or the sign in place of the words hundredths ,”

out of

every etc . Thus we shoul d say that the first nurseryman l ost1 5 per cent ( 1 5 of his trees ; the second lost 1 8 per cent ( 1 8and the third 1 6 per cent ( 1 6

Non e. — The phrase“per cent

" is from the Latin per (by) andcentum (hundred) .

Oral Exercises .

335 . a . Aman having $ 1 00 l ost 1 0. What per centof his money did he l oseb. An uncl e promised to give his nephew 5 for every

$ 25 the nephew woul d save out of his earnings . Howmuch woul d be given for $ 1 00 saved The money givenwoul d be equal to what per cent of the money saved

202

ORAL EXERCISES. 203

c. A farmer had 200 sheep, but sol d 28 of them. What

part of his sheep did he sel l How many hundredthsWhat per centd. School was in session 200 days l ast year, and Arnel ia

was present 1 80 days. What part of the time was she

present What per cent of the timee. In a certain school of 400 pupil s, 1 00 pupil s are under

1 0 years of age. What part are under 1 0 years of ageWhat per centf . In a grove of trees, 3 trees in every 5 are oak. Howmany trees in 1 00 are oak What per cent are oak

g. The rest of the trees are pine. What per cent of thetrees are pineh. Jack is catching fish, but l oses 1 out of every 4 thatbite . How many does he l ose out of 1 00 that bite What

per cent does he l ose

336 . The base and rate being giv en to find the percentage.

Il lustrative Examp le. A nurseryman set out 2520 youngtrees

,but l ost 1 5 per cent of them from drought. How '

many trees did he l ose

WRITTEN WORK .

2520Losing 1 5 per cent is the same as losing

0-1 5of his trees. of 2520 is 378.

1 2600 Ans. 378 trees .

252

337 . P ercentage is that part of a number which is foundby taking a number of hundredths of the number.

In the example, 378 trees, being 1 5 hundredths of 2520 trees, is a

percentage of 2520 trees.

338 . The base of the percentage is the whol e number a

part of which is taken as a percentage.

In the exampl e, 2520 trees is the base.

204 P ERCEN TAGE.

339 . The number of hundredths the percentage is of thebase is the rate per cent , or, briefly, the per cent .

Thus,in the exampl e, the rate p er cent is 1 5, and the per

cent of trees l ost is 1 5.

340 . The word rate used al one means the fraction bywhich the base is mul tipl ied to produce the percentage .

Thus, in the exampl e , the rate is 1 5 per cent, or 1 5or or

11656 ,

or

NOTE .— The distinction between rate and rate per cent is impor

tant ; and neglect of it is often a source of confusion. The rate maybe expressed in various forms, as 25 per cent, 25 1

2656 , 2

56 , 4,

%3, etc. But the rate per cent is onl y the numerator of the

fraction which, with the denominator 1 00 , woul d express the rate.

Thus , the rate being 12055 , the rate per cent is 25.

34 1 . From the foregoing exampl es is derived the fol

l owingRul e .

To find the percentage : Mul tip ly the base by the

342 . The rul e is thus expressed as a formula

P ercentage Base x Rate.

Oral Exercises .

343 . a . What is 4%of $ 700 ? 5%of $ 600 ?b. What is 2%of 900 bushel s 1 2%of 200 days ?0 . Find 1%of $ 2000 ; of $ 4000 ; of $ 7000.

d. Find 2%of $ 1 000 ; of $ 3000 ; of $ 5000.

e. Find 3%of $ 1 000 ; of $ 2000 ; of $ 5000.

f . Find 4%of $ 5000 ; of $ 7000 ; of $ 9000 .

g. Find 5%of $ 4000 ; of $ 6000 ; of $ 8000.

h. Find 6%of $ 9000 ; of 851 2000 ; of $ 20000.

i . Find 8%of $ 7000 ; of $ 9000 ; of $ 1 2000 .

j . What is 1%of $ 2 1 3 ? of $ 340 ? of $ 768 ?

206 P ERCEN TAGE.

v. What per cent of a number is 4 of the number 4

i ? 4 ? 116 ? 1

If? 1

15? 2

16?

215?

w. What per cent of a number is i of the number g

f 7 f ? %7 136 7 1

157 i t ?

x. What per cent of a number is 15,of the number

i t ? 216 ? i t ? r

at? it ?


347 . Il lustrative Examp le. Apiece of l and was sol d forand 27 of the price was paid in cash. How

much was paid in cash

WRITTEN WORK.

$563-84 The base, is mul tipl ied by the rate ,

to produce the percentage. The answer3946 88 is to be expressed to the nearest cent .

1 1 276 8Ans.

68

348 . What is

1 . 35 of 4 . 95 of 1 85 days2 . 57 of 9856 5 . 88 of 225 gal l ons3 . 68 of 648 tons 6 . 78 70. of 1 250 men ?7 . The annual rent of a house is fixed at 8 of its cost.

If the house cost 4500,how much is the rent

8 . In an army corps of 9250 men, 82 were reportedfit for duty. How many men were reported fit for duty

9 . A deal er purchased 2480 tons of coal , but sold 25of it the same day. How many tons did he sel l ?1 0 . A merchant bought flour at per barrel , and

sold it so as to gain 1 24 How much money did he gainon one barrel ?1 1 . Find 37} of 1 3 . Find 75 of 432 .

1 3 . Find 62; of it800 . 1 4 . Find 1 6§ of 6000.


1 5 . A merchant sol d goods to the amount of 5286 toa customer who afterwards became bankrupt, and paid onl y42 How much did the merchant get for his goods1 6 . From the amount of a bil l of goods 5 is deducted

for cash payment . How much is deducted if the bil lamounts to1 7 . Add together 1 0 of 20 of 30

of and 40 of

1 8 . Whi ch is the greater number, and how much,of 248 1

,or 60 of 2760

1 9 . Find the difierence between 874 of and

90 of

20 . The firm of Hol brook and Stevens, deal ers in l eather,being embarrassed in business, made an assignment, and

the assignee in making settl ement of claims against thefirm was abl e to pay 67 How much did one creditorreceive whose claim was $52 1 . Another creditor

,whose cl aim was

2 2 . Another creditor, whose cl aim was 8000

2 3 . A trad er bought a l ot of standing grass, and agreed

to pay for cutting it and making it into hay 30 of whathe should get for the hay when sold in the market. He

sol d 1 2 tons at 6 tons at and 1 5 tons at

per ton . How much did he pay the haymakers24 . A store which cost 6500 is rented for a year at

1 24 of its cost, through an agent who receives 5%of the

rental for services. How much does the agent receive25 . A deal er in grain bought bushel s of wheat

at He sol d 50 of it at 78 20 of it at 7 1 and

the rest of it at 65 11 per bushel . Did he gain or l ose on

the whol e, and how much

26 . The si l ver ore from a certain mine is assayed and

found to yiel d 8 of pure sil ver. How many pounds doesa ton (2240 pounds) of this ore yiel d What is the val ueof this yiel d at 634 cents an ounce Troy (Art. 267

208 PERCEN TAGE.

349 . To change a fraction to a per cent.

I l lustrative Example. What per cent of a number isof it

WRITTEN WORK.

( 1 )nagg

owzsng

( 1 ) gr, changed to hundredths c.stm.

1 6 0The resul t, is accurate , but not in

2 convenient form.

1 4 45 (2 ) A more convenient form, where only

1 75 an approximate resul t is wanted, is found by2

continuing the division one, two, three or

more places, according to the degree of approximation required .

57 8 Always give the answer to the nearest tenth,1 6 20 the nearest hundredth, or the nearest thou

1 4 45 sandth , as may be required. Thus, the answer

1 750to this exampl e, expressed to the nearest tenth,

1 734is expressed to the nearest hundredth,

1 600and to the nearest thousandth,

1 445

1 55

350 . Change the fol l owing fractions to per cents, expressing the resul ts to the nearest tenth of one per cent

2 7 . £357

. 29 . fi g. 31 . gag . 33 . m.

as1932?

so.

39 32 .

325 . 34 .

Change the fol l owing fractions to per cents, expressingthe resul ts to the nearest hundredth of one per cent :

35 . ggg. 39 . 41 . ti f36 . 38 . £325 . 42. gig.

Change the fol l owing fractions to per cents, expressingresul ts to the nearest thousandth of one per cent

44 . fig. 45 . m. as m.

21 0 P ERCEN TAGE.

( l ) The money A had at first is the base, and is 1 00%of itself .

Adding the gain, equal to 24 of the base, we have the amount equalto 1 24%of the base . (2 ) The money B had at first is the base , and

is 1 00%of itsel f . Subtracting from this the l oss, equal to 1 6%of

the base , we have the remainder equal to 84 of the base.

352 . From these exampl es are derived the fol l owingI

Rul es .

1 . To find the amount : Add the percentage to the base ;

or, Mul tip ly the base by 1 p l us the rate.

2 . To find the remainder : Subtract the percentage fromthe base ; or

, Multip ly the base by 1 minus the rate.

353 . These rul es are thus expressed as formulas

1 . Amount Base P ercentage Base x (1 Rate) .

2 . Remainder Base P ercentage Base x (1 Rate) .


48 . A tree feet high l ast year has grown 32%in

height this year. What is its height now 9

49 . What wil l be its height next year if it again grows32%50 . A merchant investing in business, gains in

three years 72%of his capital . How much has he at the

end of that time51 . A l ent B 850 on condition that B should repay the

money at the end of a year,together with a sum of money

equal to 6%of the l oan for the use of it. What amountof money should B pay to A at the end of the year52 . What is 1 2870 of53 . What is 1 35%of head of cattl e54 . If a piece of cl oth 53 yards l ong shrinks 1 370 when

wet, what is its length when wet

ORAL EXERCISES. 21 1

55 . How much remains to be paid on a bil l of goodsamounting to when 37 of it has been paid56 . What is l eft of an estate worth when 68%of it has been squandered

57 . Agrocer has charged on his books againstcustomers, but expects to l ose 1 2%of this in bad debts.

How much does he expect to col l ect58 . From a. cargo of 500 casks of mol asses the l eakage

is al l owed to be 7 The casks contain on the average 424gal l ons each. What is the net amount of molasses afterleakage has been deducted59 . A distance of 1 4 mil es is to be walked in 2 hours.

How many feet remain when 65%of the whol e distance hasbeen walked ? How many minutes remain when 65%of

the whol e time has been spent60 . A ranchman has sheep upon his ranch.

Supposing the rate of increase to be 60%each year, how

many wil l he have at the end of one year of two years61 . The rate of increase in the popul ation of a. certaincity being each year, what wi l l be the total popul ation a year hence if it is now What wil l it bethree years hence ?

Oral Exercises .

355 . The percentage and base being giv en, to find the rate .

Illustrative Examp le. What per cent of 1 80 is 90

SOLUTION .— 90 is l of 1 80 ; and 4 is 50

a . What per cent of 40 gal l ons is 1 0 gal l onsb. What per cent of 1 gross is 4 dozenc. What per cent of time is l ost, if 6 days out of 20 are

l ostd. Amil kman buysmilk at 5 If and sel ls it at 7 {7 a quart.What per cent does he gain

21 2 P ERCENTAGE.

e. Fred had 90 chickens, but during a cold rain storm 1 8of them died. What per cent of his chickens did he l osej : A young tree 1 5 feet high last year grew this year 9feet. By what per cent did its height increase

g. What per cent of 50 is 40 Of 40 is 50

356 . I l lustrative Examples. (1 ) A school boy was present 361 hal f-days during the year, and school kept 380 hal fdays . What was the boy’s per cent of attendance

(2 ) In the same school there was a girl who attended 375hal f-days . What was her per cent of attendance

WRITTEN WORK.

(1 ) (2)342 0 342 0

1 9 00 33 00

1 9 00 30 40

2 600

( 1 ) The boy was present i f} and the 2 280

girl “3of the time. Changing these frac 3200tions to hundredths (Art. we find

3040that the boy was present or 95 of

1 60the time. Ans. 95

(2 ) The girl’s per cent of attendance

was or according to the degree of approximation withwhich we wish to express it. Ans. or

357 . From these exampl es is derived the fol l owing rul e.

To find the rate per cent : Divide the percentage by the

base, carrying the division to hundredths . Or,

To find the rate : D ivide the percentage by the base.

358 . The rul e is thus expressed as a formul a

Norm. This rul e may be establ ished in another way, thus : The

percentage is the product of the base and the rate . Therefore, if

the percentage be divided by the base (one of its two factors) , thequotient wil l be the rate (the other factor) .

21 4 P ERCENTAGE.

74 . If 35 gal l ons of spirit containing 7870 pure al cohol ismixed wi th 48 gal l ons of spirit containing 83%pure al cohol ,what per cent of pure al cohol wil l the mixture contain75—88. In the fol l owing tabl e are given the popul ation in

1 890 and the popul ation in 1 880 of the ten l argest cities inthe U nited States and of a few others that increased rapidlybetween the same years. Find the per cent of increase (tothe nearest tenth of one per cent) in the popul ation ofeach city.

Crrms . P O P U LATION 1 890 . POP ULATION 1 880 .

New York,N .Y.

Chicago, Il l .

P hil adelphia, P a .

Brooklyn,'

N .Y.

St. Louis, Mo.

Boston, Mass .

Bal timore, Md .

San Francisco, Cal .

Cincinnati, Ohio

Cl eveland, Ohio

Minneapol is, Minn .

Omaha , Neb .

St. P aul , Minn .

Denver, Colo

360 . The percentage and rate being giv en, to find the base .

a . Agrocer sold 50 barrel s of flour, which was 25%ofal l he had. How many barrel s of flour had he

Sow rrox .— If 50 barrels was 25%of al l he had, 1 was 2 barrels,

and 1 00 was 200 barrels. Ans . 200 barrel s.

WRITTEN EXAMP LES. 21 5

b. In an orchard there are 60 cherry trees. How manytrees are there in the orchard if the cherry trees are 20%of the whol e numberc. If 5%of my money is $ 30, how much have Id. $ 40 is .8%of what sum 24 is 6%of what sume. Of how much time is 48 days 1 2%j : 35 is 5%of what number

g. If 360 is 20%more than the price of a piano, whatis the price of the pianoSOLUTION .

-If $ 360 is more than the price of the piano , it

must be 1 20%of that price ; 1 %of the price is The of $ 360, or 8 3

and 1 00 or the whol e price, is a300 .

h. 240 is 20%more than what sum of moneyi . 250 yards is 25%more than what distancej . If $ 360 is 1 0%l ess than the rent of a house, what is

the rent of the house

SOLUTION . If $360 is 1 0 l ess than the rent, itmust be 90 of the

rent ; 1 of the rent is 515 of $360, or $4 ; and 1 00 or the whol e

rent, is 8 400 .

k. $ 240 is 2070 l ess than what sum of money

361 . I l lustrative Example. Aman sold 208 acres of l and,which was 3270 of al l he owned. How much did he own

WRITTEN WORK .

208 650 . Ans. 650 acres.

Since 208 is a percentage , produced by mul tiplying the base by therate , (Art . the base can be found by dividing 208 by

362 . Hence the fol l owing

To find the base : Divide the percentage by the rate.

363. The rul e is thus expressed as a formul a

Base P ercentage Rate.

21 6 P ERCEN TAGE.

89 . 81 5 pounds is 5%of what weight90 . is 5%of how much money9 1 . If 1 5%of the cost of a factory is 7500, what is the

whol e cost9 2 . 54 inches is 370 of what distance93 . 428 hours is 8%of how much time94 . is 1 0570 of what sum of money95 . 8343 square feet is 9%of how many square yards96 . 1 2570 of what sum of money is equal to $1 0359 7 . 1

’s is 3%of what number

98 .

11,r is of what number

9 9 . Of what number is 1 728 cubic inches a percentage, ifthe rate is 36 701 00 . The school census of a certain city shows

chil dren of school age (5 to 1 5 years) ; if these chil drenform 20%of the whol e population, what is the population1 01 . Of what number is mil es a percentage, if the

rate is 1 3;1 02 . A horse was sol d for 258, which was an advance of

20%on the cost. How much did the horse cost1 03. A grocer sold sugar for 1 1 570 of its cost and made

4 of a cent a pound. How much did the sugar cost1 04. Ahouse is sol d for 4500, which is 1 0%l ess than it

cost to build. Howmuch did it cost to buil d1 05 . $ 540 is 1 0%l ess than the cost of a pair of horses.

How much did the horses cost1 06 . A stock of goods has been damaged by smoke and

water to the extent of and is now valued at 4500.

What was their original val ue1 07 . After a spendthrift has wasted 70%of his property

he has 9000 l eft. How much had he at first1 08. A boy is 1 4 years ol d, and hi s age is l ess thanthat of hi s brother. How old is the brother

P ERGEN TAGE.

PRO FIT AND LOSS.

365. a . By sel l ing goods at 25%above cost, howmuch v is

gained if the cost was 1 2 $ 32 $50

b. By sel l ing goods at 1 0%be l ow cost, how much is l ostif the cost was $ 20 ? 60 $ 85 ?

0 . At what price must tea which cost 40¢ a pound be soldto gain 33470 ?d. At what price must damaged flour which cost 6 a

barrel be sold to l osee. A quantity of wheat was bought for $ 750. What

would be gained if the market price should advanceWhat would be l ost if the market price should fal l1 3 What per cent is gained if sheep, costing 5 a head

are sold for $ 6 ? for $ 7 for $ 1 0 ?

9 . What per cent wil l be l ost if cheese, costing 1 5¢ a

pound, is sold for 1 2¢ a pound for 1 0¢ for 7

h. A lumber merchant paid $ 27 per thousand feet forboards at the mil l ; paid $ 5 per thousand for freight, insurance, and other expenses, and sol d the same at $ 40 per ,

thousand. What per cent did he gaini . What was the cost of cl oth on whi ch, if sold at

a yard, there wil l be a gain ofj . What was the cost of hay on which, if sol d at $ 1 2

a ton , there w il l be a l oss of 40%It . What was the cost of flour if a l oss of 24¢ on a bar

re l means a l oss ofl . What was the cost of a horse upon which, if sold, at

$ 20 above cost, there is a gain of 1 0%366 . The difference between the cost of anything and

the price for which it is sold is a profit or a l oss.

36 7 . A profit or a l oss is usual ly reckoned as a percent

age, the cost being taken for the base.

P ROFIT AND LOSS. 21 9

1 2 1 . A sheep farm in Texas which cost $ 7950 was soldso as to gain 1 7 What was received for it1 2 2 . A lot of timber was bought for $ 852 . For what

must it be sold to gain1 23 . A deal er bought 2000 pairs of boots for $7000 . For

what price a pair must he sel l them to gain1 2 4 . A deal er sold three car l oads of corn at 1 5%profit,

and gained $ 279 . How much did the corn cost .9

1 2 5 . By sel l ing a schooner for there was a gainof 1 5%on the cost. What was the cost1 26 . A tract of land, having increased in val ue

is now worth What was it worth before1 2 7 . A piano was sold at a l oss of 28570 for $ 286 .

How much did the piano cost‘

?

1 28 . A grocer pays 42 13 a pound for butter, and sel l s itfor What per cent does he gain1 2 9 . What is the per cent of gain if a lot of l ive stockcosting 5400 is sold so as to gain 1 269

1 30 . P aper bought at per ream is sol d at 25¢ perquire . What per cent is gained1 31 . Amerchant bought 320 barrel s of beef at per

barrel ; paid 65¢ per barrel for storage and insurance, and

sold the whol e for What per cent did he gain1 32 . A grain deal er bought wheat at barl ey at

corn at and cats at 46¢ per bushel . At what pricemust he sel l each kind of grain to gain1 33 . A hardware deal er bought 6 gross door knobs at

1 8 per gross, and sold hal f of them at 20¢ and hal f at 25?apiece. How much did he gain, and what per cent .

9

1 34 . TWO horses were sol d for 1 80 each . On one therewas a gain of and on the other a l oss of Was

there gain or l oss on the whol e transaction, and if any, howmuch, and what per cent

220 P ERCEN TAGE.

1 35 . If 2000 was paid for goods, hal f of which were sol dfor 645, and the rest for 585, what per cent was l ost1 36 . What was the original value of the sixteenth partof a ship whi ch, sel l ing at an advance of brings3456 How much did the whol e ship increase in value

1 37 .

'

A beef packer bought 1 50 head of cattl e at

per head, 40 head at and 80 head at He

so ld them al l for What per cent did he gain1 38 . Aman buys books at 40%bel ow the l ist price and

sel l s them.

at 1 5%bel ow. What per cent does he gain1 39 . What per cent is gained by buying coal by the long

ton (2240 lb. ) and sel l ing it by the short ton (2000at the same price per ton1 40 . A storekeeper marks his goods 25%above cost, and

then takes off 1 0%from his asking price in making sales.What per cent does he gain

COMMISSION.

369 . An agent is a person authorized to transact businessfor another person, who is cal l ed the principal .

Agents have different names according to the nature of the authority given them, or the kind of business they transact . A factor or

commission merchant buys and sel ls merchandise for others but

usual l y has temporary possession of the goods, which he is bound

to care for with reasonabl e prudence (by insuring, preserving fromdamage, etc .) as if they were his own . A broker is employed to

negotiate purchases and sal es of merchandise or other property,and usual ly has no possession of it ; but a stock-broker has possession

and care of the stocks and bonds which he buys and sel l s. Acol l ector

is an agent employed to col l ect debts .

370 . When factors, commission merchants, brokers, or

other agents are al l owed for their serv ices a percentageon the amount of the purchases, sal es, or col l ections, thispercentage is cal l ed commission.

222 P ERCEN TAGE.

1 47 . If 8240 is sent to an agent to cover the amount ofhis purchases and his commission thereupon of whatis the a mount of his purchases 9

NOTE.— The amount sent is 1 03%of the amount of the purchases.

1 48 . What per cent is an agent charging if, out of a

remittance of he reserves before makingthe purchases required of him1 49 . If out of a remittance of I keep my

commission of 3470 on the purchase, how much barl ey at

45¢ a bushel can I buy with the remainder

1 50 . A commission merchant, after sel l ing the goodsconsigned to him,

remitted the amount received l ess hiscommission of to the consignor. If the l atter received

what was the amount Of the sal es

1 51 . As commission merchant I sel l for my principal4500 thousand feet of lumber at per thousand toB . B . Fiel d CO .

, and take their notes on 1,2,and 3

months’ time, which notes I indorse, thus becoming respon

sibl e for their payment. For making the sal e my commission is and for guaranteeing payment my charge is3470 . What do I earn by the whol e transaction1 52 . A commission house, whose sol e business is to sel l

the entire product of certain cotton factories, and to guarantee payments for the same, sel l s during the year cottoncl oths to the amount of What does thehouse receive, the commission for sel l ing being and

that for guaranteeing payments more

COMMERCIAL DISCO U NT.

374 . Commercial or trade discount is a percentage deductedfrom the price of goods or from the amount of a bil l ofgoods for a cash payment, or to enabl e the purchaser to sel lat a profit without asking more than the market price, orfor any other reason . To induce earl ier settl ement, debtsand cl aims are sometimes reduced by a percentage.

COMMERCIAL DISCOUN T. 223


1 53 . What is the price of cl oth which,after a discount

of 5%has been made, costs per yard1 54 . A merchant bought goods on 60 days’ credit, but

offered cash for a 1 0% discount on the bil l of $ 2500 .

How much woul d he pay i f his offer shoul d be accepted ?1 55 . A shipper agrees to send his goods by a certain rail

road i f he can be al lowed a discount (cal l ed a rebate ) of20%upon his freight bil l s. How much w il l he save on a

bil l amounting to1 56 . What is the gross amount of a freight bil l which,

after a rebate of amounts to 9

1 57 . What is the net amount of a bil l of for

goods at l ist prices subject to trade discounts of 20%and

and 5%off for cashNOTE .

— This means that a discount of 20 is first made from the

face of the bil l ; then from the remainder a discount of 1 0 is made ;and final ly from the l ast remainder a discount of 5 is made . These

are success ive discounts .

1 58 . A bil l for hardware amounting in gross tois subject to successive discounts of and

What is the net amount1 59 . Abil l for books amounting to $ 523 is reduced to

what net amount by discounts of and 5%1 60 . A deal er bought 1 00 baby carriages at each

,

trade discounts 30%and and 5%off for cash . Whatwas the net amount of the bil l ?1 6 1 . A retail trader buys certain goods at discounts of

and and sel l s them at l ist prices. Whatper cent does he gain1 62 . Amerchant buys goods at discounts of 1 0%and

5% from l ist prices, and sel l s at 5%bel ow l ist prices.

What per cent does he gain1 63 . If the price of paper is per ream after dis

counts of 20%and what is the l ist price 9

224 P ERCEN TAGE.

1 64 . Which is better for the buyer, and how much, adirect discount of or successive discounts of 1 0 1 0

and 5%1 65 . What singl e discount is equival ent to the successive

discounts 2070 , and 5%1 66 . A deal er having bought goods at l ist prices l ess dis

counts of 20 1 0 and 5 sel l s themat 35 above net cost

prices. At what percent bel ow l ist prices does he sel l

INSU RANCE.

376 . A,the owner of a house, pays B a smal l percentage

of its value, B,on his part, contracting to make good to A

any l oss that may arise within a specified time from damageor destruction of the house by fire.

Such a contract is a contract of insurance. B is the

insurer, or under-writer, and A is the insured. Al so A’shouse is said to be insured.

377 . Insurance is a contract by whi ch the insurer under

takes to pay the insured for any l oss or damage resul tingfrom an uncertain event.NOTE .

— Among such events are fires in buildings, storms at sea,

sickness, personal injury , death , etc. Hence arise different kinds of

insurance, as fire insurance, marine insurance , accident insurance,

heal th insurance, l ife insurance and many other kinds .

378 . The written contract is cal l ed the pol icy . The

price paid for insurance is the premium.

379 . The premium is a percentage of which the amountinsured is the base .

NOTE 1 . When property is insured, the amount insured on the

property is made somewhat l ess than the ful l value of the property .

NOTE 2 . P ol icies are renewed yearl y, or once in five years, or

at other stated times, and the premium is paid in advance .

NOTE 3 . There may be a charge, usual l y of one dol l ar, for writ

ing the pol icy but no notice is taken of this in the fol lowing examples.

226 P ERCEN TAGE.

1 7 9 . A cargo of wool from Austral ia containing 1 000

ba l es, averaging 987} pounds to a bal e, and val ued at 2353 a

pound, is insured for the voyage at 470 on {a of its val ue.

What is the amount of the premium1 80 . January 1 , 1 892, a machinist took out a heal th

pol icy, paying 5 premium on the first day of eachmonth. April 5, 1 893, he fel l sick, and rece ived 1 5 a

week for 5 weeks . How much did he receive more thanhe had paid out in premiums1 81 . Aman insured his l ife for paying a pre

mium each year of per 1 000 . How much more orl ess than the total amount paid in premiums wil l his heirsreceive if he l ives to make 1 0 annual payments 9 20 annual

payments 40 annual payments1 82 . A ship was insured for f} of her value at a year.

After the annual premium, 1 275, had been paid fourtimes the ship was l ost. Find the l oss to the insurers andto the insured.

1 83 . Amarine insurance company has at riskon cotton afloat, at 24% premium. Learning that thevessel s carrying this cotton are overdue and that severe

storms at sea are reported, the agent of the company decidesto reinsure of the risk in other companies at 3 :1f%. Wil lthe company gain or l ose by the whol e business, and howmuch, supposing the cotton arrives safely1 84 . An insurance company takes a risk of on a

factory at and reinsures this risk with another com

pany at What is the amount of premium retainedby the first company1 85 . A merchant has pol icies of insurance on his stock

of goods as fol l ows : one for at $70 , one for

at one for at 470 , and one for 2500

at The goods cost him If a fire shouldtotal ly destroy the goods, what woul d be his l oss, incl udingthe sums paid for insurance

TAXES. 227

1 8 6 . The combined experience of many fire insurancecompan ies through a l ong series of years has shown, l et ussuppose, that one dwel l ing house in 800 is l ost by fire eachyear. What fractional part of the sum received forpremiumson dwel l ing house insurance must a company expect to payout for l osses, if the rate of insurance is 1 470 on five-year

pol icies What cl ear profit would the company make eachyear on at risk on dwel l ing houses

TAXES.

381 . The money necessary to meet the expenses of a.

town, city, county, or state government, is raised by l evyingtaxes on the persons and on the property of the citizens.

382 . A tax l ev ied on the person is cal l ed a pol l tax or a

capitation tax . It is a tax of so much per head (pol l ) .

A tax l evied on property (real and personal estate) is a

property tax ; and a tax on annual income , an income tax .

383 . Real estate is property in l and and buil dings, incl uding whatever is attached to and bel ongs w ith them.

It may be described general ly as immovabl e property.

Movabl e property, as money, ships, merchandise, horses,cattl e, carriages, etc.

,is cal l ed personal property . Real

estate is taxed in the county, town, or city where it is situated ; personal property is taxed where its owner l ives.

384 . The method of raising money by taxation so thateach citizen pays his fair share varies in different states,but may be described in general terms as fol l owsThe citizens assembled in town meeting, or representatives chosen

by the citizens assembl ed in city council , or in a county board,

determine by vote what sum of money shal l be raised to meet theexpenses of the local government. This sum is cal l ed the tax l evy .

Ofi cers cal led assessors estimate the value of each piece of realestate, recording their estimates against the owners

’names. The sum

of al l these estimates is cal led the valuation of real estate. The

228 P ERCENTAGE.

value of the personal property owned by each citizen is estimatedand recorded in l ike manner ; and the sum of these estimates is the

valuation of personal property . The sum of these two valuations

is the total val uation .

That part of a person ’s annual income which is subject to taxation

is usual l y included in the estimate of his personal property.

The assessors are now ready to apportion the tax levy. The amount

that is to be raised by the pol l taxes is found by counting the number

of citizens required by law to pay a pol l t ax, and mul tipl ying the

amount of one pol l tax by this number. This result being subtracted

from the tax l evy l eaves the amount that must be raised on property.

This is cal l ed the property tax l evy .

To provide a margin for abatements , uncol l ected taxes, etc . , the

property tax l evy'is increased in some states by adding a. smal l per

centage cal l ed overl ay .

The property tax l evy increased by the overlay and divided by the

total valuation gives the tax rate.

Final ly, each property owner’s tax is computed by mul tip lying theassessed va lue of his property by the tax rate .

In some states, the state tax is apportioned among the towns and

cities in proportion to their several valuations . The county tax is

apportioned in the same way among the towns and cities in the county.

Each town and city includes its share of the.

state tax and of the

county tax in its own tax l evy ; so that the citizen pays his state,

county, and town or city tax in one sum to the town or city col l ector

of taxes. In other states the county and not the town levies and col

lects state and county taxes .

Learn how the state, county, town or city, and school-district taxes

are levied and col l ected in your own state.

385 . I l lustrative Example. The fol l owing item is takenfrom a newspaper.

“The assessors of Oldtown have com

puted the val uation of property subject to taxation in the

town, and report : P ersonal property, real

estate, total , increase over l astyear, as town tax l evy, state tax, $ 6590 ;county tax, 8675 rate, 35 per thousand ; pol l s, 281 5 ;popul ation , Oldtown Hera ld.

How was the tax rate of this town computed 9

230 P ERCEN TAGE.

386 . Examp l es for W ri tten W ork .

1 87 . In Oldtown what is the amount of A. G. Brown’s taxbil l

,who owns a house and garden val ued at $ 5400 ; per

sonal property, 1 265 ; and who pays one pol l tax 91 88 . What is the amount of D . L. Goodrich’s tax bil l ,

who owns a farm and buildings valued at horses,cattl e, tool s, farm products, and other personal propertyval ued at and who pays one pol l tax 91 89 . What is the amount of M. J . Wood’s tax bil l , who

l ives in another town, but owns in Ol dtown 265 acres ofwoodland val ued at $ 25 per acre 91 90 . The rate of taxation in Norwood being per

thousand, what is the val uation of a farm on which a tax ofis paid ?

1 9 1 . Mr. Corey finds by his tax bil l that he has beenassessed on his real estate, and on his per

sonal property. The rate be ing per thousand, whatvaluation have the assessors pl aced on his real estate and

what on his personal property 91 9 2 . In Newton the val uation of real estate for the year

1 894 was and that of personal property wasP ol l s

, 7763, taxed at 2 . The city tax l evy wasstate tax, county tax,

Find the tax rate al l owing an overl ay not exceeding1 93 . The total val uation of real and personal property in

Boston, in 1 894, was The total tax l evy wasThe number of pol l s, taxed $ 2 each, was

Al l owing 4%overl ay, what was the rate 9 Takingthe actual rate, find the actual overl ay, and the

total warrant to the col l ector.

1 94 . Find the tax l evy, total val uation, number of pol l s,and pol l tax in the town or city where you l ive ; computethe tax rate ; and then find the tax bil l s for personssupposed to own given amounts of property.

CU STOMS OR D U TIES. 231

CUSTOMS O R DUTIES.

387 . The expenditures o f the U nited States are in partmet by taxes upon merchandise imported into the country .

These taxes are cal l ed customs or duties .

The custom house is a building belonging to the U nited States,

where al l business connected with incoming and outgoing vessels and

their cargoes is transacted with officers of the government . A port of

entry is a place where a custom house has been establ ished . It is

usual l y a seaport, but may be any place on or near the boundarywhere merchandise is brought into the country . The duties l evied on

imported merchand ise are paid at the custom house .

The rates of duty to be paid on the various kinds of merchandiseimported are fixed by an Act of Congress, usual ly referred to as The

Tariff Act ." To l earn the rate of duty on each kind and qual ityof merchandise , an importer must refer to the schedules contained in

the last Tariff Act.

"

388 . Duties are either specific or ad val orem; the formerbeing based on the quantity, the l atter on the val ue of the

merchandise imported .

A few of the specific duties l ev ied by the Tariff Act of

1 894 are as fol l ows : 1 ¢ per pound on w indow gl ass, 50

per cubic foot on rough marbl e, 3 ¢ per dozen on eggs, $ 2

per ton on hay, 1 415per yard on cotton cl oth,40 ¢ per ton

on coal , and per pound on wrought iron bars.

Some of the ad va lorem duties are 1 5%on straw, 20%on beef

, pork, and mutton ; 25%on watches, cl ocks, andwooden furniture ; 3570 on stained gl ass, porcel ain, crockery,paper, confectionery, and jewel ry ; 40%on carpets, wool encl oth, and cl othing ; 50%on silks, vel vets, l aces, cl oaks,ul sters, stockings, and shirts.

Sometimes the duties l evied on an articl e are both specific

and ad va lorem. Thus the duty on col l ars and cuffs is 30¢

per dozen and 30%ad val orem; on col ogne water

per gal l on and 50%ad val orem ; on cigars per poundand 2570 ad val orem ; on spirit varnish 25%and

232 P ERCEN TAGE.

per gal l on for the al cohol in it ; and on pearl buttons Ifper l ine 1

‘s inch) button measure per gross and 1 570 ad

val orem.

NOTE 1 .— In estimating specific duties, certain al lowances are

made : ( 1 ) for waste or impurities, cal led draft ; (2) for weight ofboxes, casks, etc. , cal led tare (3) forwaste of l iquids, cal led l eakage ;(4) for breaking of bottl es, etc. , cal l ed breakage. The weight of goods

before al lowances are made is the gross weight ; and the weight

after deducting al l owances is the net weight .

NOTE 2 .— The value of imported goods appears from the invoice ;

which is a l ist of the goods with statement of the quantities and of

the prices in the country fromwhich they were imported.

389 . Exammes for W ri tten W ork .

NOTE.— For rates of duty not given, see Articl e 388.

1 9 5 . How much is the duty on 200 tons (2240 lb. to a

ton) of wrought iron bars.

1 96 . How much is the duty, at per pound, on 1 5

boxes of raisins, 24 l b. to a box, tare 6) lb. to a box 9

1 9 7 . At 35%how much is the duty on 50 dozen watchcrystal s, invoiced at per dozen, breakage1 98 . What is the duty on 1 0 bbl . spirit varn ish

, 32

gal l ons to a barrel , invoiced at $ 6 per gal l on, al l owancefor l eakage being1 99 . Find the amount of duties on 1 000 yards Engl ishcarpet, invoiced at 4 shil l ings a yard in London, and im

ported at New Orl eans.

NOTE .— The pound sterl ing

200 . How much is the duty on 25 tons of hay and 1 0

tons of straw (the l atter val ued at 6 per ton) brought fromCanada into the U nited States 920 1 . Watches val ued at 400 francs each in Geneva woul d

be subject to how much duty in Boston 9NOTE.

— The franc202 . How much must be paid in New York for duties on

5 velvet cl oaks, costing 600 francs each in P aris 9

234 P ERCEN TAGE.

2 1 7 . A house worth it7500 is insured for g of its val ueat 470 . What is the premium 92 1 8. What amount of insurance may I procure on my

house for $ 25, if the rate is 370 9

2 1 9 . Whatmust be paid a col lector for col l ectingif he has been promised 24%9

2 20 . A co l l ector is paid commission . What amountdid he col l ect, if his commission was 9

2 2 1 . How many shares of rai l road stock, now sel l ing at

1 5%advance on the par val ue, $ 1 00, can be bought forbrokerage 470 9

2 2 2 . Insurance was procured on the ship Al ice to CapeTown and return in the sum of at and on her

return cargo, val ued at at What was the

amount of the premium paid ?2 23 . What is the duty at 50%ad val oremon 527 yards of

silk val ued at per yard 92 2 4 . What is the duty at 302 per dozen and 30%

ad val orem on 500 dozen l inen col l ars val ued at per

dozen 9

2 25 . A specul ator bought manufacturing stocks at $ 80

per share and sol d them at $ 1 04 . What per cent did hemake on his investment 9

What per cent is gained or l ost

2 2 6 . By buy ing at 1 00 and sel l ing at 1 20 9

2 2 7 . By buying at 1 20 and sel l ing at 1 00 9

2 28 . By buying at 90 and sel l ing at 1 1 0 9

2 2 9 . By buy ing at 1 1 0 and sel l ing at 90 9

2 30 . When 80 shares of stock, par value $ 1 00, are sol dfor 8 1 00

,at what per cent above par does it sel l 9

2 3 1 . Amerchant sol d a l ot of wool for $ 1 886, gainingFor what shoul d he have sol d it to gain 25%9

2 32 . A grocer sold 400 bbl . of flour at 9470 profit, gainingWhat did the flour cost per barrel ?

MISCELLANEOU S EXAMP LES. 35

233 . A specul ator bought 300 shares of mining stock at

bel ow par, and sol d them at above par. How

much di d he gain, the par value of a share being 1 0 9

234 . If 200 shares of a rail road are bought at 1 054 andsold at 1 1 54per share, what is the amount gained 9 what

per cent is gained on the investment 9235 . A watch is sold for $ 1 90 at a l oss of What

shoul d it have sol d for to make the l oss on ly 5%9

236 . If I buy 20 shares of stock, the par val ue of whichis 1 00 per share, at 1 4%above par, and sel l it at 4%bel owpar, what per cent do I l ose 923 7 . A carpet mil l valued at is insured for 4 ofits value by two companies, the first taking g of the risk at

11570 , and the second the rest at g%. What was the whol e

premium 9238 . A provision deal er

’s sal es in one year amounted to

Two thirds of this amount was received for beef,

on which his profit was and one third for pork, on

which his profit was What was the cost to him ofthe whol e amount of beef and pork sold 9239 . If an insurance company takes a risk of $ 9000

at and reinsures one hal f of it in another companyat how much does the first company make if no

l oss occurs 9240 . The capital stock of a bank is and the

net earnings for the l as t quarter are Whatis the largest even rate of dividend that the directors can

decl are, and what sumwil l be carried to the surpl us 9

2 41 . A rail road corporation increases its capital stockfrom to The stockhol ders are

al l owed to purchase at 90 per share as many shares of thenew stock as they hold of the ol d stock, and the remainingshares are sold in open market at an average price ofper share . How much can a holder of 250 shares of theol d stock make by sel l ing his new stock in the market 9

SECTION XIV

INTEREST.

39 1 . I l lustrative Examp les. (1 ) Amerchant borrowed$ 3000 for a year, and used it in his business. At the

end of the year he paid back the money and paid for its usea sum equal to 6%of the money borrowed. How much didhe pay for its use 9 Ans. 1 80.

39 2 . The price paid for the use of money is interest .

The money for the use of which interest is paid is theprincipal .

393 . The rate of interest is the fraction which a year’sinterest is of the principal . It is usual ly expressed as a

rate per cent per annum.

In the example, what is the principal ? the interest 9 the rate of

interest 9394 . The rate of interest fixed by l aw is the l egal rate.

Interest at a higher rate than the l egal rate is usury .

(2 ) If $ 1 80 is paid for the use of $ 3000 for one year,how much ought to be paid for the use of the same sum ofmoney for two years 9 three years 9 hal f a year 9

(3) If 6%of $ 3000 is paid for the use of $ 3000 one

year, what per cent of that sum ought to be paid for itsuse two years 9 three years 9 hal f a year 9

395 . Interest is reckoned as a percentage, the principal

being the base and the rate being the rate per annum in

creased or diminished in p roportion to the time.

Thus, in exampl e (3) above, the rate per annum is 6 the rate for

two years is 1 2 for three years, 1 8 for hal f a year,

236

238 INTEREST.

INTEREST FOR MONTHS

398 . I l lustrative Examp les. ( 1 ) At 6%per annum, whatis the interest of for 2 months 9 (2 ) Offor 8 months 9

(1 )WRITTEN WORK .

( 1 ) The rate for 2 months, or A} of a

3year, is l of or and 1 %of the

P rmel pal , $ 2 principal is found by moving the decima lInterest, 3 63 point two places to the l eft. Ans.

(2 ) (2 ) The interest for 8 months is 4 times

the interest for 2 months . Mul tiply ingthe principal by 4, and then moving the

P rinCIpal , 523-1 8 decimal point two places to the l eft , gives

4 the same resul t as finding two months’

Interest, interest (as in Ex . and mul tiplyingthat by 4 . Ans.

39 9 . From these exampl es we derive the fol l owing

WRITTEN WORK .

Rul e .

To find the interest,at for any number of months

Multip ly the principa l by ha lf the number Of months, and

move the decima l point two p laces to the left.


400 . At 6%a year, or 1%for 2 mo.,find the interest

9 . Of 320 for 2 mo. 1 9 . Of 8000 for 8 mo.

1 0 . Of $ 430 for 4 mo. 20 . Of 927 for 1 mo.

1 1 . O f for 2 mo. 2 1 . Of $625 for 3 mo .

1 2 . Of 526 for 1 0 mo . 22 . Of $ 350 for 7 mo.

1 3 . Of $ 365 for 1 4 mo . 2 3 . Of for 5 mo.

1 4 . Of for 1 6 mo. 2 4 . Of 2500 for 9 mo.

1 5 . Of for 1 8 mo . 25 . Of 9000 for 1 5 mo.

1 6 . Of 500 for 20 mo . 2 6 . Of $ 5000 for 5 mo.

1 7 . Of 3000 for 1 6 mo. 2 7 . Of 8000 for 1 7 mo .

1 8. Of 6000 for 6 mo. 28 . Of 800 for 25 mo.

IN TEREST FOR DAYS. 239

INTEREST FOR DAYS

40 1 . I l l ustrative Examp les. ( 1 ) At 6%a year, what is theinterest of $ 2400 for 6 days 9 (2 ) Of 630 for 1 6 days

9

( 1 ) ( l ) The rate for 60 days (2 months)WRITTEN W ORK being the rate for 6 days is I

ll, of

P rincipal , 3 2400 which is One thousandth of the

Interest principal is found by movi ng the decimal

point three places to the l eft . Ans. a

(2 ) (2 ) The interest for 1 6 days might be

WRITTEN WORK . found by first finding the interest for 6P rinci 630

days (moving the decimal point) , thenpal ,

1 6dividing that interest by 6 , and lastly multipl ying the quotient by 1 6 . But it usual ly

3780 takes fewer figures to reverse the order of63 these steps, and the resul t is the same.

1 0080 So the principal is mul tipl ied by 1 6, the

Interest, product divided by 6 , and the decimal pointmoved three places to the left . Ans. 3 l

402 . From these exampl es we derive the fol l owingRul e .

To find the interest, at for any number of days Mul

l iply the p rincipa l by the number Of days, div ide by 6 , and

move the decima l po int three p laces to the left.

NOTE . If the number of days happens to be divisibl e by 6 , a step

is saved by mul tiply ing the principal by one sixth of the number of

days and then moving the point three pl aces to the l eft .


403 . At 6%a year, or for 6 da . ,find the interest

29 . Of 920 for 6 days . 35 . Of $ 328 for 93 days.

30 . Of $ 426 for 1 2 days . 36 . Of 972 for 1 23 days.

31 . Of $ 2365 for 1 8 days. 3 7 . Of 525 for 90 days.

32 . Of 3840 for 1 5 days . 38 . Of 632 for 42 days.

33 .

-Of $ 927 for 63 days. 39 . Of for 1 day .

as Of $ 231 for 33 days. 40 . Of 8 for 2 days.

240 INTEREST.

6%per annum find the interestOf 9280 for 3 days. 45 . Of

Of $ 2800 for 4 days . 46 . Of

Of 352250 for 5 days . 47 . Of 35

Of for 53 days. 48 . Of

INTEREST FOR YEARS, MONTHS, AND DAYS

Oral Exercises .

404 . a . The rate of interest being What is the ratefor 2 years ? for 3 years 9 for 4 years ? for 5 years ?for 6 years 9 for 8 years 9b. If the interest for 2 months is or of the

principal , What part of the principal is the interest for 4months ? for. 6 months 9 for 8 months ? for 5 months ?for 7 months ? for any number of months 9The interestfor any number Of months is one ha lf as many

hundredths Of the principa l as there are months .

0 . If the interest for 6 days is of the principal ,What part of the principal is the interest for 1 2 days 9 1 8

days ? 24 days ? 9 days ? 1 5 days 9 1 6 days ? 1 7 days ?1 9 days 9 any number of days 9The interest for any number Of days is one sixth as many

thousandths Of the principal as there are days.

405 . I l lustrative Examp les. (1 ) What is the rate, at 6%per annum,

for 3 years, 8 months, 24 days 9

(2 ) For 2 years, 7 months, 1 5 days 9

SOLUTION SOLUTIONThe rate for 3 y . The rate for 2 y .

for 8 mo . for 7 mo.

for 24 d.

Ans. 0.224

242 INTEREST.

At 6%per annum,what is the interest

56 . Of $ 300 for 2 y. 5 mo. 24 d. 9

57 . Of for 1 y. 3 mo. 27 d. 9

58 . Of for 2 y. 4 mo. 28 d. 9

59 . Of for 3 y. 6 mo. 1 9 d . 9

60 . Of for 1 y. 8 mo. 1 8 d. 9

6 1 . Of for 2 y. 9 mo. 1 3 d. 9

409 . The sum of the principal and the interest cal l edthe amount .


4 1 0 . Find the amount, at 6%per annum,

6 2 . Of $ 900 for 2 y. 6 7 . Of $ 5000 for 2} y.

63 . Of for 7 y. 68 . Of 85 for 5 y.

6 4 . Of for 5 y . 69 . Of for I} y .

6 5 . Of 5000 for 3 y . for 1 4 y.

66 . Of 1 500 for 4 y. 7 1 . Of for 34y .

Find the amount, at 6%per annum,

7 2 . Of $ 92 for 4 mo. 79 . Of $ 800 for 7 mo.

7 3 . Of $320 for 6 mo. 80 . Of $ 650 for 1 1 mo.

7 4 . Of $ 460 for 2 mo. 81 . Of for 6 mo .

7 5 . Of 85 for 8 mo. 82 . Of for 7 mo.

7 6 . Of 620 for 1 0 mo. 83 . Of 85 for 8mo.

7 7 . Of 230 for 3 mo. 84 . Of for 9 mo.

78. Of 500 for 5 mo. 85. Of for 3 mo.

Find the amount, at 6%per annum,

86 . Of for 57 days ; for 1 6 days .87 . Of for 35 days ; for 27 days.

88. Of for 48 days ; for 7 days.

89 . Of for 3 days ; for 1 day.

90 . Of for 2 days ; for 5 days.9 1 . Of for 33 days ; for 63 days.92 . Of for 93 days ; for 1 23 days.

IN TEREST AT AN Y RATE OTHER THAN 243

INTEREST AT ANY RATE OTHER THAN 6 70 .

41 1 . I l lustrative Examp le. What is the interest$ 6280 for 1 year 6 months 25 days at 7 70 9 at 5%9499 at

WRITTEN WORK .

P rincipal , 3 6280 The interest is computed first at

because this is a convenient rate .

0 O O

lO O

1 046§Th is Interest moreased by 3 of i tsel f 1 8

1 2the interest at 7 and diminished by

25 0gof itsel f is the interest at

56520 In the same way the interest at 6%diminished by 4 of itsel f becomes the

fi l tered ; atinterest at and increased by 4 of

(2) 39985 6 1 9“ itsel f becomes the interest atInterest at 7 Ans. s 8 9 3942 443 ;

Interest at $492 .805gi 7392 089

Oral Exercises .

41 2 . What part of the interest at 6%should be added

or subtracted to obtain the interest

a . At 8%r 0 . At 5%r e . At 75705. At a. At 4%? f . At 4.1,%9If the part to be added or subtracted is not an easy one to find,

the interest at 6%may be divided by 6 and the quotient mul tipl iedby the given rate per cent ; or

,better, the interest at 6%may be

mul tipl ied by the given rate per cent and this product divided by 6 .


41 3. What is the interest93 . On 500 for 3 years at 5%99 4 . On 325 for 2 years at 4%995 . On 926 for 4 years at 7 9

96 . On 840 for 6 years at 9

9 7 . On 8 for 2 years at 9

244 IN TEREST.

What is the interest98 . On 325 for 2 months at 4%99 9 . On 351 296 for 8 months at 5%91 00 . On for 9 months at 3%91 01 . On 33 for 7 months at 9

1 02 . On for 5 months at 2 15-70 91 03 . On 1 600 for 1 4 months at 7f%91 04. On 35900 for 1 month at

1 05 . On 853.60 for 3 months at 7%91 06 . Find one day’s interest, at on 3482 .85 ; on

on on 35 and on

Add the resul ts. Add the five given sums of money and'

reckon one day’s interest on the whol e at How doesthe l ast resul t compare with the sum of the first five resul ts 9

What is the interest

1 0 7 . On for° 5 days at 570 91 08 . On for 1 8 days at 4%91 09 . On 940 for 63 days at 7 9

On 862 for 93 days at 4570 91 1 1 . On for 33 days at 9%9

On 35 for 1 23 days at 8%91 1 3 . On 35 for 35 days at 9

1 1 4 . On 35 for 28 days at 35%91 1 5 . On for 1 day at 1 70 9

What is the interest

1 1 6 . On for 1 y . 3 mo . 24 d. at 4%91 1 7 . On 85 for 2 y. 5 mo . 1 5 d. at 7%91 1 8 . On 35 for 1 7 mo . 1 3 d. at 4470 9

On for 2 y . 3 mo . 1 6 d. at 9

1 2 0 . On ior 3 y . 6 mo. 28 d. at

1 2 1 . On for 4 y . 2 mo. 1 6 d.

1 2 2 . On for 1 y. 6 mo. 1 7 d. at

1 2 3 . On for 8 mo. 1 8 d. at 970

246 IN TEREST.

1 38 . What is the interest on from February 7 ,1 894, to December 1 , 1 898, at 8%91 39 . What is the interest on from October 25,

1 895, to July 28, 1 899, at 1 070 91 40 . What is the interest on from August 1 3,

1 892,to April 2 , 1 894, at 5470 9

1 4 1 . What is the interest on 625 from the 20th ofJanuary, 1 891 , to the 1 st of May, 1 895, at 9

1 42 . What is the interest on $ 2340 from March 1 5th,1 894, to February 28th, 1 895, at 5%91 43 . What is the interest on 35 from l 0th April ,

1 893, to 2d October, 1 894, at 2470 91 44 . What is the interest on fromApril 1 9, 1 890,

to May 30, 1 893, at 7 9

1 4 5 . What is the interest on from May 1 5, 1 893,to January 6 , 1 895, at 9

1 46 . What is the interest on from June 30, 1 892,to October 31 , 1 894, at 9

1 4 7 . What is the interest on from January 30,1 893

,to March 1 9 , 1 895, at 9

1 48 . On the 1 2 th of October, 1 894, I gave two notes, onefor 450 with interest at and the other for 960 withinterest at What amount wil l be requi red to pay thesenotes November 5th, 1 894 91 49 . Mr. Wheel er owes 500 payabl e October i 5th, 1 894,

but is tol d that interest at 5%wil l be deducted if he wil lpay the debt March 25th, 1 894 . How much wil l he save

by paying at the earl ier date 91 50 . Messrs. Fowl e Belknap bought goods at various

dates during the year, agreeing to pay interest at 4%on the

amount of each purchase from its date to the l st of thefol l owing January. The purchases were as fol l ows : April1 5th, May 20th, June 25th,August i 9th, September 9th, and No

vember 3d, What amount was due January l st 9

ACCURATE IN TEREST. 247

AOOU RATE INTEREST .

41 6 . The common methods of computing interest set forth in the

foregoing rul es and exampl es, are based on the assumption that the

year consists of 1 2 equal months of 30 days each , or 360 days . This

assumption, though convenient, and for common purposes accurateenough, is not accurate enough when appl ied to large sums of money .

To be accurate , the time shoul d be reckoned in years and days, and

one day should be regarded as 343 of a year . The governments of theU nited States and G reat Britain pay accurate interest. Also bankers

and merchants are adopting the custom of computing accurate i nterest

when the principal is large . The use of Interest Tabl es makes thismethod as easy as any other.

41 7 . I l lustrativeExamp le. What is the accurate interest,at of for 7 days 9

Su mmon — Since 7 days equals fi x of a. year, the interest for 7days equals six of a year

’s interest. Ans. 8

NOTE .— By the common method, the answer woul d have been

or more .

41 8 . From this exampl e is derived the fol l owing

Rul e .

To compute accurate interest for any time l ess than a

year : Count the exa ct number Of days from date to date, andtake as many 365ths Of a year

’s interest as there are days.


1 51 . Find the accurate interest, at on fromJanuary 1 5th to June 1 5th, 1 895.

1 52 . Find the accurate interest on from Jeuuary 20th, 1 894, to March 28th, 1 895, at1 53 . Find the accurate interest on from June

1 2th to September i 2th, 1 895, at1 54 . Find the accurate interest on 4860 from July l st,

1 895, to December l st, 1 896, at 7

248 IN TEREST.

1 55 . Find the accurate interest on fromDecember 1 5th, 1 893, to October 25th, 1 894, at1 56 . Find the accurate interest on from Feb

ruary l st to May l st, 1 894 , at

1 57 . Find the accurate interest on 35 fromDecember 2oth, 1 895, to September 30th, 1 896, at1 58 . Find one day’s accurate interest

,at on each of

the fol l owing sums : Add

the resul ts, and compare the sum with one day’s accurateinterest on the whol e of the money.

1 59 . Find the difference between the accurate interestand the interest by the common method

,on

fromApril 4th to July 1 7th of the same year, at

PARTIAL PAYMENTS.

420 . I l lustrative Examp le.

This is the written promise of one person, Charl esHammond, to pay another person , Wil l iam Mason ,

“

or any

one to whom W il l iam Mason may order it paid, a cer

tain sum of money, for value received . Such a

promise is cal l ed a promissory note, a note of hand, or simplya note. The sum named in the note

, as is the

face of the note.

250 IN TEREST.

Find what was due on this note December 20, 1 896, interest at

WRITTEN WORK.

P rincipal fromJune 25, 1 894Interest to Oct. 1 7 1 894 (3 mo . 22 d. )Amount .

First payment Oct.New principal from Oct.

Interest to May 2 1 , 1 895 (7 mo . 4

Second payment May 2 1 , 1 895

Interest on from Oct. 1 7 1 894, toAug . 1 9 , 1 895 (1 0 mo . 2 d. )

Second and third paymentsNew principal fromAug. 1 9 , 1 895

Interest to July ( 1 0 mo . 1 2 d. )Amount .

Fourth paymentNew principal from July 1 , 1 896Interest to Dec . 20

,1 896 (5 mo. 1 9

Amount due Dec. 20,1 896

424 . The above is in accordance with

The U nited States Rul e for P artial

1 . Find the amount of the principal to the time when the

payment or the sum of the payments equa ls or exceeds the

interest ; take from this amount a sum equal to the payment

or payments.

2 . With the rema inder as a new principal , proceed as

before, to the time Of settlement.

4'This payment is too smal l to cover the accrued interest , and so it

is set aside until the third payment is made, when it is added to that

payment and with it subtracted fromthe amount than due.

PARTIAL PAYMEN TS. 251


1 60 . November 1 5,1 893, I gave my note promising to

pay on demand,with interest at 490. March 7

,1 894,

I paid 225. What remained due August 28, 1 894 91 6 1 . I hel d a note

, secured by a mortgage on real estate,

for 800, bearing interest at 570 from June 1 0, 1 89 1 . July1 5, 1 892 , I received $ 240 ; May 22 , 1 893, I received $ 1 50.

What remained due June 1 0, 1 894 91 62 . A note for $ 1 500, dated November 1 2 , 1 89 1 , was

indorsed as fol l ows : January 9 , 1 892 , 1 80 ; June 1 8, 1 892 ,$ 425 ; October 3, 1 892 , $ 500 . What amount was dueApril 1 6, 1 894, interest at 6%91 63 . What bal ance was due August 1 , 1 895, on a promis

sory note for 1 235 on interest at 6%from October 1 , 1 889 ,and indorsed, January 20, 1 890, April 25, 1 89 1 ,

$ 350 ; May 1 0, 1 892 , $ 450 ; and July 1 5, 1 894, 200 9

1 64 .

1 200 CLEVELAND , May 1 9 , 1 892 .

For val ue received, We jointly and several ly promise to

pay Wil l iamHunter, or order, Twel ve Hundred Dol l ars, ondemand, with interest at four per cent.

DAVID LAMSON .

HU GH BRICKETT .

Indorsements : October 1 2 , 1 892, 250 ; February 23,

1 893, $ 250 ; July 1 4, 1 894, $ 250 ; December 20, 1 894 ,

$ 250. What was due Jul y 1 , 1 895 9

1 65 . July 1 6, 1 894, John Al l en borrowed of Al bertNichol s $ 3500, and gave his note for that sum with interest at 7 September 22 , 1 894, a payment o f $ 1 250 was

made, and December 1 0, 1 894, another payment of 1 250.

January 1 , 1 895, a new note was given for the bal ance thendue, with interest at Find the bal ance due and writethe new note in proper form.

252 IN TEREST.

426 . When partial payments are made upon notes thathave but a short time to run, interest is often computedby the fol l owing, cal l ed

The Merchants’Rul e for P artial P ayments .

1 . Compute interest on the principa l from the time it begins

to draw interest to the time of settlement, and a l so on each

paymentfrom the time it is made to the time Of settlement.

2 . Take the difl‘

erence between the sum Of the p rincipa l and

its interest and the sum Of the payments and their interests

this difi'

erence wi l l be the ba lance due.

427 . I l lustrative Example . Find how much was due on

the fol l owing note May 1 5, 1 895 .

ROCHESTER, N . Y. , January 1 5, 1 895.

Four months after date, I promise to pay to the order ofAl fred King CO ., One Thousand Dol l ars, with interest atsix per cent, value received. STEPHEN FOSTER .

Indorsements January 28,1 895, 1 20 ; February 1 6,

1 895, $ 1 50 ; March 3, 1 895, $ 1 00 ; March 25, 1 895, $ 80 .

WRITTEN WORK .

P rincipal on interest from J an. 1 5, 1 895

Interest to May 1 5, 1 895 (4 mo. )Amount of note

P ayment, J an. 28

Interest toMay 1 5 (3 mo . 1 7

P ayment Feb. 1 6

Interest toMay 1 5 (2 mo . 29 d.P ayment March 3Interest toMay 1 5 (2 mo. 12 d.P ayment March 25 .

Interest toMay 1 5 ( 1 mo. 20 d.

Ans. 3

254 IN TEREST.

PROBLEMS IN INTEREST.

429 . From exampl es thus far giv‘

en, we l earn that interestis the product of three factors, principal , rate per annum, and

time . For a short expression of this fact, l et l etters standfor numbers as fol l ows

i = the interest, in dol l ars.

p the principal , in dol lars .r the rate per annum, a decimal fraction .

t : the time, in years.Then we may say

i p r t,

meaning that interest equal s principal multip l ied by rate

mul tip l ied by time. For in al gebraic notation two or morel etters written in immediate succession, as p r t, denote the

product of the numbers the l etters stand for.

As an exampl e, suppose the interest on $ 480 for 3 yearsat 5%is required.

r t

2400

480‘i = p r t

Otherwise,$ 480 x x 3 :

I) X 7' X t : i

430 . In the formula1: p r t

are represented four numbers, any three of which beinggiven, the fourth can be found. Hence arise four probl ems :

1 . Given p , r, and t, to find i .2 . Given p , r, and i , to find t.3. Given p ,

i,and t

,to find r.

4 . Given i,r, and t, to findp .

Working this example in the or

dinary way, we firstmul tiply 0.05, the

rate per annum r) by 3, the num

ber of years and obtain the

rate for the whol e time r t) . We

then mul tipl y the principal p ) byr t) , and obtain the inte rest

p r0 .

P ROBLEMS IN IN TEREST. 255

The first of these probl ems represents al l the exampl esheretofore given which required the interest to be found.

The second, third, and fourth require further il lustration.

431 . P rincipa l . rate, and interest given, to find the time .

If each member of the equation1: p Tt

is divided by p r, the resul t wil l bei

p r

which means that if the given interest is divided by the product of the principa l and the rate, the quotient wil l be the time.

t)


432 . I l lustrative Examp le. In what time wil l 380 yield

interest at 5%9

SOLUTION .

91 I}. Ans. 1 4years, or 1 y . 6 mo.

380 x 38

In what time wil l1 69 . 600 gain 8 30 at 670 1 7 2. 3200 gain 8 1 60 at 5%71 70 . 8 400 gain 8 48 at 4% 1 7 3 . 81 6 gain 6 at 71 96 ?1 71 . 800 gain 84 at 7 9 1 74 . gain at

1 7 5 . In what time wil l it4820 amount to at 4%9NOTE . To find the interest, subtract the principal from the

amount . Then find the time as in the previous exampl es.

1 76 . In what time wil l 326 , at 7 amount to 9

1 7 7 . How l ong must a note for run to amount toat 870 9

1 78 . In what time wil l 1 00 amount to 200 at 6%91 7 9 . In what time wil l 55500 amount to 1 000 at 5%91 80 . In what time wil l any sum of money at interest

doubl e itsel f at 5%9 at 6%9

2 Thus,

256 INTEREST.

1 81 . In what time wil l a sum of money at interest doubl eitse l f at 2%9 at 470 91 82 . In what time wil l a sum of money at interest doubl e

itsel f at 3%9 at 1 0%91 83 . Show in general that the time in which a sum of

money on interest at the rate r wil l doubl e itsel f is 3years.

43 . P rincipal , interest, and time given, to find the rate .

If each member of the equation

i p r t

is divided by p t, the resul t wil l be

1,

p t

which means that if the interest is divided by the product Ofthe principa l and the time, the quotient wi l l be the rate.


434 . I l lustrative Examp le. At what rate wil l 360 gaininterest in 1 year 5 months 9

8 360 x 51 0

1 8 4 . At what rate wil l as gain 5000 in 1 0 years 91 85 . At what rate wil l 1 2 gain 4 in 3 y. 4 mo. 9

1 86 . At what rate wil l 450 gain in 1 y. 7 mo. 9

1 8 7 . At what rate wil l 250 gain 35 in 2 y. 9 mo. 1 8 d. 9

1 88 . At what rate wil l gain in 1 y. 6 mo. 9

1 89 . At what rate wil l 3575 gain in 3 y . 6 mo. 9

1 90 . At what rate wil l 1 000 gain 500 in 6 y . 8 mo. 9

1 9 1 . The amount of 1 50 for 2 y. 6 mo. was

What was the rate 9Norm. The interest is found by subtracting the principal fromthe

SOLUTION .

258 IN TEREST.

202 . How much money on interest at 6%for 8 mo. 1 7 d.

wil l gain 9

203 . How many 5%bonds of 1 000 each must be purchased to secure an income of $ 1 000 semi-annual ly 9204 . A certain fund is invested in bonds and yields297 quarterly. How much is invested 9

437 . The amount has been defined as the sum of the

principal and the interest. Letting a stand for the amount,we may say a =p i . But as i =p r t, we may al so say

a =p + 1 ) r t ( 1 )Thi s equation may be written in the form

a p ( 1 r t) (2 )For if p is mul tipl ied by 1 +r t, the resul t wil l be p+p r t.As an exampl e suppose the amount of $420 on interest

3} years at 6%is required.

This may be sol ved in two ways

( 1 )

r t

420

840

i =p r t

p a

a 508.20 =p +p 1‘ i

438. Amount, rate, and timc given, to find the principa l .

If each member of the equation

a p ( 1 r t)is divided by 1 r t, the resul t wil l be

a

1 + r tp ,

which means that if the amount is divided by 1 plus the produot of the rate and the time, the quotient wi l l be thep rincipal .

(2 )

1 7 5

420

840

420

p ( 1 r t)

P RESEN T WORTH AND TRUE DISCOUN T. 259

439 . I l lustrative Examp le. What sum of money put on

interest at 5%wil l amount to 299 in 3 years 9

SOLUTION .

a 8299 rt $ 299

r 0-05 1 + rt 326°

Ans. 8 260 .

205 . What sum of money put on interest now at 5%wil l amount to 1 000 in 1 0 years 9206 . What principal on interest at 670 wil l amount to

$ 1 740 in 7 y . 6 mo. 9

207 . What is the principal on interest at 470 that wil lamount to in 20 years 9208 . Aman who is under obl igation to pay it2000 at the

end of 8 years prov ides for his obl igation by putting on interest now,

at such a sum of money as wil l amount to2000 in 8 years. How much does he put on interest 9

PRESENT WORTH AND TRUE DISCO U NT.

440 . The present worth of a sum of money due at somefuture time is that sum which, put on interest now

,will

amount to the given sum in the given time .

NOTE . The l ast five exampl es are exampl es of present worth, theanswer to each being the sumwhich, put on interest at the given rate,

wil l amount to the given sum in the given time.

The difference between a sum of money due at a futuretime and its present worth is the true discount .

209 . What is the true discount at 6%of $4200 due in2 y. 6 mo . 9

21 0 . A young man has the option of receiving a l egacy of$ 5000 at the age of 25 or its present worth at the age of 2 1 .

The rate of interest being 5%, 4what is the val ue of hi s

l egacy at the age of 2 1 9

260 IN TEREST.

2 1 1 . Aman purchased a tract of woodl and for

paying cash, and agreeing to pay the rest in four

equal annual payments at the end of one,two, three, and

four years, with the option, however, of paying the presentworth of the l ast three instal lments at the end of the firstyear. How much wil l it cost him to settl e the whol e cl aimat the end of the first year, the rate of interest being 7 9

OOMPO U ND INTEREST .

441 . I l lustrative Example. A l oan of 800 was made

with interest at 6%payabl e annual ly, unpaid interest to beadded to the principal and draw interest at the same rate .

No interest was paid for four and one hal f years, at the endof which time the l oan with accumul ated interest was paid.

What was the whol e amount paid 9

WRITTEN WORK.

P rincipal 8Interest for l st yearAmount, or principal for 2d yearInterest for 2d yearAmount, or principal for 3d yearInterest for 3d yearAmount, or principal for 4th yearInterest for 4th year

Amount, or principal for 5th yearInterest for hal f a yearAmou nt

Ans . Whol e amount paid 8Deduct l oan

Leaving accumulated interest 0

44 2 . Interest ac cumul ated by adding unpaid interest tothe principal at stated interval s, the whol e to form a new

principal , is compound interest .

In the exampl e, what is the principal 9 the amount 9 thecompound interest 9

262 IN TEREST.

22 1 . What is the difference at the end of five yearsbetween the amount o f 1 000 at interest compoundedannual ly, and the amount of the same sum of money at the

same rate Of interest but compounded semi-annual ly ?222 . Show that the amount of $ 1 at compound interest

for 5 years, interest compounded annual ly at is equalto dol l ars ; al so that the amount of any number ofdol l ars, say It50 for the same time and at the same rate, isequal to 50 x2 23 . Show in general that, if a represent the amount

, p

the principal , r the rate, and n the number of years (or

periods) , thata p (i

NOTE. If p 8 l , the formul a becomes

a ( 1

and this is the formul a used in the making of the Compound InterestTable given in the next Articl e .

445. The work of computing compound interest may beshortened by the use of the

COMPOUND INTEREST TABLE.

Showing the amount of 3 1 at compound interestfrom 1 year to1 0 years, at 3 , 4 , 5, 6 , and 7 per cent.

COMP OUND IN TEREST. 263

446 . I l lustrative Examp le. What is the compound interLest of $ 1 000 for 2 y . 4 mo. at 7%9

WRITTEN WORK.

Amount of 3 1 at 7 for 2 yearsAmount of $ 1 000 at 7 for 2 years

Amount of 8 for 4 mo .

1 1 7 1 6 1 43Amount of 0 1 000 for 2 y . 4 mo .I

1 000

.

Compound interest Ans.

The amount of 8 1 000 for 2 years is first found, and the amount ofthat amount for fourmonths is then fou nd by mul tiplying by


224 . What is the compound interest of 450 for 2 y.

6 mo. at 5 interest payabl e annual ly 922 5. What is the compound interest of 1 200 for 5 y. at

4 70 , interest payabl e annual ly 922 6 . What is the compound interest of 2500 for 4 y.

at 8 interest payabl e semi-annual ly 922 7 . What is the compound interest of $ 300 for 3 y.

2 mo. at i nterest payabl e semi-annual ly 922 8 . What is the compound interest of $ 380 for 1 y.

1 0 mo. 22 d. at interest payabl e semi-annual ly 922 9 . If at the age of 25 years, a person puts $ 1 000 on

interest, compounding annual ly at 670 , what wil l be theamount due him when he is 40 years old 92 30 . What is the difference bet ween the compound interestof for 5 y. at 5 interest payabl e annual ly

,and

the simpl e interest on the same sum of money for the same

time at the same rate 9231 . Find the compound interest of 5000 for 1 7 y. at

5 interest payabl e annual ly .

NOTE .— First find by the tabl e the amount for 1 0 years, then find

the amount of that amount for 7 yearsmore.

264 INTEREST.

BANKDISCO U NT.

448. Il lustrative Examp le.

Suppose Davis wants the money promised by this note at once,

August 2oth . He can offer the note at a bank for the money . If the

ofi cers of the bank are satisfied that Stone’s credit is good, so that he

can be depended on to pay the note when it becomes due, or that

Davis’ credit is good, so that he can be depended on to pay the note

if Stone fail s to do so , the bank wil l pay Davis 6 l ess the inter

est cn that sum for 63 days at the current rate . Supposing the cur

rent rate to be 7 per annum,Davis wil l get for the note l ess

8 or 8 The note is then said to have been discounted by

the bank.

Before giving the note to the bank Davis must indorse it, that is,write his name across the back of it, and so become responsibl e to

the bank for its payment when it becomes due if Stone should fai l to

pay it at that time .

The bank reserves interest for 63 days instead of 60 , as specified in

the note, because the promisor, or maker, is al lowed three days, cal l eddays of grace, beyond the time specified in the note, within which to

make the promised payment. This note is nomina l ly due October

1 9th, but not l ega l l y due until October 22d. On this last day it is

said to mature.“

In New York, New J ersey, Connecticut, Wisconsin, Vermont,and some other states, days of grace have been abol ished by statute.

In these states a note is nominal ly and l egal ly due on the same day.

266 IN TEREST.

3 . To find the proceeds : Subtract the discount from the

face Of the note.

4 . To find the discount and the proceeds of an interestbearing note Firstfind the amount Of the note at itsmaturity,and use this amount as the basis Of discount as if it were the

face Of a'

nOn-interest bearing note.

NOTE 1 .— The common, or 360-day , method of computing interest

is usual l y employed for di scount.NOTE 2 .

— The term of discount is usual l y reckoned by counting

the actual number of days from the day of discount to and including

the day of maturity . In this book it is always so reckoned, except

in the case of a note made payabl e a certain number of months afterdate and discounted on the day of its date . In such a case, the dis

count wil l be reckoned by months and days.


451 . Find the bank discount and proceeds of the fol l owing notes, discounted at date, with days of grace, at 6%

2 32 . A 30 days’ note for2 33 . A 60 days’ note for2 34 . A 20 days’ note for235 . A 90 days’ note for

Find the discount and proceeds of notes as fol l ows

Face . Date . Time . Date of Discount.2 36 . 7 May 9 , 1 895 60 days June 3, 1 8952 37 . Aug. 4

, 1 894 90 days Oct. 6, 1 894

2 38 . Nov . 8, 1 894 3 months J an. 2, 1 895

2 39 . J an . 1,1 895 1 5 days J an . 5, 1 895

240 . Oct. 7 , 1 896 2 months Nov . 1 , 1 896

2 41 . July 3, 1 895 1 00 days Sept. 1 0, 1 8952 42 . Sept. 1 2 , 1 894 30 days Oct. 1 , 1 894

2 43 . Dec. 1, 1 896 4 months Dec. 1 4

,1 896

BANK DISCOUN T. 267

592 13050

BOSTON, February 1 0, 1 894.

Three months after date , I promise to pay to the orderof James McBride, Five Hundred N inety-two and 1

255

6Dol

l ars at the Manufacturers’ National Bank, val ue received.

Discounted, at date, at WILLIAM ( “M ON

245 .

1 240 P ORTLAND, June 1 6 , 1 894 .

Sixty days after date, I promise to pay to the order of

Stephen Gardiner, Twel ve Hundred Forty Dol l ars, at theAtl as National Bank, in Boston, Mass.

,va lue received.

Discounted July 1,1 894, at 570 .

BENJ AMIN HAMMOND .

246 .

WORCESTER, MASS . , December 3 1 , 1 894 .

Six months after date, I promise to pay George Dexter,or order, Six Hundred Forty-eight and T2541, Dol l ars, for val ue

received. HIRAM BU TTERWORTH.

Discounted February 2 1 , 1 895, at 7

247 .

P HILADELPHIA, P A. , June 6 , 1 894.

Four months after date, for val ue received, I promise topay P eter Francis, or order, Eight Hundred Forty-six and

11655Dol l ars, at the office of the Girard Trust Co .

Discounted June 8, 1 894, atW ILLIAM WHARTON

248 .

3 1 048 ALBANY , N . Y Jul y 6 , 1 894 .

Thirty days after date, for value received, we promise to

pay to the order of Ingal l s, P eters CO . , One ThousandForty-eight Dol l ars, at the Commercial Bank .

JACOB VAN D U ! ER CO .

Discounted July 8, 1 894, at No grace.

68 IN TEREST.

CHICAGO , ILL. , December 30, 1 894 .

Three months after date, for ~

val ue received, we promiseto pay to the order of David Kimbal l , Nine Hundred Fiftysix and

15040Dol lars

,at the P eopl es

’ Bank.

BENNETT, HU RD Co.

Discounted January 1 5, 1 895, at

3500 NEW YORK, January 1 5, 1 894.

N inety days after date, for val ue received, I promise to

pay to the order of Edwin Carl eton, Three Thousand FiveHundred Dol lars.

WM. F. ROSCOE.

Discounted January 20, 1 894, at Accurate interest.No grace .

2500 DETRO IT, MICH May 1 2 , 1 894.

Six months after date, for value received, I promise to payto the order of Henry G. Davidson, Two Thousand FiveHundred Dol lars

,with interest at eight per cent per annum.

Discounted June 8, 1 894, atSAMU EL F' CRANE"

285311050

BOSTON, November 20, 1 894.

Four months after date, we promise to pay John G.

Farl ow, or order, Two Thousand Eight Hundred Fifty-threeand 1

1065 Dol l ars, at our ofiice, with interest at five per cent

per annum,value received.

NORRIS BROTI RS Co.

Discounted January 9, 1 895, at 4 1 70 .

270 IN TEREST.

and their name across the face of the bil l . The Engl ish merchantscoul d now have the bil l discounted, just as if it were a promissorynote ; or they could keep it til l maturity and receive ful l payment

from the Engl ish bankers at that time. So the American merchant

paid his debt, and the Engl ish merchants received theirmoney, and

yet no money was sent across the ocean.

But there remains the debt of the American bankers to the '

Engl ish

bankers . How wil l this be adjusted ? There wil l be no need of

sending the money ; because this particul ar debt wil l enter as merel yone item into an open account which the two banking houses are

keeping with each other . They are constantl y drawing bi l ls on each

other, so that this particular bil l wil l very soon be offset by bil ls

which the Engl ish bankers will draw on the American bankers in

favor ofAmericanmerchants who have been sending goods to Engl and .

453 . There is another way in which the transaction just describedmight have been effected . The American merchant , whom we w il l

cal l John Robinson, instead of sending a bil l of exchange to the

Engl ish merchants, might have waited to be drawn upon by them.

Then the bil l woul d have been in this form

The Engl ish cloth merchants, Brown , Smith , CO., after having

seen the cloth placed safely aboard ship and having received the bil l of

l ading for it, write a bil l of exchange as above and give it, with the bil l

of l ading attached, to the Barings , receiving from that house the face

BILLS OF EXCHANGE. 27 1

of the bil l l ess a discount. The Barings send this bil l of exchange

with bil l of lading attached to their correspondents, Kidder, P eabody,Co . of Boston , with instructions to col l ect the bil l and place the

amount to the Earings’credit. The Boston bankers notify Robinson

that by accepting the bil l of exchange he can have his bil l of lading,

which is his evidence that the imported cl oth belongs to him. Rob

inson pays the bil l of exchange at maturity (or before maturity ifal l owed a discount) to the Boston bankers, and so the transaction iscompl ete as far as the merchants are concerned. The bankers take

care of their part of the transaction in the manner al ready explained.

Bil ls of exchange drawn by bankers on bankers , l ike the one first

given above, are cal led bankers’ bi l l s and those drawn by merchantson merchants, l ike the second one above given , are cal l ed commercia l

bi l l s.

NOTE 1 . As a precaution against l oss, bil ls of exchange are usu

al ly drawn in dupl icate or in tripl icate , and sent by different mails .

The second of exchange bears the words first and third unpaid,”and the third of exchange bears the words first and second

unpaid.

” The payment of any one bi l l of a set makes the othertwo void .

NOTE 2 .— Bil ls of exchange may be on time ”

or at sight.”

If the bil l is to be payable on presentation, the words at sight”

or on demand are used.

Write the other two bil ls of the set to which each of the bil ls above

given bel ongs.


254 . Suppose an Engl ish merchant has sent you carpets,for which you have agreed to pay £ 600. Write one of aset of exchange, payabl e sixty days after sight, such as an

American banker woul d draw on a London banker in favorof the Engl ish merchant, and sel l to you for the purpose ofmaking the payment. What woul d this exchange cost youat $ 48 5; to the pound sterl ing 9 What woul d it cost youif written so as to be payabl e on demand, demand bil l sbeing quoted at to the pound sterl ing 9 Why shoul ddemand (or Sight) bil l s cost more than 60 days

’ bil ls 9

272 INTEREST.

255 . Suppose an Engl ish merchant has sent you cutl ery tothe value of 1 75, and draws upon you for that amount at30 days after sight. Write the first of exchange which hewoul d draw for this purpose .

2 56 . Write a 60 days’ bil l of exchange such as you mightreceive from an Engl ish merchant to whom you had sent

goods to the value of 4000.

NOTE . This woul d be an order by an Engl ish banker directing an

American banker to pay you a4000 .

2 57 . Write a 60 days’ bil l of exchange such as you woulddraw upon an Engl ish merchant to whomyou have exported5000 bushel s of wheat at 73%cents a bushel , in favor of theAmerican banker to whom you sel l the bil l with the bil l ofl ading attached. Woul d the banker pay you the ful l faceof the hil l 9 Why not 9

455 . Exchange with al l parts of the world may be

effected through London. Thi s is because London is the

great commercial center of the worl d ; and its bankers drawupon and are drawn upon by bankers in al l countries.Exchange with continental Europe can al so be effectedthrough the l arge commercial centers, as P aris, Vienna,Berl in, Hamburg, Amsterdam,

etc. The whol e commerceof the world, with insignificant exceptions, is carried on bybil l s of exchange . The commercial deal ings between twocou ntries may amount to thousands of mil l ions of dol l ars,and yet the bil l s of exchange drawn in each country may beso nearly offset by bil l s drawn in the other that only a

trifling amount of money is needed to pay the balance.

456 . The rates of exchange between this country and theprincipal commercial centers are given daily in the news

papers. The fol l owing is taken from a New York paper“ The market for sterl ing Opened firmer in tone and stil l higher

rates bid, the posted rates however remaining unchanged at i and

274 IN TEREST.

The par of sterl ing exchange is -6 7 . The actual business rate

of demand sterl ing bil ls at which gold can be exported to London

without loss on regul ar commercial account is about for bars and

for coin,and . the rate for which it can be imported without loss

is

It wil l be seen by comparing this with the former extract, that therate for sterl ing exchange that day was nearly up to the gol d-exporting

point .

°

Indeed, the same paper contained the statement that considerable amounts of gold had been purchased in New York for shipment

to Europe.

458. Exampl es for W ri tten W ork.

258 . Find the cost of a demand bil l of exchange on Londonfor 2000 at the posted rate. At the l owest rate for actualbusiness above quoted.

259 . Find the cost of amdays’ bil l of exchange on Lon

don for 865 at the l owest rate for actual business above

quoted.

2 60 . Howmuch could have been obtained in New Yorkfor commercial bil l s on London amounting to 682 1 03 . at

the highest rate above quoted for such bil l s2 61 . My friend tel egraphsme fromEurope that he wishes

me to have 1 000 placed to his credit at a London banker’s

with the l east possibl e delay. Howmuch wil l it cost me todo this at the highest rate above quoted for cabl es2 62 . Find the cost of a 60 days’ bil l of exchange on P aris

for 2500 francs. Of a 60 days’ bil l on Antwerp for the same

amount. Of a demand bil l on either pl ace for the same

amount.2 63 . FromBerl in, in Germany, I have imported books to

the val ue of 1 200marks, and I have agreed to send a 60 days’

bil l of exchange for this amount. How much wil l the bil lcost me2 64. The rate for sterl ing exchange in New York having

risen to how much shal l I save by purchasing gol d inbars at for the purpose of remitting

mum's. 275

265 . My agent in Amsterdam tel egraphs me that he haspurchased on my account goods to the amount of 8500guilders, and requests me to send him demand exchange forthat amount. How much wil l the exchange cost me in NewYork

DRAFT8 .

459 . I l lustrative Examp le.

Living in Boston and wishing to pay John Lafarge, of New

Orl eans, $ 2000 , I purchase of Richardson, F l int CO . , bankers, of

Boston , their draft ( in formas given above) on Wm. Manderson CO.,

bankers, of New Orleans. This draft I send to Lafarge, who presentsit for acceptance , and, when the draft matures, receives the moneyfromWm. Manderson Co . or he receives the money at once if he

al lows a discount. By comparing this draft with the bil l of exchange(page 269 ) it w il l be seen that they are in substance the same . Indeed ,

drafts are often cal led inl and or domestic bil ls of exchange, for the

reason that they are used to effect exchange between different parts of

the same country, just as foreign bil l s of exchange are used to effect

exchange between different countries. Drafts are not usual ly writtenin dupl icate or tripl icate, so that the ir language is simpl ified by the

omission of the words required for that purpose .

The rates for domestic exchange vary sl ightl y according to the

pressure of supply and demand, the drafts being at a premiumwhen

276 IN TEREST.

the supply is l ess than the demand and at a discount when the suppl y

is greater than the demand. For example, the statement New York

exchange at Chicago to-day was premium (page 273) means

that Chicago bankers have not an abundance of funds standing to theircredit in New York to draw against ; whil e at the same time the

demand for drafts on New York 1 8 good . So the Chicago bankers

put up the rate a. l ittl e , charging 35 cents for a _1 000-dol lar draft and

for other amounts, at the same rate . If the rate should go much

higher, it would be cheaper to send money by express to New Yorkthan to buy drafts.

What conditions of business would cause New York exchange at

Chicago to be at a discount When New York funds are at a

premium in Chicago, how should Chicago funds be in New York

460 . Exampl es for W ritten W ork.

2 66 . Find the cost of a Chicago draft for on New

York, payabl e at sight, when New York funds are quotedat 25¢ premium.

2 67 . Find the cost in New York of a sight draft foron San Francisco when San Francisco funds are

at a discount of2 68 . Find the cost of remitting by sight draftfrom Boston to N ew Orl eans when New Orleans funds arequoted in Boston at 75¢ premium.

269 . What is the cost at Detroit of a sight draft foron New York when New York funds are quoted

in Detroit at a discount of 1 5¢

STOCKS AND BONDS.

A partnership or firm is formed by two or morepersons entering into an agreement to carry on businesstogether, furnishing capital , and sharing profits and l ossesaccording to the terms of their agreement .In the absence of legal notice to the contrary , the law hol ds each

partner l iabl e for the debts of the firm, not merely to tne extent of

the capital he has put in, but beyond this, if necessary, to the extent

of al l his other property .

278 IN TEREST.

fer on the books of the corporation . So the members of acorporation are constantly changing, some going out and

others coming in . P ersons who make a business of buying and sel l ing stocks for their customers are cal l ed stock

brokers ; and their compensation is cal l ed brokerage.

When a stock is quoted at 90 , the meaning is that a share, the par

value of which is 8 1 00 , can be bought for 1590 . When quoted at 1 082,the share woul d cost In the first case the stock is at 1 0 70discount, and in the other at 82%premium.

46 7 . The government of the U nited States, the governments of the several states, and of counties, cities, towns,or school districts not infrequently borrow money, issuingas ev idence of their indebtedness interest-bearing promissory notes cal l ed bonds .

46 8 . Government bonds are usual ly offered by the government publ icly for sal e to the highest bidders.

The price obtained depends on the rate of interest offered, on

peopl e’s confidence in the government

’s good faith and financial

resources, and on the abundance or scarcity of money seeking invest.

ment .

When government bonds have passed into private hands they are

bought and sold in the stock market.

46 9 . Business corporations obtain l oans by sel l ing bondsin the same way that governments do ; but a corporationusual ly secures its bonds by a mortgage on its property .

By consul ting the newspaper the prices paid in the open market

for different stocks and bonds can be l earned and these prices show

what people think of them as investments .

It may be seen , for example, that a second mortgage bond is sold

for l ess than a first mortgage bond of the same corporation ; and

shares of common stock are sol d for l ess than shares of preferred

stock. Again, by comparing the prices of bonds which pay the same

rate of interest and have about the same number of years to run, it is

easy to see which corporation or which government has the better

credi t in the money market.

STOCKS AND BONDS. 79

The fo l l owing quotations are taken from a daily paper

At Brokers Board .

Bonds . Ra ilroads.

Atch. 4s

Cen. P . l sts

Erie 2nds

Gen. El . 58

N. J. G en . g . 58

St . L. S. F. gen .

Tex . P . l sts

do 2nds

U . P ac . l sts

Wabash l st 53

G overnments.

U . S. 48 reg. 1 1 32 Industria l s.

do 48 c. Am. Cot. Oil

do 58 1 1 83 U . S. Cordage

At Auct ion.

2 Merchants Nat. Bank1 Merrimac Mfg . CO .

2 Mass . Cotton Mil ls1 Kan . City StockYards CO .

8 1 000 New Hampshire 1 895

$2000 Boston 48 , 1 923

$ 1 000 Duluth 48 , 1 92 1

1 000 Minneapol is 43 , 1 9 1 7

470 . Exampl es for W ri tten W ork .

2 70 . At the quotations above given find the cost ofAtchison, Topeka, and Santa Fé R.R. 4 per cent

bonds ; Central P acific BR . First Mortgage Bonds ;and Erie Railway Second Mortgage Bonds ; brokerage l%~

Atchison

Bal t . Ohio

Boston AlbanyC. C. C . 85 St. L.

do

Harl emN . J . G en .

N . Y. , N . H. , H.

U nion P ac .

Express Companies.

Adams

Fargo

280 IN TEREST.

271 . Find the cost of eight shares in the Boston andAl banyR R ; 1 6 shares in the preferred stock of the Cl evel and,Cincinnati, Chicago, and St. Louis R.R. ; and 20 shares inthe New York, New Haven, and Hartford R.R. ; brokerage

l2 72 . Find the cost of twenty thousand dol l ars Of the

General El ectric Company’s 5 per cent bonds ; of

the New Jersey Central R R. 5 per cent bonds, payabl e in

gol d ; of the St. Louis and San Francisco RB .

General Mortgage Bonds ; brokerage27 3 . Among the securities coming into his hands the

administrator of an estate finds certificates for 1 0 shares inthe Harl em R.R. ; 25 shares in the New Jersey Central ;1 5 shares in the Wel l s and Fargo Express Company ; 1 2shares in the Adams Express Company ; 1 0 shares in

the American Cottonseed Oil Company ; 1 0 shares in the

U nited States Cordage Company ; 5 shares in the MerchantsNational Bank ; 3 shares in the Merrimac ManufacturingCompany ; 5 shares in the Massachusetts Cotton Mil l s ; and1 0 shares in the Kansas City Stock Yards Company. At

above quotations, what is this property worth2 74 . At the prices above quoted, what per cent on the

investment would be yiel ded annual ly by Wabash FirstMortgage 5s by General El ectric 58 ? by New JerseyCentral gol d 58 by U .S. Government 58 ?2 75 . At prices above quoted, compute the rate of income

on money invested in U S . 4% coupon bonds ; in New

Hampshire 68 ; in Boston 48 ; in Minneapol is 48 .

2 76 . Which gives the l arger return on the investmentAtchison, Topeka, and Santa Fé 43 or City of Duluth 4s, asabove quoted Why, then, do the l atter command a higher

price in the market27 7 . How much was gained on 500 shares U nion P acific

bought at and sold at 1 7g, brokerage for eachtransaction

282 INTEREST.

AVERAGE OR EQUATION OF PAYMENTS.

47 1 . I l lustrative Examp le. A debtor owes to one personthe fol l owing sums at the dates specified : Oct. 1 , 262 ;

Oct. 1 0, $ 220 ; Nov . 6, $ 250. At what date can he pay

the whol e debt without l oss of interest to either party

Interest Method .

WRITTEN WORK.Exn axarrom— To do

Due. Items. Days. Interest. this example,wemay supOct. 1 , $ 262 0 pose al l the items to be

a 1 0,

220 9 0.33 paid at the earl iest date atNov 6 250 36 1 50

which any item becomes

due, viz. Oct. 1 .

1 day’smt. of 732This would involve a

Oct. 1 1 5 d. Oct . 1 6 . Ans. loss to the debtor of interest on $220 from Oct.

1 to Oct. 1 0 (9 days) , and on 8250 fromOct. 1 to Nov. 6 (36 days) .

That no loss may resul t, the total Of the items, 8 732 , shoul d be

paid as many days after Oct. 1 as wil l be required for 8 732 at 6%to

gain of interest . To find this time, we divide by the

interest of 8 732 for 1 day at and have for a quotient 1 5.

Fifteen days after Oct. 1 is Oct. 1 6 . Ans . Oct. 1 6 .

472 . The process of finding the time when the paymentof several items, due at difierent times, may be made at

once, without l oss of interest to either party, is average, or

equation of payments

473 . The date at which several sums due at differenttimes may be paid at once is the average date or equated

time of payment .

474. From the foregoing operation may be derived a

To find the average time for the payment of several sumsdue at difierent times

1 . Select some convenient da te ; for example, the earl iest

date at which any itemmatures.

EQUATION OF RAYMEN TS. 283

2 . Compute the interest on each i temfrom the selected date

to the date of its maturity.

3 . Add the interests thus found ; divide their sum by the

interest of the sum of the items for one day ; the quotient wi l l

be the number of days from the selected da te to the average

date of payment.4 . Add this number to the selected date ; the result wi l l be

the average date required.

P roof .

F ind the sum of the interests on al l items due before theaverage date, from the date at which they are severa l ly due to

the a verage date ; al so find the sum of the interests on a l l

items due after the average date f rom that da te to the dates at

which they are severa l ly due . If these sums a re equa l , or

difl'er by less than ha lf a day

’s interest on the sum of a l l the

items, the result is correct.

NOTE 1 .— Any date may be sel ected from which to average an

account. The l ast day Of the month previous to that in which the

earl iest maturing itembecomes due is a convenient date .

NOTE 2 . When any item contains cents, if l ess than 50, disregardthem ; if 50 ormore , increase the units of dol l ars by 6 1 .


288 . What is the average date for paying three notes dueas fol l ows : March 31 , $ 400 ; April 30, 300 ; May 30,200

26 9 . What is the average date of maturity of three notesof $ 800 each, due respectivel y Nov . 5, Dec. 8, and Feb. 3

290 . What is the average date ofmaturity of the fol l owingitems of account, viz. : $ 900, due Sept . 1 0 ; due

Oct. 2 1 ; and due Oct. 28

29 1 . Find the equated time for paying $ 430, due in 5

months ; 270, due in 9 months ; and $ 300, due in 8months .

84 INTEREST.

292 . Average the above, having the first item due in 3

months , the others in 9 months each.

293 . Aman purchased a farm for $ 3600, agreeing to pay$ 600 down, and the remainder in five equal semi-annualinstal lments. What is the average date of the five instal lments29 4 . Thomas Brown has borrowed sums of money from

John Rich as fol l ows : September 1 2 , 1 894, $ 500 ; December 22 , 1 894, 300 ; January 1 6, 1 895, 600 ; and June 20,1 895, 400 . What is the proper date for a note to cover

al l these l oans29 5 . What is the average date of the fol l owing ao

coun t

Mr . T. I . BAKER TO I . DOWNER CO. ,Dr .

1 8 9 6 .

Ap ril 1 0 To Mdse on 30 days’credit

May 1 6( C i f 60 i t a

J une 3 30

J uly 1 8

NOTE . First find at what time each item fal ls due by adding the

time of credit to the date Of the item.

296 . What is the equated date of maturity of the fol l owing items

Mr. W. I . HAMMOND To PENNSYLVANIA IRON 00 Dr.

286 IN TEREST.

In this case the bal ance of interest l ost woul d be in favor of the

creditor as before, but the bal ance of the account, $ 1 00, is owed by

the debtor, and so he may be required to pay it a sufficient l ength

of time before March 1 8 for the if 1 00 to earn of interest, which

is 1 25 days . One hundred and twenty-five days before March 1 8 is

Nov. 1 3. Ans . Nov . 1 3, 1 894 .

Observe in this case that the balances of the account and of the

interest are on opposite sides of the account.

4 77 . From these il lustrations is derived the fol l owing

Rul e.

To find the average or equated time for the settl ement Of

an account when there are both debit and credit items1 . F ind the interest on the severa l items of the account

from the earl iest date at whi ch any item becomes due to their

severa l maturities.

2 . Find the ba lance of interest of the debit and credit sides

of the account, a lso the ba lance of the items.

3 . D ivide the ba lance of interest by the interest of the

ba lance of the items for one day. The quotient wi l l be the

time in days between the selected date and the average time ofsettlement.

4 . Count this time FORWARD from the selected date, when

the ba lances of the interest and of the account are on the same

side of the account, and BACK when on opposite sides. The

result wi l l be the date of settlement.

NOTE I. When at the time of the averaging of an account the

average time for paying the balance is already past, interest is al l owed

on the bal ance .

NOTE II.— Twelve per cent is a rate of interest much used in

averaging accounts. It is convenient because it is equivalent to l percent a month.

Nora 1 1 1 .— It is usual to disregard cents in the items of the

account, increasing the dol lars by 1 if the cents disregarded are 50

or more.

AVERAGE OF ACCOUN TS. 87


29 7 . At what date can the bal ance Of the fol l owing l edgeraccount be paid without l oss to either party

DAVID G . SAUNDERS.

1 8 9 5 . 0 t” 1 8 9 5 .

1 00 0 00 Aprtzu By Mate .

July 8 1 1 8 98 Aug . 1 0 Rea l Estate

298 . What is the average date of maturity for the fol l owing accoun t

Dr . BLADE BROS. 8c 00 .

1 89 6 . 0 7 1 8 9 6 .

Apri l 6'

To Sundries on 2 mo. 4 00 00 June 1 Mdse on 6 0 d .

Aug . 6 Mdsc 1 6 00 00 July 8 Mdsc 80

76 Mdse 1 2 00 00 Aug . Cash

299 . Find the average date of maturity of the fol l owingaccount :

HENRY DRAPEB GO.

1 8 94 . 0 t 1 8 9 4 . [on 9 0 d . 8 aJ an. 6 To Mdse on 30 d. 6 00 00 J an . 1 By Rea l Estate 6 00 00

Feb. 7 so 4 2 0 00 Mar . 1 6 Cash sea 00

300 . Average the fol l owing account

Dr . CHARLES BATES.

1 89 6 . 8 it 1 8 9 6 .

Aug . 2 0 ToMdse, 60 d. 1 73 1 6 Aug . 25 By Mdse, 80 d.

Oct. 1 4 Cash 3 1 4 68 Sep t. 1 2 Mdse, 801 8 Cash 280 0 0

80 Mace, 1 mo. 81 26'

288 IN TEREST.

Average the fol l owing account

WILLIAMS, BIOE 00 .

1 8 9 5 . 6 f” 1 8 9 5 .

J an . 6 To Mdse, 3 mo. 83 9 9 2 J an . 1 By Bal . of acct.

2 6 Mdse, 8 0 d . 6 82 2 0 1 6 Rea l Estate,

Feb. 2 1 Mdse, 8 mo . 85 1 2 3 mo .

Mag 2 9 32 00 00 Feb. 7 Mdse, 2 mo.

479 . Mi scel l aneous Exampl es .

Find the amount of a $ 400 note given the 26th ofAugust, and paid the 29 th of the next December, interest8%per annum.

303 . A trader borrowed $ 8000 at 5%per annum. He

used this money in a way to yiel d him 35 during theyear. How much was the cl ear gain to him on the bor

rowed money304 . A due-bil l is given for with interest atWhat w il l the bil l amount to if it runs 72 days305 . A note given May 7 , 1 895, on demand at 6%for

$ 825, was indorsed July 6 , 1 895, 1 00 ; Oct. 9 , 1 895, $ 1 00.

What sum remained due April 1 , 1 896306 . J an. 1

,1 895, I gave a note for $ 7 1 2 at When

wil l this note amount to 800

307 . The use of money being worth 5%a year, what isthe present value of $4000 due four years hence308 . If a merchant sel l s coal at 5 per ton, and has to

wait 6 months for his pay, at what price could he afford tosel l for cash, money being worth to him a month309 . Write a note on 60 days’ time for such a sum that,

when discounted by a bank at the proceeds may be8 600.

31 0 . What is the amount,at compound interest, of $ 1 00

from April 1 , 1 895, to J an . 1, 1 898, at 6%per annum,

interest payabl e semi-annual ly

SECTION XV .

PROP ORTION .

480 . I llustrative Examp le. Two persons bought cl oth of

the same kind. One bought 9 yards for 1 5 dol lars, and theother 1 2 yards for 20 dol l ars. Were the amounts of money

paid proportional to the quantities of cl oth boughtTo answer this question we ask, first, what part 9 yards is of 1 2

yards and then what part 8 1 5 is Of 8 20 . Finding that 8 1 5 is the same

part of 8 20 that 9 yd. is of 1 2 yd. , we say that the amounts of moneypaid were proportiona l to the quantities of cl oth bought . We maysay, too, that the quantities of cloth bought were proportional to the

amounts of money paid.

481 . If four numbers, as9 yd. 1 2 yd. 1 5 20,

are such' that the first is the same part Of the second thatthe thi rd is of the fourth, the numbers are proportional .

The l ast two numbers are proportional to the first two,and the first two to the last two.

482 . To find what part one number is of another, thefirst number is divided by the second. The div idend and

the divisor in such a case must be numbers of the same kindor denomination, and the quotient is cal l ed their ratio.This is the measuring form of division (Art.

The ratio of two numbers is a number expressing the relative magnitude of one number as compared with or measured by the other.

In the above exampl e, what is the ratio Of 9 yd. to 1 2 yd.What is the ratio of 1 5 to $ 20

ORAL EXERCISES. 291

The terms of a ratio are the two numbers comparedor measured the one with the other. One is the antecedent

(dividend) and the other the consequent (divisor) . Theyare usuall y written with a col on to indicate division ; thus,

9 yd. : 1 2 yd.

The terms of a ratio must be numbers of the same kind or

denomination .

484 . A proportion is an equal ity of ratios. The proportion which the two equal ratios above mentioned form is

9 yd. 1 2 yd. $ 1 5 $ 20,

and is read, Nine yards is to twel ve yards as fifteen dollars is to twenty dol l ars.

”

485 . a . What is the ratio of 4 pounds to 1 4 poundsOf 1 6 cents to 56 cents ? What proportion, then, can bewritten ?b. Two men work, one 1 2 days for $ 2 1 , and the other 4

days for 7 . What is the ratio of 1 2 days to 4 days ? Of

8 2 1 to 7 Is the pay proportional to the time ? Whatproportion can be written to express thi sc. Read these proportions

4 lb. 1 4 lb.

1 2 days 4 days $ 2 1 $ 7 .

1 0 ft. 30 ft. 25 ft. 75 ft.35 99 1 65.

30 mil es 75 mil es 6 hours 1 5 hours.d. How many terms are there in a proportion ? Whichof these are antecedents Which are consequents ?e. Two men bought wheat. One bought 1 2 bushel s for

$8, and the other 30 bushel s for $ 24 . Were the amountsof money paid proportional to the quantities of wheatbought Why not

292 P ROP ORTION .

f Can a ratio be found between 7 bushel s and 2 1 daysWhy not ? Between 8 inches and 2 feet ? Between 8

inches and 24 inches ?

9 . Write any two equal fractions, asH» and H, and tel lwhether the terms (numerator and denominator) of the one

fraction are proportional to the terms of the other.

486 . The terms of equa l fractions are proportiona l , and

the f ractions themselves are equa l ratios. Hence the equation1 0 1 6

1 5 24

is another form,cal l ed the fractional form,

of writing a

proportion. The numerators are the antecedents and the

denominators are the consequents of a proportion, which isread “ 1 0 is to 1 5 as 1 6 is to

487 . As the antecedent of a ratio is the dividend and the

consequent the divisor, it fol l ows thatWhen the antecedent is mul tipl ied or the the ratio is multipl ied

consequent is divided by any number, by that number.When the antecedent is divided or the con the ratio is divided by

sequent is mul tipl ied by any number, that number.When both terms of a ratio are mul tipl ied

}the value Of the ratio is

or divided by the same number, not changed .

Oral Exercises .

a . Mul tipl y the ratio 2 : 3 by 2 . d. Divide the ratio by 2 .

b. Mul tiply the ratio 5 : 6 by 3 . 6 . D iv ide the ratio by 3 .

c. Mul tiply the ratio by 4 . f . Divide the ratio by 5.

9 . Change the ratio 1 8 : 30 to an equal ratio in smal l er

terms.

h. Change the ratio 4 : 9 to an equal ratio in l arger terms.

i . In what respects is a ratio l ike a common fraction

489 . Of the four terms of a proportion, the first and lastare cal l ed the extremes, and the second and third are cal led

the means.

294 P ROP ORTION .

Oral Exercises .

49 2 . In the fol lowing proportions find the value of theterm represented by x.

a . 6 : 4 = 1 5 : x. e.

b. 1 2 : 9 = x : 6 . f:

c. 27 : x : g. 1 2

d . x : 1 6 5 : 8. h. x : 4 hours = 1 5 mil es 1 2 mi l es.

NOTE. In Exercises e, f , g, and h, disregard the denominationswhil e finding the numerical value Of x, and afterwards apply to

value its proper denomination.

49 3 . Exampl es for W ritten W ork.

1 . Given 9 1 50 1 05 x, to find the val ue of x.

2 . Given 850 75 x 65, to find the value of x.

3 . Given 375 x 225 255, to find the value of x.

4 . Given x 1 45 9 29, to find the val ue of x.

Find the value and the denomination of x from the fol

l owing prOpertions :5 . 35 hats 200 hats x.

6 . 750 acres 3 acres x $ 1 1 4.

7 . 202 mil es x mi les inches inches.8 . 3§ inches 41} inches x 1 38 inches.

9 . $ 500.

1 0 . $ 400 x.

Appl i cations of P roporti on.

494 . I l lustrative Example . If 1 4 yards of cl oth cost 98cents, how much wil l 1 50 yards of the same cl oth cost

WRITTEN WORK.Letx represent the

cost of 1 50 yards .

1 4 yard

zs. 1 5

f5grards 98 cents x cents

Then 98 and x are

X rtional to 1 44” 1 050 cents

1 4 and 1 50.

351 0-50 Now in the ratio

98 cents : x cents, the consequent is greater than the antecedent, because


1 50 yards of cloth cost more than 1 4 yards. Therefore the other ratio

is to be made with the consequent greater than the antecedent, thus ,

1 4 yd. 1 50 yd. The value of x is found to be 1050 cents . Ans .

From this exampl e is derived the fol l owing

Rul e .

To sol ve questions by proportion

1 . Make the number that is of the same denomination as

the answer the third term and the answer itself ; represented

by x, the fourth term.

2 . Determine from the statement of the question whether

the answer is to be greater or less than the third term.

3. Make the other two gi ven numbers the first and second

terms of the proportion, taking the greater number for the

second term if the answer is to be greater than the third term,

and the less number for the second term if the answer is to be

less than the third term.

4. Mul tip ly the third term by the second term, and divide

the product by thefirst term.

NOTE .— This rul e is often cal l ed The Rul e of Three, from the cir

cumstance that three quantities are given to find a fourth.


1 1 . If 1 7 yards of silk cost how much wil l 24yards cost1 2 . How much wheat can be bought for 250, when 1 5bushel s can be bought for1 3 . If feet of l ead pipe weigh 93 pounds, what is theweight of a piece measuring 84 feet1 4 . A farmer raises tons of hay on 2§ acres of l andAt the same rate of production, how many tons shoul d heraise on 1 23acres1 5 . A father’s age is to that of hi s son as 7 to 2 , and the

son is 1 4 years old. What is the age of the father

296 P ROP ORTION .

1 6 . If 400 pounds Of coal are required to run an engine1 2 hours, how many pounds wil l be required for 27 days of1 0 hours each ‘7

1 7 . If 35 men can dig a trench in 1 4 days, how manymen wil l it take to dig it in 5 days1 8 . The amount Of 1 on interest for 3 y . 9 mo . 1 2 (1 .

being how much money put on interest for the

same time wil l amount to 1 000

1 9 . A force of 220 laborers wil l cl ear the streets of snowin 8 hours. How many men wil l it take to do the same

work in hours20 . Ral ph and Arthur compare their speed in rowing,

and find that Ralph can row 480 yards whi l e Arthur rows500 yards. If it takes Arthur hours to cross a l ake, howl ong wil l it take Ral ph2 1 . A has a horse that trots 8 mil es whil e B’

s horse trots7 mil es. Both horses are driven over the same road, A’

s

horse doing the distance in 2 h. 20 m. How l ong does ittake B’

s horse22 . The rates Of two steamboats are as 1 1 to 1 3 . If the

faster boat makes her trip in 58 hours, in how many hoursdoes the other make the same trip23 . A contractor requires 64 days to pave a street when

his men work 9 hours a day. How many days wil l herequire when his men work 8 hours a day24 . Turn to page 9 1 , and do Exampl es 1 3 and 1 4 by pro

portion .

Additional exampl es which can be done by proportion may be foundon pages 94, 1 1 0, and 259 . See also page 1 43, Example 78, and page

1 45, Exampl e 1 0 1 .

497 . Draw a triangl e, ABC, and across it draw a l ine DEparal l el to the side BC. Measure accurately AD and AB,

al so AE and AC. Find the ratio of AD to AB, al so thatof AE to AC. Are these two ratios equal They should

298 P ROP ORTION .

found equal ? They shoul d be, for if two triangles aremutual ly equiangular their corresponding sides are proportiona l .

That is,AB : A'B'

BC : B'C' CA : C’A’.

These are simil ar triangl es.

499 . Draw two triangl es, one with sides measuring, say,1 8

,24

, and 27 units, and the other with sides measuring 42 ,56, and 63 units. Are the sides of these two triangl es proportional Measure the angl es. Are the triangl es mutual ly equiangul ar They should

,be, for if two triangles have

their corresponding sides p roportional , they are mutua l lyequiangular. These are simil ar triangl es .

500 . Simila r triangl es have these two properties

( 1 ) Their corresponding angles are equa l ;

(2 ) Their corresponding sides areproportiona l .

Fromwhat has just been said, it appears that triangl es having one

of these properties have also the other ; so that similar triangles maybe recognized either by their mutual ly equal angles, or by their proportional sides.

2 7 . There are two simil ar triangl es, one with sides measuring 1 1 , 8, and 7 inches, and the other with its l ongestside measuring 33 inches. Find the other two sides. Drawthese triangl es to a convenient scal e.

28 . The l ongest side of a triangl e is 87 ft. , and thetriangl e is simil ar to another whose sides are 6 ft. 8 in.,

7 ft.

, and 9 ft. 8 in. Find the other two sides of the firsttriangle.

2 9 . The sides of a triangl e are 39, 52, and 65 inches inl ength. The l ongest side of a simil ar triangl e measures95 inches. Find the other two sides.30 . A right triangl e with base 90, perpendicul ar 48,

and hypotenuse 1 02, is crossed by a straight l ine paral l el


to the base whi ch cuts off J} of the perpendicul ar. Findthe two parts into whichthe hypotenuse is div ided.

Find the l ength DE,of the

paral l el l ine itsel f.31 . Is the triangl e whosesides are 27, 36, and 45

similar to the triangl ewhose sides are 42 , 56, and 70 ? Why32 . The sun is shining at the same moment on two

upright objects, a tree and a stake for exampl e, whi ch castshadows on l evel ground. Imagine a l ine drawn from the

top of each object to the end of its shadow. These l inesrepresent the direction of the sun

’s rays

, and are para l l el .

With the upright Objects and their shadows they form twosimilar triangl es. If the shadowsmeasure 95 ft. and 90 ft.,and the stake is 7 ft. high, how high is the tree33 . What is the height of a church Spire which casts a

shadow 238 ft. l ong, if at the same moment a cane, 35 in .

“

l ong, held vertical l y, casts a shadow 6 ft. 5 in. l ong

501 . Two pl ane figures are simil ar if a l l the l ines of one

arep roportional to the corresponding l ines of the other.

It is not enough that the two figures shoul d have the same shape inthe sense of having their angl es equal each to each ; they must have al l

their corresponding l ines proportional . Thus, a square and an Oblong

are not similar figures, for al though each has four square corners, their

800 P ROP ORTION .

breadths are not proportional to their l engths. But two squares are

similar figures. WhyTwo oblongs are similar if their breadths are proportional to their

l engths ; that i a, ifAB : A'B ’ AD : A'D ’

.

Their diagonals , AC and A’C’, are also proportional to their l engths

or to their breadths.

34 . An Obl ong 77 ft . l ong, 36 ft. wide, has a diagonal 85ft . l ong . Find the l ength and the diagonal of a similarobl ong which is 42 ft. wide35 . The pl an of a house shows a room which measures

1 6 in . in l ength by 1 0 in . in width. If the room itsel f is tobe 32 ft. l ong, how wide is it to be ? How l ong is thewhol e house to be, if it measures 42 in . on the pl an ? A

chimney which is to be 32 in. square wil l be shown by howl arge a square on the pl an ?

36 . Two maps of the same region are simil ar pl ane

figures. To test this take two maps of any region, say theNew Engl and States, one smal l

, the other l arge . Measure

the distances from Boston to P rovidence, from P rovidenceto N ew Haven

,from N ew Haven to Springfield, from

Springfiel d to Montpel ier, fromMontpel ier to Bangor, etc.,

first on the smal l er map and then On the l arger. Div ideeach distance measured on the l arger map by the corresponding distance measured on the smal ler map. You

should find the same quotient (ratio ) in each case.

3 7 . On a city pl an drawn to a scal e of 1 00 ft. to theinch, what is the breadth of a street 80 ft. widea Whatare the dimensions of a house l ot fronting 40 ft. on the

street and extending back 1 25 ft 9

38 . On a township map drawn to a scal e of one fourthof a mil e to the inch

,what are the dimensions of a farm

420 rods l ong and 200 rods wide39 . On a map drawn to a scal e of 40 mil es to the inch,

what woul d be the distance from Boston to Al bany, whichis about 200 mil es

302 P ROP ORTION .

504 . Two sol ids are simil ar if al l the l ines (edges,diagonal s, etc. ) of one are proportional to the

ing l ines of the other.

It is not enough that the two sol ids shoul d have the same shape in

the sense of having their corners al ike, as a cube and a brick, but the

corresponding l ines must be proportional .A cube and a brick are not similar, because the length, breadth, and

thickness of the former are not proportional to the length, breadth,

and thickness of the l atter. But two cubes are similar sol ids. Why

47 . A common brick is 8 in. l ong, 4 in. wide, and 2 in .

thick. Find the l ength and width of a stone 1 ft. 2 in.

thick and simil ar to a brick in shape.

48 . Is a rectangul ar bl ock of marbl e with dimensions4 x 1 3. x 1 71{ ft. similar to one with dimensions 5 x IQx 1 }ft. If not, change the width and thickness of the firstbl ock to make it simil ar to the second.

49 . Amonument is 1 80 feet high and 25 feet square at

the base. If a model Of the monument is made 1 5 incheshigh, how l arge is its base50 . The three edges of a pyramid meeting at the top are

each 1 5 in . in l ength . The base of the pyramid is a

triangl e, each side of which measures 6 inches. Find thecorresponding l ines of a pyramid similar to this, but higherin the ratio of 5 to 3.


51 . The heights of two simil ar cones are as 4 to 2 . The

circumference of the of the larger one is 1 4 inches.

What is the circumference ofthe base of the smal l er one

52 . The diameters of two simil ar cyl inders are as 1 0 to 5.

If the height Of the l arger is 24 inches, what is the heightof the smal l er

53 . Suppose the diameters, CD and C'D', Of the bases of

two simil ar cones (see figures above) to measure 26 inchesand 1 3 inches respectively, and suppose the sl ant heightCA of the larger cone to measure 40 inches ; what doesthe sl ant height C

'A'of the smal l er cone measure

304 P ROP ORTION .

PARTNERSHIP OR DISTRIBUTIVE PRO PORTION.

505. I l lustrative Examp le. Three men, A,B, and C,

form a partnership, agreeing to share profits or l osses in

proportion to the parts of the capital put in by them.

A put in B put in and C put in

They gained What was each partner’s share

WRITTEN WORK .

s a A’s share.

The proportions

B’s share . are formed on the

a s C’s share.

P l u mp“ that the

whol e capital is to

A’s share x the P3“ Put in by

one partner as the

927 x WhOl e gain is to thatB’s share

N a h um share of

x4 1 22 4C 8 share .8

$ 927


54 . A and B formed a partnership, A putting in 6000,

and B 8000 . They gained 4431 . What was the shareOf each55 . Three merchants

,A,B, and C, shipped goods from

Cal cutta to N ew York by the same vessel . The val ue Of

A’s part of the cargo was $ 3650 ; of B’

s, $ 8265 ; and

of C’

s, 9840. During a storm a part of the cargo valued

at 6925 was thrown overboard. What was each merchant’s share Of the l oss56 . A bankrupt owed $ 950 to A, $ 875 to B, $ 1 260 to

C, and 1 000 to D . His whol e property was sold for

of which was used to pay the expensesof the sal e and settl ement. What was each person

’s shareof the remainder

306 P ROP ORTION .

The distribution is to be made on the double basis of money andtime. The use of $ 2500 for 8 months is equival ent to the use of

for 1 month ; the use of 0 3200 for 1 0 months is equivalentto the use of for 1 month ; and the use of 32000 for 1 2

months is equivalent to the use of for 1 month . There

fore, the distribution should be made in proportion to 3and or, what woul d be the same, in proportion to

the numbers 20, 32 , and 24 .

Am. H’s share , B’

s, 3 780 ; C’s, $585.

63 . X, Y, and ! engaged in a land specul ation, agreeingto share the profits in proportion to the money investedand the time it remained invested. X put in and

withdrew it . at the end of 3 years ; Y put in

and withdrew it at the end of 4 years ; and ! put in

which remained til l the end of the 6th year, whena distribution of profits amounting to was

made . Find each man’s share of the profits.

64 . Two men,A and B,

received 2500 for buil ding a

road. A furnished 1 0 laborers for 48 days and 5 horsesfor 42 days

, and B furnished 8 men for 45 days and 4

horses for 30 days. Supposing a day’s l abor of a man

as compared in val ue with that of a horse is as 7 to 5, whatshoul d be the share of the money for each contractor65 . Four men hired a pasture the whol e season for 500.

The first put in 4 horses for 20 weeks, 3 cows for 1 2 weeks,and 1 0 cal ves for 24 weeks ; the second put in 5 cows for1 6 weeks ; the third put in 2 horses for 1 5 weeks ; and thefourth p it in 6 cows for 1 8 weeks and 1 5 calves for 2 1weeks . Supposing that a cow eats 4 times, and a horse6 times, as much as a cal f, how much shoul d each man payfor the animal s he pastured

SECTION XVI .

P OWERS AND ROOTS .

INVOLUTION.

508. Il lustrative Examp les. What is the product of thetwo equal factors 3 x 3 ? of 5 x 5 ? of 7 x 7 ? of 9 x 9 ?

of 1 0 x 1 0What is the product of the three equal factors 2 x 2 x 2

of 3 x 3 x 3 ? of 4 x 4 x 4 ? of 5 x 5 x 5 ?

What is the product of the four equal factors 2 x 2 x 2 x 2of 3 x 3 x 3 x 3 ? of the five equal factors 2 x 2 x 2 x 2 x 2 ?

509 . A power is a product of equal factors. The productof two equal factors is a second power; the product of threeequal factors is a third power ; the product of four equalfactors is a fourth power ; and so on.

What is the second power of 3 the third the fourth the fifth

What is the thi rd power of 5 the fourth the fifth

What is the second power of 1 0 the third the fourth

NOTE .— The names square and cube are often used instead of

secondp ower and thirdpower, because the area of a square is foundby multipl ying together two equal factors ( length and breadth) , andthe volume of a cube is found by mul tipl ying together three equalfactors (l ength, breadth , and height) .

51 0 . A power is indicated by writing one of the equalfactors with a smal l figure, cal l ed the exponent , just aboveand to the right of it.

Thus, 53 indicates and is read “ the third power of or

t ‘5 to the

third power,”or

‘t the cube of or“ 5 cube .

”

307

308 P OWERS AND ROOM .

51 1 . Involution is the process of finding the power of anumber, or raising a number to a power. It is performedby mul tipl ication .

Oral Exercises .

51 2 . a . Give the second powers (squares) of al l the integral numbers from 1 to 1 0 incl usive.

b. Give the third powers (cubes) of al l integral numbersfrom 1 to 1 0 inclusive .

c. What is the square of .3 of of of g?d . What is the square of of ofe. What is the cube of 5 ? of of of g ?

j : What is the cube of of of

9 . What is the square of 20 of 40 of 90 ‘7

h. What is the cube of 1 0 of 30 of 50

Examp l es for W ri tten W ork .

51 3 . Find the powers of the fol l owing numbers as in

dicated

5 . 92 9 . 1 3 . 1 7 .

z. 273. e. 992 1 0 . 1 4 . 1 8 .

7 . 9992. 1 1 . 1 5 . 1 9 .

4 . s. 993. 1 2 . 1 6 . ao. ( l .o6)t

NOTE.— The resul ts of the l ast four exampl es read as dol lars are,

respectively , the amounts of 1 dol l ar at compound interest at 5%for 3years, at 3%for 4 years, at 6%for 2 years, and at 6%for 5 years.

See the tabl e (p .

EVOLUTION.

51 4 . One of the equal factors which mul tipl ied together

produce a number is a root of that number.

One of the two equal factors of a number is the second

root , or square root ; one of the three equal factors is the

1 0 P OWERS AND ROOTS.

SQ UARE ROOT .

51 8. To find the number of figures required to express the

square root of a number.

What is the smal lest integral number expressed by one figure

What is the square of this number What is the largest integra lnumber expressed by one figure What is the square of this num

ber Any number from 1 to 81 is expressed by how many figuresIf, then, a root is expressed by one figure, by how many figures is its

square expressedWhat is the smal lest and what is the l argest integral number

expressed by two figures What are the squares of these numbersAny number from 1 00 to 980 1 is expressed by how many figures If

,

then ,a root is expressed by two figures, by how many figures is its

square expressedIn a simil ar way it can be shown that if a root is expressed by three

figures, its square is expressed by five or six figures ; and that if the

root is expressed by four figures, the square is expressed by seven or

eight figures, and so on indefinitely.

In general , therefore , thefigures Of a number are twice as many, or

One l ess than twice as many, as thefigures of its square root.Accordingly , if thefigures Of a number are separated intoperiods of

twofigures each, beginning with the units’

figure, there wi l l be asmanyOf theseperiods as the square root Of the number hasfigures .

NOTE.— The l ast ( l eft-hand) period may have one or two figures.

Howmanyfigures has the square root of 1 1 56 of of

5 1 9 . To find the parts of the second pow er of a number.

To find what parts the second power of a number is madeup of, raise a number, 34 for exampl e, to the second powerthus

34 30 4 t u

34 30 4 t+ a

1 36 1 20 1 6 tu as

1 02 900 1 20 t2

tu

1 1 56 t2+ 2 tu + u

3

The work in the second form is done without carrying, and so shows

al l the partial products obtained by mul tiplying each term (Art. 32)

SQUARE ROOT. 31 1

of the mul tipl icand by each term of the mul tipl ier. The work at the

right is in algebraic form, t and u standing for the tens and the units

of any number whatever.

The product in either case shows that the second power of any num

ber consisting of tens and units is made up of three parts

( 1 ) the square Of the tens,

(2 ) twice the product of the tens by the units,

(3) the square of the units.

Formul a

520 . To extract the square root .

I l lustrative Examp le. (1 ) What is the square root of1 1 56

The figures of the given numberWRITTEN WORK being separated into periods of two

t’ 2 tu u”

(34 figures each, show that the square52 root consists of tens and units .

6 256If the root is represented by t + u,

its square, or the given number, wi l lbe represented by t2 2 t u u2

2 t uand the values of the several parts of

2 tu as

this formula are to be sought in the

given number.Since the square of the tens Of the root, P , is hundreds, the 1 1 hun

dreds oi the given number contains the square of the tens of the root.

The greatest square in 1 1 (hundreds) is 9 (hundreds) , the square rootof which is 3 (tens) . That is,

t2 = 9.00 , t = 30.

So the first fig ure of the root is 3.

Subtracting 900 from the given number, and t2 from the formula,

gives the equal remainders2 t“ u2 256 ,

which, by separating the first into factors,‘ may be written

(2 t u)u 256 .

t'It is easy to see that if 2 t u is mul tipl ied by u, the product

Wil l be 2 tu “3.


If the value of the first factor, 2 t u, were known, 256 divided byit would give the second factor, u . But 2 t is much the larger part of

the first factor and is known ; so it can be used as a trial divisor forfinding a probable value of u, which can be verified afterwards.

Dividing 256 by t) gives 4 . So 4 is the second figure of theroot, provided the remaining terms of the formula, 2 t u u”, do not

exceed 256 .

But 2 t 60

and u 4,

adding, the sum is 2 t u 64,

which mul tipl ied by u 4

gives 2 t u u2 256 .

So the value of u is verified . The true divisor, 2 t u) ,mul tipl ied by the quotient, 4 ( u) , gives a number not exceeding 256 .

"t

Since there is no remainder, the given number is a perfect square,and 34 is its square root.

Ans . 34 .

Examp l es for W ri tten W ork .

52 1 . Find the square root

2 1 . 1 681 . 2 4 . 1 936 .

2 2 . 2704 . 2 5 . 372 1 .

2 3 . 324 . 2 6 . 84 1 .

Find the square root of the l argest square in each of the

fol l owing numbers ; and find the remainder :

33 . 730 . 3 7 . 1 450. 4 1 . 2450 .

34 . 4300 . 38 . 5800. 42 . 7760 .

35 . 300 . 39 . 1 770 . 43 . 9900 .

36 . 250 . 40 . 360 . 44 . 490 .

it If the value of u had proved too l arge, it would have been diminiehed, and the process of verification repeated.



524 . Find the square root of49 . 81 225. 52 . 3841 6 . 55 . 39526369 .

50 . 1 1 9025. 53 . 767376 . 56 . 52345225.

51 . 242064 . 54 . 327 1 84 . 57 . 851 55984.

525. From the foregoing exampl es and expl anations maybe derived the fol l owing

To find the square root of a number

1 . Beginning at the units’

p lace, separate the figures Of the

given number into p eriods Of twofigures each.

2 . F ind the greatest square in the number expressed by the

left hand period, and write its square root as the first term Ofthe root.

3 . Subtract this square f rom the corresponding part Of the

given number, and with the rema inder unite the next two terms

Of the given number for a dividend.

4 . Double the part Of the root a l ready found for a trial

divisor, and by this,regarded as tens, divide the dividend,

writi ng the quotient as the next term Of the root, al so add ing it

to the tria l divisor to make the true di visor.

5. Mul tip ly the true divisor by the quotient and subtract the

product from the dividend . (If this product should be too

large, make the quotient less and repeat the work. )6 . With the remainder lastfound unite the next two terms

Of the given number for the next dividend, take double thepart

Of the root nowfound, regarded as tens, for a trial divisor and

proceed as before.

NOTE 1 .— The last step is to be repeated until al l the terms of the

given number have been used, or until the root has been found to any

desired number of decimal places. In the latter case two zeros are

annexed to each successive remainder,

SQUARE ROOT. 31 5

Non : 2. When a dividend does not contain a trial divisor, place

a zero as the next figure of the root ; place also a zero at the right of thetrial divisor ; for the next dividend unite the next two terms of the

given number to the previous dividend, and proceed as before .


526 . Find the square root of58 . 2039 1 84 . 63 . 68 .

59 . 1 036 1 96 1 . 64 . 69 . 6457

60 . 81 432576 . 65 . 70 . 65 51 2836 .

6 1 . 94225849 . 66 . 71 .

62 . 49 1 1 2064 . 6 7 . 0 43652449 . 7 2 .

527 . I l lustrative Examp le. Find the square root of 1 5.

WRITTEN WORK .

544

767) 56005369

7742 ) 231 001 5484

7744) 761 60

69696

64640 Ans.

61 952

26880

528 . Find to seven pl aces the square root of73. 40. 7 7 . 5. 81 . 85 .

78. 8. 82 . 86 .

79 . 1 0. 83 . 87 .

76 . 3 . so. 20. 84 . 88.

After four figures of the root have beenfound by the rul e, three more figures canbe found by l ong division . This is because

the trial divisor, 2 t 77440 , has now

become so l arge in comparison with thequotient, u 9 , that the error committedby using the trial divisor instead of the

true divisor does not affect the quotientfor at l east three places.


529 . The square root of a common fraction can be foundby taking the square roots of both numerator and denominator.

NOTE .— If the denominator is not a perfect square , the fraction

should be changed to an equival ent fraction the denominator of which

is a perfect square . Thus, 3should be changed to g, and 4 to it

530 . Find the square root of

as. as. 9 2 . 9 5 . s. 98 . 1 2g.

90 . 93 . i tgg. 9 5 . g. 99 . 1 2g .

9 4 . H. 9 7 . 33. 1 00 . 1 63g-g.

531 . App l i cations of Square Root .

1 0 1 . A square garden contains square feet. Howmany feet l ong is each side1 02 . There are howmany rods in each side of a square fieldcontaining 6724 square rods ?1 03 . How many feet are there in each side of a square

court covered by pav ing bl ocks, each 8 inches square1 04 . Find the side of a square that shal l contain as many

square feet as an obl ong measuring 420 by 1 05 feet.1 05 . There are four obl ongs measuring 24 by 2 1 6 , 48 by

1 08, 54 by 96, and 2 7 by 1 92 feet respectivel y. F ind thedimensions of a square which is equal to each of the obl ongs,and of a square which is equal to al l four of them together.

1 06 . How many rods must one walk to go around a ten

acre l ot of l and l aid out in square form ?

1 07 . There are 272 i square feet in a square rod. How

many feet are there in the side of a square rod ?

1 08 . Find how many feet of fence it requires to surroundan acre of l and l aid out in square form.

1 09 . Find how many feet of fence it requires to surroundan acre of l and l aid out in Obl ong form, the l ength being tothe breadth as 3 to 2 .


OU BE ROOT .

533 . To find the number of figures required to express the

cube root of a number.

What is the smal l est number expressed by one figure The largest

What are the cubes of these numbers Any number from 1 to 729 is

expressed by how many figures If, then, a root is expressed by one

figure, by how many figures is its cube expressedWhat is the smal lest number expressed by two figures The largest

What are the cubes of these numbers it Any number from 1 000 to

970299 is expressed by how many figures If, then, a root is ex

pressed by two figures, by how many figures is its cube expressed

In general , thefigures Of a number are thrice as many, or one or

two less than thrice as many, as thefigures of its cube root.Accordingl y, if thefigures of a number are separated into periods of

three figures each, beginning with the units’

figure, there wi l l be as

manyperiods as the cube root of the number hasfigures.

534 . To find the parts of the third pow er of a number.

To find what parts the third power of a number is madeup of, raise a number, 34 for exampl e, to the thi rd power,thus

900 +

34 30 4

4624 3600 960 64

3468 27000 7200 480

39304 27000 1 0800 1 440 64

ig+ 2 l u -l-u

2

t u

t2u 2 tu2 u

8

is 2 tzu tu2

+ 3 t2u -3 tu” -u

8

The work in the second form is done without carrying, and

shows al l the partial products obtained by mul tipl ying each termthe mul tipl icand by each term of the mul tipl ier.

CUBE ROOT. 31 9

The work in the third form is algebraic, t and u standing for the

tens and units of any number.

The product in either case shows that the third power of any

number cons isting of tens and units is made up Of four parts

(1 ) the cube of the tens,

(2) three times theproduct of the square of the tens by the units,

(3) three times theproduct Of the tens by the square of the units ,

(4) the cube Of the units.

Formul a : (t+ u)3= ta 3 t’ u 3 tu2 + u

3.

535. To extract the cube root .

Il lustrative Example. (1 ) What is the cube root of39304

WRITTEN WORK.

t3+ 3 t2u + 3 tu

2+ u

3

t3

3 ti — 2700 1 2304

ui

3 t2 + 3 tu + u2

3 t2 u + 3 tu3+ u

3 1 2304

If the root is represented by t u, its cube , or . the given number,wil l be represented by t

a 3 t2 u+3 tu2 us, and the values of the

several parts of this formul a are to be sought in the given number.

Since the cube of the tens, t3, is thousands, the 39 thousands of the

given number contains the cube of the tens of the root.

The greatest cube in 39 (thousands) is 27 (thousands) , the cube

root of which is 3 (tens) . That is,

$3 27000, and t 30.

So the first figure of the cube root is 3 .

Subtracting 27000 from the given number, and ts from the formula,gives the equal remainders,

8 t9 u + 3 tu3 + u3 1 2304

The figures of the

given number being

separated into periods

of three figures each ,

show that the cube root

is expressed by two fig

ures ; that is, consistsof tens and units.

320 P OWERS AND ROOTS.

which, by separating the first into factors ,“may be written

(3 t“ 3 i n u2) u 1 2304

If the value of the first factor, 3 t2 3 tu u‘l,were known, 1 2304

divided by it woul d give the second factor, u. But 3 ta is much the

largest part of the first factor, and is known ; so it can be used as a

trial divisor for finding a probabl e val ue of u, which can be verified

afterwards.

D ividing 1 2304 by 3 t? ) gives 4. So 4 is the second figure of

the root, provided the remaining terms of the formula, 3 t2u+3 tu2+u”,do not exceed 1 2304 .

But 3 t? 2700

and 3 tu 360

and u? 1 6

adding, the sum is 3 t2 3 tu u2 3076

which mul tipl ied by u 4

gives 3 t2 u 3 ta2 u8 1 2304.

So the value of u is verified. The true divisor, 3076 (=3 t9+3 tu+mul tipl ied by the quotient, u) , gives a number not exceeding

1 2304 .I

Since there is no remainder, the given number is a perfect cube , and34 is its cube root. {730304 34. Ans. 34.


536 . Find the cube roots of

1 34 . 1 2 1 67 . 1 37 . 226981 .

1 35 . 32768. 1 38 . 274625.

1 36 . 1 57464 . 1 39 . 38901 7 .

i t It is easy to see that if 3 te 3 tu u2 are mul tipl ied by u, the

product wil l be 3 t? u 3 tu‘t u”.

i If the value of u had proved too l arge, it would have been diminished, and the process of verification repeated.


2 . F ind the greatest c ube in the number expressed by the

left-hand period, and write its cube root as thefirst term Of the

root

3 . Subtract this cube from the corresponding part Of the

given number, and with the rema inder write the next three

terms Of the given number for a divide nd.

4 . Take three times the square Of the part Of the root a lready

found for a tria l divisor, and by this regarded as hundreds

divide the dividend, writing the quotient as the next . term Ofthe root.

5. TO the tria l divisor add three times the product Of the

tens by the units and the square Of the units Of the root for the

true divisor.

6 . Mul tip ly the true divisor by the quotient (units Of the

root) and subtract the p roduct from the dividend. (If this

product should be too large, make the quotient less and rep eat

the work. )7 . With the remainder last found write the next three terms

Of the given number for the next dividend, take three times the

square Of the part Of the root now found, regarded as hun

dreds , for a trial divisor and proceed as before.

NOTE 1 .— The last step is to be repeated until al l the terms of the

given number have been used, or until the root has been found to any

desired number of decimal places. In the latter case“

three zeros areannexed to each successive remainder.

NOTE 2 .— When a dividend does not contain a trial divisor, pl ace

a zero as the next figure of the root, place al so two zeros at the right

of the trial divisor for the next trial divisor ; for the next dividend

unite the next three terms of the given number and proceed as before.


539 . Find the cube root of

1 43 . 36 1 400569867 1 46 . 1 49 .

1 44 . 1 7 1 4 1 6328875. 1 4 7 . 5732349 1 0 443 . 1 50 . 725845560803.

1 45 . 95006081 547 . 1 48 . 7 46823. 1 51 .

CUBE ROOT. 323

540 . I l lustrative Examp le. Find the cube root of 1 5.

WRITTEN WORK.

1 456 5824

1 1 76000

1 09 1 94696

1 8243468 38693040

36486936

22061 040

1 8243468

Find to six places the cube root of

1 52 . 40 . 1 54 . 2 .

1 53 . 75. 1 55 . 4 .

542 . The cube root Of a common fraction can be found bytaking the cube roots Of both numerator and denominator.

NOTE . If the denominator is not a perfect cube , the fractionshould be changed to an equivalent fraction , the denominator of whichis a perfect cube . Thus, g should be changed to a , and 1

9, to

543 . Find the cube root Of

1 60 . 1 63 .

1 61 . m.

1 62 . mg. 1 65 .

By the omission of the formula,and of unnecessary figures, the workof finding a cube root may be writ

ten in this condensed form.

Afterfour figures of the root havebeen found by the rul e , two morecan be found by l ong division . This

is because the trial divisor 3 t21 8243468 has now become so l argein comparison with the quotient,that the error committed by using thetrial divisor instead of the true di

visor does not afiect the quotient forat l east two places. Ans.


1 69 . What is the l ength of one edge of a cube whi ch contains 1 25000 cubic feet1 70 . How l ong, wide, and high is a cubical pil e of woodcontaining 4 cords1 7 1 . Find the dimensions of a cube which shal l contain as

much as a common brick measuring 8 x 4 x 2 inches.

1 7 2 . Find the dimensions Of a cubical cistern which shal lhold 3000 gal l ons.

1 73 . A bin whose l ength, breadth, and depth are to be

proportional to the numbers 3, 2, and 1 is to be madecapabl e of holding 500 bushel s of grain . Find the dimansions.

1 74 . Acubic foot of water weighs 1 000 ounces. Find thedimensions of a cubical cistern that woul d hold 1 0 tons

lb. ) of water.

1 7 5 . Iron weighs times as much as water. Find thedimensions of a cube of iron that woul d weigh a ton.

1 7 6 . A bl ock of marbl e weighing a ton is as wide as itis l ong, and hal f as thi ck as it is l ong . Marbl e weighs

times as much as water. Find the dimensions of thebl ock .

1 7 7 . The number of cubic inches in a Sphere is found bycubing the

'

diameter, and mul tiplying this by i of thenumber Find how many inches there are in the

diameter of a gl obe that hol ds just 1 000 ounces of water.

1 78 . Gold weighs times as much as an equal v olumeOf water. Find thediameter of a sphere Of gol d that wouldweigh a ton (2000

1 7 9 . Find the diameter of a sphere of iron that woul dweigh a ton .

326 unnsvmTION .

550 . A l ine is the boundary or l imit Of a surface . It has

one dimension, l ength, but no breadth and no thickness.

Notice the edges of a geometric solid. These are l ines, bounding or

l imiting the faces (surfaces) .

551 . A straight l ine is the path Of a point moving con

stantl y in the same direction.

The point of your pencil moving against the edge of your ruler doesnot change its direction, and so makes a straight l ine.

A thread or a string hel d by the two ends and stretched tight is a

good il lustration of a straight l ine . How does a carpenter use a chalk

l ine, and what for

552 . A curved l ine or a curve is the path of a pointmoving in a constantly changing direction .

The point of the pencil attached to one leg of your compasses when

turning about the other l eg moves in a constantly changing direction,

and so makes the curve cal l ed the circumference of a circl e.

Another curve can be made in thisway . Stick two pins through the

paper securely into the board be

neath , at the points F and F ’. Take

a piece Of l inen thread somewhatl onger than twice the distance fromF to F' tie the ends together ; and,

with the point Of a pencil at P ,

stretch the thread tightly into a tri

angul ar shape . Now move the pencil

around, keeping the thread stretched

tight. The pencil wil l mark out an ell ipse .

Find examples of straight and of curved l ines among the edges of

geometric sol ids.

Cut a round piece of wood, l ike a broomstick or a rol l er, squareacross . What sort Of a curve is the edge Of the cut surface Cut it

obl iquely. What sort Of a curve is the edge of the cut surface

553 . A pl ane surface is touched by al l points of a

straight l ine laid upon it in any direction.

W ith your rul er test the top Of your desk to see if it is a perfect

pl ane. Test a warped board . How does the carpenter test the surfaceof a board he has planed

EXERCISES. 327

554 . Straight l ines which have the sanie direction are

paral l el l ines . Two paral l el l ines nowhere meet, howeverfar extended either way.

0 — D

Fro . 2 . Paral lel Lines.

555. When one straight l ine, AB, is met by another, CD,

two angles are formed. If the angl es are equal (Fig. 3)

Fro . 8. Fro . 4.

they are right angles , and the l ines are said to be perpendieul et to each other ; if unequal (Fig . they are Obl iqueangl es, the smal l er, BDC, being an acute (sharp) angl e, andthe l arger, C

'DA, an obtuse (blunt) angl e.

Exercises .

556 . a . Which is the l arger, a rightangl e or an acute angl e ? A right angl eor an obtuse angl e W ith a square drawa right angl e (Fig .

b. Draw a straight l ine, and anothermeeting it as in Fig. 4. With a protractor,(see p. measure the angl e BDC’.

Measure the angl e C'DA. What is their

sum 1 80°equal s howmany right angl es

c. Draw a straight l ine, AB (Fig.

and above it mark a point, E . With one edge Of the square

placed al ong the l ine AB measure the distance of the pointE from the l ine AB .

328 MENSURATION .

By distance from a point

to a straight l ine is always

meant the perp endicular dis

tance .

d. From a given point P

(Fig. 7 ) to draw a l ine per

pendicul ar to a given l ineAB.

With one point Of the

compasses placed at P , drawarcs cutting the given l ine

in two points 0 and D, conveniently far apart. Then withthe compasses open somewhatmore than hal f the di stancefrom C to D,

and from C and D

as centers, draw two short arcscutting each other at E .

A straight l ine drawn throughthe points P and E with a

'

rul er

wil l be perpendicular to the givenl ine . When the work is done,test its accuracy with a square .

e. At a given point P ,

in a given l ine AB (Fig.

to draw a l ine perpendicularto the given l ine .

With compasses, pl ace two

points 0 and D at equal distances from P in the given l ine.

From 0 and D as centers, andwith the compas ses Open aboutas far as from 0 to D,

drawtwo short arcs cutting each other at E. Draw with a rul er

a l ine through the points P and E . This l ine wil l beperpendicular to the given l ine . When the work is done,test its accuracy with a square .

330 MEN SURATION .

i . Draw a straight l ine ; and, from two points M and N

in it as centers (Fig. and withthe same Opening of the compasses,draw two short arcs as representedin the figure. P lace a rul er withits edge just touching these two

arcs and draw a straight l ine. This l ine wil l be paral l elto the first l ine.

j . Draw two paral l el l ines, AB and CD (Fig. and

across them a straight l ine EF. How many angl es are

Fro . 1 2 .

formed How many are acute How many are obtuseCut the paper al ong the l ines AB , CD,

and EF,and Show,

by l aying one angl e On another,that the four acute angl es

are equal to each other,

EGD EHB AHE CGF ;

and that the four obtuse angles are equal to each other,

B GE BHF AHE OGE.

k. Lay a rul er, AB (Fig . on the paper. Against itbring the wooden triangl e GHI, and sl ide it so that theedge GE may have successively the positions CD,

EF,

GH,etc .

,which are marked by drawing a pencil against

the edge GH in each position . As the rul er was not

changed in direction, each side of the triangl e, though

TRIANGLES. 331

moved,kept the same direction ; therefore the l ines CD,

EF,

GH, etc .,have the same direction, and are paral l el .

557 . A pl ane figure is a portion Of a pl ane surfacebounded by l ines either straight, or curved, or both.

558 . A polygon is a plane figure bounded by straightl ines .

What is the l east number Of straight l ines that wil l incl ose a

portion Of a plane What, then , is the l east number of sides a

polygon can have How does the number of sides in any polygon

compare with the number Of angles

559 . P olygons are cl assified ac cording to the number Oftheir Sides or angl es, into triangl es, hav ing three angl es

(or Sides) ; quadril ateral s, having four sides (or angl es) ;pentagons, hexagons , octagons, decagons, dodecagons, etc .

having respectively five, six, eight, ten, twel ve, etc.

, angl es

(or sides) .

TRIANG LES.

560 . Triangl es are cl assified accordingto their sides into equil ateral triangl es,having three equal sides ; isoscel es triangl es, hav ing two equal sides ; and scal ene

triangl es, hav ing no equal sides.

332 MENSURATION .

56 1 . Triangl es are al so cl assified according to theirangl es into acute-angl ed triangl es, having al l three angl es

acute ; right-angl ed triangl es, having one right angl e ; andobtuse-angl ed triangl es, having one obtuse angl e .

Exercises .

56 2 . a . Draw triangl es Of various forms . Notice in

each the position of the l argest angl e rel ativel y to thel ongest side, and the position of the smal l est angl e rel a

tivel y to the shortest side .

In any tri angl e the longer of two sides is opposite the larger

angle, and the larger of two angles is opposite the longer side.

b. If two sides of a triangl e are equal , what is true Of the

angl es Opposite them Draw such a triangl e on paper, cut

it out, tear Off the two angl es that are opposite the equal

sides, and compare them,thus l earning that

The angles of an isosceles triangle which l ie opposite the

equa l sides are equa l .

334 MENSURATION .

q. Can an equi lateral triangl e be either right-angl ed or

obtuse-angl ed

563. Any side of a triangl e may be taken for the base.

Then its al titude, or height , is the perpendicular distance ofthe vertex (top) from the base .

The al titude Of a triangle is best measured with a square, as in

Art. 556 , c. The perpendicul ar fromthe vertex, 0 ,may fal l on the base,

as in Fig. 25 , or on the prolongation of the base, as in Fig . 26 .

Fro . 25. FIG . 26.

a . Draw an isoscel es triangl e and the perpendicul ar fromthe vertex to the base . Cut out the triangl e and fol d one

part upon the other, making the crease come upon the per

pendicul ar, thus l earning that

The perpendicular from the vertex divides an isosceles tri

angle into two equa l right-angled triangles.

Q UADRILATERALS.

564 . Quadril ateral s are cl assified, according to the rel ativedirections of their sides, into paral l elograms , hav ing both

pairs of Opposite sides paral l el , Figs. 2 7, 28, 30, 31 ;

Fro . 28. Square .

trapezoids, having one pair of opposite sides paral l el ,

Fig. 29 ; and trapeziums , hav ing no paral l el sides, Fig . 32 .

QUADRILATERALS. 335

565. P aral l el ograms are either rectang ular (al l theirangl es right angl es) , Figs. 27 28

,or obl ique, Figs. 30, 31

and they are either equil ateral , Figs. 28, 31 , or not equil at

eral , Figs. 27, 30.

Fro . 31 . Rhombus.

566 . A square is a rectangul ar and equil ateral paral l el ogram.

567 . An oblong is a rectangular but not equilateralparal l el ogram.

568 . A rhombus is an obl ique and equil ateral paral l el ogram. This form is al so cal l ed a diamond.

569 . A rhomboid is an obl ique but not equilateral parallel ogram.

570 . Any side of a paral l el ogram may be taken for a

base. Then its al titude, or height , is the perpendicul ar distance between the base and the Opposite side . It is bestmeasured with a square, as in Art . 556, c.

571 . The base of a trapezoid is one Of its two paral l elsides. The al titude is the perpendicul ar distance betweenthem. It is best measured with a square .

572 . A diagonal Of a polygon is a straight l ine drawnbetween two corners not joined by a side.

573 . a . Can a triangl e have a diagonal Why not

b. How many diagonal s has a quadril ateralc. Draw a paral l el ogram and one of its diagonal s on

paper. Cut out the paral l el ogram,and cut it in two by the

336 MENSURATION .

diagonal . Lay one of the triangl es thus made upon the

other, and so l earn that

1 . T he opposite sides of a para l lelogram are equal .

2 . T he opposite angles of a para l lelogram are equa l .

3 . T he diagona l of a para l lelogram divides it into two

equa l triangles.

d. Draw any quadril ateral and one of its diagonal s.

See whether the sum of al l the angl es of the two tri

angl es is the same as the sum Of the four angl es Of the

quadril ateral . Then how many right angl es (or degrees)is the sum Of the four angl es Of a quadril ateral equal to

The sum of the four angles“

of a quadri latera l is equa l to

four right angles.

e. One angl e Of a paral l el ogram is Find the otherangl es.

j : Draw a paral l el ogram and its two diagonal s. Measurethe parts into whi ch the diagonal s divide each other, andso learn that

The diagona ls of a para l lelogram bisect each other.

g. Draw a rhombus and its two diagonal s . Cut out the

rhombus and Show by fol ding thatT he two diagona ls of a rhombus bisect each other at right

angles, and divide the rhombus into four equa l right-angled

triangles.

h. Draw any pol ygon. Draw al l the diagonal s that canbe drawn from one corner. This divides the polygon intohow many triangl es ? How does the number of triangl escompare w ith the number of Sides ? Try thi s with different polygons (four, five, six, etc.

, sides) , and l earn that, in

generah

T he number of triangles into which a polygon is divided by

diagona ls drawn from one corner is two less than the numberof sides.

838 MEN SURATION .

draw a regul ar dodecagon.

First draw a circl e and divide its circumference into six equal arcs as for a regular

hexagon , and then divide each are into two

equal parts, making twel ve parts in al l . Jointhe successive division points by straight

lines .

577 . The Sides of a polygon takentogether form its perimeter .

Fm 132 — M M Dodemson Which difiers l ess from the circumference of the circumscribed circl e, the peri

meter Oi the regular hexagon or the perimeter of the regular

dodecagon

If regul ar pol ygons Of 24 , 48, 96 , etc. , sides were to be drawn in

the circle , the perimeter of which of these would come nearest to the

circumference If the number Of sides Of the polygon became infinite,the perimeter would come infinitely near the circumference. Hence,A circl emay be considered to be a regular pol ygon with an infinite

number of sides.

AREA OF POLYG ONS.

578. P aral l el ograms . The area of any paral l el ogram isequal to that of a rectangl e Of the same base ;

and height.For i f we cut off a right triangl efrom one end, and put it on at the

other end, as in Fig. 36, we changethe form to a rectangl e hav ing thesame base and height as the parall el ogram, though we do not change

the area. But the area of a rectangle is equal to the productof its base and height. (Art . Hence,

To find the area of any paral lelogram, mul tip ly the baseby the height.

Fro . 36.

AREA OF P OLYG’ON S. 339

579 . Triangl es . A triangl e is hal f a paral l el ogram ofthe same base and height.For two equal triangl es may be

pl aced, as shown in Fig . 37, so as tomake a paral l el ogram. Hence

,

To find the area 0f a triangle,

multip ly the base by the height, and d ivide the product

by two.

580 . The fol l owing is a convenient rule for finding thearea of a triangl e when the three sides are given, but thereasons for it cannot be expl ained without more geometrythan can wel l be introduced here :To find the area of a triangl e when the three sides are

given : F ind ha lf the sum of the three sides ; from this ha lfsum subtract each side in turn ; multip ly together the three

rema inders and the ha lj l sum, and extract the square root oftheproduct.

Exercises .

581 . a . Cut out from paper, pasteboard, or a board, a

paral l el ogram. Measure the base and the height and findthe area.

b. U sing the same paral l el ogram as in the last exercise,take one of the other pair of Sides for a base,measure it andthe height, and find the area . Compare the resul ts. Shoul dthey be the same Whyc. Draw a paral l el ogramwith one side measuring 8 inches,

another 4 inches, and having an angl e of 50 degrees. Findits area .

d. Draw a triangl e w ith three sides of different l engths.

Measure the three bases (sides) and the three heights, andwith these find the area of the triangl e in three differentWays. Compare the three resul ts. Should they agree Why

340 MEN SURATION .

582 . Trapezoids . A trapezoidmay bedivided into two triangl eshav ing the parall el sides of the trapezoid for bases and thedistance between them for their height.Hence, To find the area of a trapezoid,

mul tip ly ha lf sum of the paral lel sides by the distance

between them.

583 . Draw a trapezoid ; find themiddl e

point Of each Of the non-paral l el sides ; andmeasure the distance between these two

points . How does this distance comparewith hal f the sum Of the paral l el Sides Of the trapezoid

The l ine j oining the two middle points of the non-paral lel

sides of a trapezoid is para l lel to the two para l lel sides and

in length is equal to half their sum. Hence,To find the area of a trapezoid, multip ly the distance be

tween the paral lel sides by the distance between the middle

points of the non-para l lel sides.

Fro . 89 .

584 . Trapezium. A trapezium can be divided by a

diagonal into two triangl es ; then the areas of these triangl es can be found separately and added.

Fro . 40. Fro. 41 .

Draw a trapezium, ABOD (Fig. and divide it into two triangles

by drawing the diagonal BD , and find their areas. D ivide the sametrapezium into two other triangl es by drawing the diagonal AO (Fig.

and find their areas. Compare the sum Of the areas of the first

two triangl es with the sum of the areas of the second two. Should

these two sums be equal ? Why

342 MEN SURATION .

590 . Exampl es for W ri tten W ork.

1 . How many square feet are there in a rhomboid whosebase measures 4} feet, and whose height is feet2 . Find the area Of a triangl e, the base measuring 28 feet

and the al titude feet .3 . What is the area of a right-angl ed triangl e, the Sides

which incl ude the right angl e measuring 1 40 and 1 60

feet4 . Find the area of a rhombus, the diagonal s Of whi ch

measure 40 and 60 feet.5 . How many square feet are there in a board 1 4 ft . l ong,

1 8 in . wide at one end, and 26 in. wide at the other

6 . There is a house-l ot with four straight sides, two of

which are paral l el , 90 feet apart, and measuring 1 1 8 and 1 29

feet. What is its val ue at 1 5 cents a square foot7 . Draw a quadril ateral , ABC’D ; draw the diagonal AC,

and draw perpendicul ars to it from the points B and D .

Compute the area,supposing AO 1 06 feet, the perpen

dicul ar from B 62 feet, and that from D 54 feet.8 . Compute the area of a triangl e whose sides measure

1 5, 1 8, and 23 yards.

9 . The four sides of a quadri l ateral , P QRS, measure as

fol l ows : P Q= 30 yards, QR= 28 yards , RS = 38 yards,and SP : 42 yards. The diagonal , PR,

measures 48 yards .

Find the area. (Art.

1 0 . Draw al l the different pl ane figures mentioned in theabove exampl es, Of any size you pl ease, make your own

measurements, and compute the areas . Whenever you can

see different ways Of measuring and computing the area ofthe same figure, use them,

and see how nearly the resul ts

agree . P erfectly accurate measurements woul d give equalresul ts ; but perfect accuracy in measurement is unattainabl e. Stil l

, the more accurate your measurements the l esswil l your resul ts differ.

CIRCLES. 343

CIRCLES.

59 1 . The ratio of the circumference to the diameter of acircl e, expressed to the nearest ten-thousandth, isHence the fol l owing

Formul as rel ati ng to the Circ l e .

1 . Circumference Diameter x

Circumference2 . D tel ame r

3. Circumference 2 Radius x

Circumference4 . R d3 ms

2 x

Norm.— Instead of the val ue or is much used. It

is accurate enough for ordinary purposes.

592 . The rul e in Art . 589 may be expressed thus

Circumference x Radius2

P utting in the place of the circumference its value as givenin Formul a 3, we have

Area 2 x Radius x x Radius.

2

Hence the formul as

5. Area Radius”x Diameter”x

6 . Radius

7 . Diameter

344 MEN SURATION .

593 . By using l etters in place of words, and the Greekl etter tr (pronounced pt) to represent the ratio of circumference to diameter, these formul as may be written

1 . C= 1 rD . 5. A. 1 rR2 = i 1 rD’.

2 . D 9 .

1 r

3. 0 : 2 1 rR.

4 . R_2 1 r

1 1 . What is the distance around a circular piece of groundwhich measures 200 feet ac ross the middl e1 2 . Find how far a carriage has gone when one of its

wheel s, measuring 3 feet 8 inches in diameter, has made1 500 revol utions.

1 3 . What is the distance across a circul ar piece of groundwhich measures 200 feet around ?1 4 . Find the inner and the outer circumference of a walk

7) feet wide running around a circul ar grass plat thatmeasures 1 50 feet in diameter.

1 5 . There are two circl es drawn from the same center ;the circumferences measuring 88 feet and 1 32 feet respec

tively . Find the width of the ring.

1 6 . What is the area Of a circl e 1 0 inches in diameter1 7 . What is the area of a circl e of 1 0 inches radius1 8 . What is the area of a circl e of 1 0 inches circumference ?1 9 . How many square feet of l and are incl osed by a cir

onlar race track measuring around the inner edge 1 mil e20 . Find the diameter Of a circl e whose area is 1 00 square

feet. Of another, whose area is 50 square feet . Of another,whose area is 25 square feet. How do the diameters Of thefirst and l ast compare

346 MEN 8 URATION

24 . In a circl e whose radius is yards, what is the areaof a sector of 72° Of 22

°30'

25 . If eight radii are drawn from the center of a circl eso as to form eight equal angl es, what is the area of eachsector if the circumference measures 1 00 feet26 . In a circl e whose radius is 1 5 inches, what is the

l ength of an arc of 60° 20°1 5

°

2 7 . In a circl e whose circumference is 40 inches, whatangl e at the center is subtended by an are measuring 5inches 8 inches 1 2 inches28 . The radius Of a circl e is 1 0 inches . What angl e at

the center is subtended by an arc of 1 0 inches ‘7

2 9 . The radius Of a circl e is 1 0 inches . How l ong is anarc of of of30 . An arc Of 1

°on the circumference of a given circl e

measures 1 inch . Find the diameter.

31 . An arc of 1 ° on the surface of the earth at the

equator measures statute mil es. From this find thecircumference and the diameter of the earth at the equator.

RIG HT TRIANG LES.

598 . In a right triangl e, the side Op

posits the right angl e is the hypotenuse,and the other two sides are the base andthe perpendicular. Fig. 43 .

599 . Suppose ABC (Fig. 44) is a right triangl e , whosesides are 3, 4, and 5 feet respectively. A square formed

upon the hypotenuse, AO, wil l containhowmany square feet A square formedupon the base, B0 , wil l contain howmanysquare feet A square formed uponthe perpendicul ar, AB, wil l contain howmany square feet IS the square uponthe l ine AC’ equival ent to the sum of the

p m, 44,two squares upon AB and

‘

BC

RIGHT TRIAN GLES. 347

Exercises .

NOTE . Figures drawn to a scale are recommended as a means of

making the fol lowing exercises clear.

600 . How does the square on the hypotenuse comparewith the sum Of the squares on the other two sides ineach Of the right triangl es whose sides are as fol l ows

a . 6, 8, and 1 0 d. 1 8, 24, and 30

b. 20, 2 1 , and 29 e. 33, 56 , and 65

c. 39, 52 , and 65 j : 39, 80, and 89

g. From one corner Of a square floor,measure 6 feet

al ong one side and 8 feet al ong the other side . N owmeasure across f rom the end Of the 6 feet to the end

of the 8 feet. How many feet is this l ast distance ? It

wil l be 1 0 feet if the corner Of the room is exactly square

and you have measured accurately.

h. Measure any distances you pl ease al ong the two Sl deSOf the square corner Of a floor ; then measure across fromthe end of one Of these distances to the end of the other.

See if the square Of this l ast distance is equal to the sum

Of the squares Of the first two distances.

i . Draw on paper any right triangl e you pl ease, and

measure the three sides as accurately as possibl e . Square

the three sides, and see whether one Of these Squares isequal to the sum Of the other two .

The foregoing are il l ustrations Of the most cel ebrated and the most important Of al l the truths Of geom

stry . It was taught nearly 2500 years ago by P ythagoras,and is commonl y cal l ed

The P ythagorean P roposition .

The square upon the hypotenuse of a right tri angle is

equal to the sum of the squares upon the other two sides.

348 MEN SURATION .

602 . Hence we have the fol l owing rul e1 . TO find the hypotenuse, hav ing the base and per

pendicul ar given : Square the base and the perpendicular,

add the squares, and extract the square root of the sum.

2 . To find the base or p erpendi cul ar, havi ng the hypot

enuse and the other side given : Square the hypotenuse and

the given side, subtract the latter square from the former, and

extract the square root Of the rema inder.

603 . If the l ength Of the hypotenuse is denoted by h,that of the perpendicul ar by a

,and that of the base by b,

the P ythagorean P roposition is expressed‘by this formula,

1 .

2 b2,

and the above rul es by these formul as,

4 . b


604. In the fol l owing exampl es t wo sides Of a rightangl ed triangl e being given , find the third (represented byone Of the l etters a , b, or h) .

32 . 48, 55, h. 35 . 25, 60, h. 38 . 48, 90, h.

33 . 60, b, 87 . 36 . 36 , b, 1 1 1 . 39 . 1 8, b, 82 .

34 . a, 84, 1 1 6 . 37 . a

, 77, 85. 40 . a , 1 1 0, 1 46 .

In the fol l owing exampl es find the thi rd Side to two

places of decimal s

41 . 6 , 1 0, h. 43 . 5, b, 1 0.

= b.

4 2 . 1,l, k.

0

4 7 . Find the diagonal of a square, side Of whichmeasures 1 0 inches.

350 MEN SURATION .

59 . Find how l ong the rafters must be for a barn 40 feetw ide

, the ridge-pol e being 1 8 feet above the l evel of the

eaves, and the eaves projecting 2 feet from the wal l s.

60 . A regul ar hexagon is inscribed in a circl e whoseradius is 1 0 inches. Find the apothem.

~ Find the ratioOf the apothem to the radius. Find the area of thehexagon .

6 1 . If six radi i are drawn from the center Of a circl e so

as to div ide the circumference into six equal arcs, each arc

subtends an angl e of how many degrees at the center If

the chords are drawn to these six equal arcs, what kind Of a

polygon wi l l be inscribed in the circl e How many tri

angl es wil l be formed What kind Of triangl es If the

radius Of the circl e is 1 0 inches, what is the area of eachOf these triangl es The area of each segment6 2 . Given a rhombus the shorter diagonal of which is

equal to one side,which measures 1 0 inches. Find the

l onger diagonal and the area Of the rhombus63. An equil ateral triangl e is inscribed in a circl e whose

radius is 1 0 inches. Find one side of the triangl e .

64 . An equil ateral triangl e, each side Of which is 1 0

inches,is inscribed in a circl e . Find the radius of the circl e .

65 . The corners of a square are cut Off so as to make of ita regul ar octagon . The side of the square measuring 1 0inches, what is the l ength Of each side Of the octagon66 . Assuming any number you choose as the l ength Of

the base (side) Of an equil ateral triangl e, find the height.Find al so the area.

If a represents the l ength of one side Of an equilateral triangl e , itsheight is See whether this is not so in the exampl e you have

chosen, or in any other example.

The base (side) of an equil ateral triangl e being represented by a

and the height by 5a one hal f the product Of these two dimensions,or l a

? is the area. See whether this is not so in the exampl e youhave chosen , or in any other exampl e.

SOLIDS. 351

67 . The area Of an equilateral triangl e being given as 1 00

square feet, find one side.

Let a represent the length in feet of the side to be found . Then

ia’VS 1 00 , fromwhich the value Of a can be found .

68 . How many feet of fence are required to surround anacre Of l and in the form of a regul ar hexagon

69 . How many feet Of fence are required to surround anacre Of l and in the form Of a regu l ar octagon .

Compare the resul t of Exampl es 68 and 69 with the resul ts Of Exam

pl es 1 08 and 1 09 , page 3 1 6 , and Exampl e 2 1 , page 345, and what do youl earn If you were to be given as much l and as you coul d walk

around in a day, what form Of path woul d give you the mOst l and

SOLIDS.

FIG . 45 .— P rl sm. FIG . 46 .

— Rectangular Sol id. Fro . 47.— Cube.

605 . Aprism is a sol id bounded by two equal and paral l elpolygons and three or more paral l el ograms .

The equal and para l lel polygons are the bases of the prism, and the

perpendicular distance between the bases is the a l titude or height.

The paral l elograms taken together form its convex surface.

When the bases are regular polygons and the paral l elograms are

perpendicular to the bases, the prism is a regularprism.

606 . A rectangul ar sol id is a sol id bounded by six reetan

gl es.

607 . A cube is a sol id bounded by six equal squares.

Is a cube a prism Is a rectangul ar sol id a prism How manyfaces Of a rectangul ar sol id can be squares, and yet the sol id not be acube

352 MEN SURATION .

608 . A cy l inder is a prismthe bases Of whichare circl es. The paral l el ograms forming theconvex surface are infinite in number and in

finitely narrow, but taken together they formthe curved convex surface. Fig. 48.

609 . A pyramid is a sol idbounded by one polygon, whichis the base

,and three or more

triangl es which terminate in one

point at the top cal l ed the vertex. Fig . 49 .

The triangles form the convex surface Of the pmmid . When the base of a pyramid is a regul ar pol ygon,and a l ine drawn from the vertex to the center of

the base is perpendicul ar to the base, the pyramid isa regul ar pyramid.

6 1 0 . A regul ar pyramid the base Of which isa circl e is a cone. The triangl es forming theconvex surface aré infinite in number and theirbases are infinitely short ; but these triangl estaken together form the curved convex surface,and the bases of the triangl es taken togetherform the circumference Of the base of the cone.

FIG . 51 .— Frustumof a P yramid. Fro . 52 .

— Frustum of a Cone.

6 1 1 . If the upper part Of a pyramid or of a cone is cutOff by a pl ane paral l el to the base, the part that remains is afrustum Of the pyramid or Of the cone.

354 MENSURATION .

61 6 . If a prism or cyl inder is 1 inch high, its convexsurface contains as many square inches as there are inchesin the perimeter of the base. If the height is increased to2, 3, or any number Of inches, the convex surface wil l beincreased in the same proportion . Hence

,

To find the convex surface of an upright prism or cyl inder,

mul tip ly the perimeter of the base by the height.

6 1 7 . A pyramid or a cone is equival ent to a. of a prismor a cyl inder Of the same base and height. Hence,To find the volume of a pyramid or cone, mul tip ly the area

of the base by the height, and divide the product by three.

The rul e may be il lustrated in this way . Make a pas teboard model Of a prism and Of a pyramid having its bas e Ofthe same form and size and having the same height. Leave

Off the base of the pyramid and one base Of the prism. Now

fil l the pyramid with sand and pour it into the prism. Howmany times can this be done before the pri sm is ful l ?In the same way, with pasteboard model s, compare the

vol ume Of a cyl inder with that Of a cone.

6 1 8 . The convex surface Of a regul ar pyramid or cone iscomposed Of triangl es whose bases form the perimeter Of

the base Of the sol id, and whose height is the slant heightOf the sol id. Hence

,

To find the convex surface of a regular pyramid or cone,

mul tiply the perimeter of the base by the slant height and

di vide the product by two.

6 1 9 . The convex surface Of the frustum Of a regular

pyramid is made up Of trapezoids whose paral l el sides formthe perimeters of the bases, and whose height is the sl antheight of the frustum. The same statement appl ies to thefrustum Of a cone

,because the bases, which are circl es, are

regul ar polygons Of an infinite number Of sides. Hence,

VOLUME AND SURFACE AREA OF SOLIDS. 355

To find the convex surface of a frustum of a regularpyra

mid Or cone, mul tip ly the sum of the perimeters of the two

bases by the slant height and divide the p roduct by two.

620 . It is proved by geometry that the surface of a

Sphere is equival ent to that of four great circl es of thesphere . Hence,To find the surface of a Sphere, find the area of a great

circl e of the sphere, and multip ly it by four.

NOTE . The curved surface of a hemisphere has twice the area Ofthe flat surface.

62 1 . A sphere may be regarded as composed of pyramidswhose bases taken together form the surface of the sphere,whose tops are al l at the center, and whose height is theradius. Hence,To find the volume of a sphere, multip ly the convex sur

face by the radius and divide the product by three.

622 . From the l as t two rul es may be derived the fol

lowingP ormul as rel at ing to the Sphere .

4 Radius”x Diameter”x

4; Radius3 x 11. Diameter

3 x

4 . Diameter3 . 1 4 1 6

623 . By using l etters in pl ace of words, and the Greekletter 1 r to represent the ratio of the circumference to thediameter, these formulas may be written as fol l ows

1 . S 4 7rR2WBS

.

s= £D=3R

6 1 r 7

356 MENSURATION .

Frus tum of a P yramid or Cone .

62 4 . The volume Of the frustum of a cone can be foundby supposing the part cut off to be restored, then finding

the volume Of the whol e cone, the volume

of the restored part, and the differencebetween the two . This difference is thevolume of the frustum.

The heights of the two cones are foundfrom the given dimensions of the frustuminthis way . Draw (Fig . 55) the axis, AR,

and a sl ant height, AC, of the whol e cone,

and between them the radii OR and BQ.

Draw al so the perpendicul ar BD, paral l el

and equal to QR. The two right-angl edFro . 55 triangl es ARC and BDC have the same

angl e at C and equal ang l es at A and B . Hence the triangl es are simi lar and their Sides are proportional . That is,

CD : DB : CR : RA.

The first three terms of this proportion are known ; forCD is the difference between the given radii of the twobases of the frustum, DB is the given height Of the frustum, and OR the given radius Of the l arger base .

The val ue of RA found from the proportion is the heightof the whol e cone. The height O f the smal l cone, QA, is

found by subtracting the given height of the frustum RQfrom the height Of the whol e cone .

W ith the given radii Of the two bases and the two

heights RA and QA, the volumes of the two cones are

found.

625. Il lustrative Examp le. A common tin milk pan, measuring 1 2 inches in diameter on the bottom,

1 5 inches indiameter across the top, and inches deep, contains howmany cubic inches How many quarts

358 MEN SURATION .

74 . The base of a pyramid is a square, and its faces are

four equil ateral triangl es. Find the convex surface and the

volume of such a pyramid, supposing each edge of it tomeasure 1 foot.75 . A regul ar hexagonal prism contains how many cubic

inches if its height is 6 inches and each side of its basemeasures 2 inches7 6 . Find the convex surface and the vol ume Of a regul ar

hexagonal pyramid, supposing its height to be 6 inches andeach side of the base 2 inches.

7 7 . How many square feet are there in the surface of a

pyramidal roof covering a house 40 feet square, if theeaves project 2 feet beyond the wal l s of the house, and theapex of the roof is 1 6 feet above the l evel of the eaves ?

78 . There is a’

sol id bounded by four equil ateral triangles.

Find the whole surface and the volume of such a sol id,supposing each edge Of it to measure 1 foot.79 . How many cubic inches are there in a right cyl inder

whoseheight is 6 inches and the diameter o f the base 4 inches?80 . What is the vol ume of a right cone of the same base

and the same height as the cyl inder just described8 1 . How many gal l ons Of oil can be stored in a cyl indricaliron vat 4 feet in diameter and 9 feet deep 982 . How many square feet of sheet iron are there in a

piece of stove pipe 6 inches in diameter and 3 feet l ong,al l owing 1 inch for l apping at the joint83 . Asection of cast-iron water pipe 1 2 feet l ong is Of uniform thickness, 1 inch, throughout. How many cubic feetof material are there in the pipe, supposing the interiordiameter to be 24 inches What does it weigh, cast ironbeing 7 71[ times as heavy as an equal bulk of water84 . How many cubic inches are there in a gri ndstone 4feet in diam. , 4 inches thick, and having a hol e at the

center 4 inches square What is the weight of this stone

if it weighs times as much as an equal bul k Of water ?


85 . How many cubic inches are there in a bushel measure,

cyl indrical in form,inches in diameter and 8 inches

deep ?

86 . A two-quart measure is to be made Of tin in cyl in

drical form, the diameter to be two thirds of the depth .

Find the dimensions, one quart containing 574 cubic inches.

87 . How many cubic inches are there in a box 1 2 inchessquare on the bottom, 1 6 inches square at the top, and 8inches deep (interior measurements)88 . A piece ‘

Of granite 20 feet l ong, 4 feet square at one

end and 2%feet square at the other, contains how many cubicfeet ? How much does it weigh, a cubic foot of graniteweighing pounds89 . Measuring a common water pai l , I find the diameterof the bottom to be 84inches, diameter of the top 1 1 7} inches,and depth 8 inches. How many quarts does this pail hol d,al l owing cubic inches to a quart90 . Measure a common tumbl er, tin dish, or other vessel

made in the form Of the frustum of a cone, and find the con

tents in cubic inches. Now measure the waterwhich fil l s thevessel in quarts, and see how nearly the two resul ts agree .

9 1 . How many square inches are there in the surface ofa bal l 4 inches in diameter9 2 .

-How many cubic inches are there in this bal l93 . Assuming the earth to be a sphere 79 1 2 miles in

diameter, how many square mil es are there in its surface94 . Suppose a cube, a cyl inder, a sphere, and a cone al l

to have the same dimensions ; that is, the edge of the cubeis equal to the height and to the diameter of the cyl inder ;equal to the diameter of the sphere ; and equal to the heightand to the diameter of the base Of the cone . If the edge Of

the cubemeasures 6 inches, what is the volume of each sol id95 . Show that if a cyl inder, Sphere, and cone have thesame dimensions (see last exampl e ) , their vol umes are pro

portional to the numbers 3, 2 , and 1 .

360 MEN SURATION .

96 . Find the diameter of a sphere whi ch contains 1 000cubic inches.

9 7 . Find the di ameter of a sphere whose surface contains1 000 square inches.

98 . Find the volume of a sphere whose surface measures1 000 square inches .

9 9 . Find the surface Of a sphere which contains 1 000cubic inches.

1 00 . If the height of a right cy l inder is equal to hal f thediameter of its base, how does the area Of its convex surfacecompare with the united area Of its two bases1 01 . The volume Of a right cone is 1 000 cubic inches.

Find its height and the diameter Of its base, the formerbeing to the l atter as 5 to 4 .

SIMILAR POLYG ONS.

628. Similar polygons are polygons which have the same

shape, though they may not have the same size . (Art. 50 1 .

Al l the corresponding sides or other l ines of simi lar polygons

are proportiona l .

629 . The areas Of simi l ar figures are thus compared

In Fig . 56 are represented three similar triangl es . The second hasits sides twice as long, and the third has its sides three times as long,

as the corresponding sides of the first. By dividing each side of the

second triangle into two , and each side Of the third into three equalparts, and drawing l ines, we find that the second triangl e is made

Fro . 56 .

up of four and the third triangl e Of nine triangl es, each equal tothe first. The corresponding sides of these similar triangl es are pro

portional to the numbers one, two , and three, whil e the areas are proportional to the numbers one, four, and nine, which are the squaresof the former numbers .

362 MEN SURATION .

1 08 . A horse is tethered to a stake so that he can grazeover 300 square feet Of ground. Another horse tetheredby a rope twice as l ong can graze over how much1 09 . If the diameter of a circl e is diminished by if , in

what ratio wil l the area be diminished ? If the area isdiminished by in in what ratio wil l the diameter be diminished

1 1 0 . From a regul ar hexagon, each side of whi chmeasured1 0 inches, was cut Off a strip al l around it wide enough totake Off 4 Of the area. What is the l ength Of each side ofthe hexagon that is l eft How wide is the strip

If a pipe 2 inches in diameter di scharges 500 gal l onsof water in a given time, what amount Of water wil l a pipe5 inches in diameter discharge in the same time, the vel ocityof the running water being the same in the two pipes1 1 2 . Draw four concentric circl es, the inmost with a radius

of 5 inches, and the others with such radii that the area ofeach ring shal l be equal to that Of the inmost circl e . Whatare the l engths of these radii ?1 1 3 . The al titude of a triangl e is 1 0 inches. By a l ine

drawn paral l el to the base, the triangl e is divided into twoequival ent parts. What is the al titude of the triangl e thuscut Off

1 1 4 . By l ines drawn paral l el to the base of a triangl e, itsarea is divided into three equival ent parts. Find the di stance Of each paral l el l ine from the base, supposing the

al titude of the whol e triangl e to be 1 0 inches.

SIMILAR SOLIDS.

632 . Simi l ar sol ids are sol ids which have the same shape,though they may not have the same size . (Art.

Al l the corresponding edges or other l ines of simi lar solids

are proportiona l .

The corresponding p lane faces of simi lar sol ids are simi lar

plane figures.

SIMILAR SOLIDS . 363

633 . The volumes of similar sol ids are thus compared.

From the cubes, which are

simil ar sol ids, represented in

Fig . 58,we see that when the

corresponding edges are pro

portional to the numbers one,

two, and three, the volumes are

proportional to the numbers FIG . 58.

one, eight, and twenty-seven.

W ith cl ay make 2 7 cubes, al l of the same size . P ut 8 of these

together to form a larger cube with edges doubl e the edges of a singl e

cube . Put the 27 cubes together, and what is formed

NOTE.— This exercise may also be worked with wooden blocks ;

but the next one should be worked in clay .

Make 27 triangular pyramids al l of the same shape and size. Put

8 of them together to make a similar pyramid with edges twice the

edges of a singl e one. P ut 2 7 pyramids together, and what is formed

634 . From these and simil ar il lustrations we may l earnin general that,The volumes of simi lar sol ids are p roportiona l to the cubes

of their corresponding l ines.

635 . Examp l es for W ri tten W ork .

1 1 5 . A prism 1 0 inches high contains 235 cubic inches.

How many cubic inches does a simil ar prism contain whichis 1 5 inches high ?1 1 6 . Apyramid 1 0 feet high contains 1 25 cubic feet. How

high is a similar pyramid containing 500 cubic feet1 1 7 . The weight of a cube of copper each edge of which

measures 4 inches is wepounds. What is the weight Of a

a cube of copper each edge of which measures 3 inches1 1 8 . The weight of a l ead bal l 1 inch in diameter beingounces

,find the weights Of l ead bal l s 2, 3, and 4 inches

in diameter.

364 MEN SURATION .

1 1 9 . A bushel measure is a cy l indrical vessel inchesin diameter and 8 inches deep. Find the dimensions ofa. peek measure Of simi l ar form.

1 20 . A farmer has three stacks of hay of simil ar form,their

heights being respectively 1 5, 20, and 25 feet. How manytons of hay are there in al l three stacks if the smal l er

one contains 8 tons1 2 1 . The mean diameter of the earth being 79 1 2 mil es, andthat of the moon 2 1 60 mil es, find how many bodies Of the

size Of the moon coul d be made out of the earth'

.

1 2 2 . The vol ume of Saturn is about 700 times, and that ofJupiter 1 233 times, that of the earth . Find the ratio Of

the diameter of each of these pl anets to that of the earth.

1 2 3 . A hemispherical bowl 1 6 inches in diameter holds acertain quantity of water. What is the diameter of a simil arbowl that hol ds hal f as much water

0

1 24 . A cone is cut into two equival ent parts by a pl ane

drawn through it paral l e l to the base. How far bel ow thetop o f the cone is this pl ane drawn, if the height Of thewhol e cone is 1 0 inches ?1 2 5 . Gold

,Sil ver

,and copper weighing respectivel y 1 95

1 05, and 8-1-95 ,times as much as an equal vol ume of water

,

it is required to find the diameter of a sphere of sil ver,and another Of copper, each to weigh as much as a sphereof gol d 4 inches in diameter.

1 2 6 . A cone 1 0 inches high is cut into three equival ent

parts by pl anes drawn through it paral l el to the base . Findthe height Of each frustum and of the smal l cone at the top.

1 2 7 . A tin pai l measures 7 inches in diameter on the bot

tom, 1 0 inches in d iameter across the top, and is 8 inchesdeep. How deep is the water in it when it is hal f ful l

366 MISCELLANEOU S EXAMP LES.

1 0 . If Mr. White gets 8 35 a month for rent of a houseworth 356000, on which he pays 8 70 a year for taxes, howmuch does he l ose a year, money being worth 6%1 1 . A man, wishing to get $ 400 from a bank for 3

months, added to the 400 the discount for 3 months 3days at 6 and drew his note for the amount. If the notewas immediately discounted, how much did the proceedsfal l short of 400

1 2 . The rate of discount at a bank being 547“ whatwoul d be the total proceeds on the 1 4th of June, 1 896, ofthe fol l owing notes

85500, given for 6 months, datedMay 1 7 1 896 .

750, given for 3 months, dated May 23, 1 896 .

254, given for 4 months, dated May 31 , 1 896 .

435, given for 2 months, dated June 6, 1 896 .

1 3 . A col l ector received as his commission at 5%on the amount of his col l ections for 1 day . If the debtorswere al l owed 1 0%discount from the face of their bil l s,what was the face of the bil l s col l ected1 4 . How much water must be mixed with a barrel of

ink (3 1 gal l ons) , which cost that the mixture maybe sol d at a gal l on and 25%be gained1 5 . A grocer l oses 1 0%by bad debts, and pays for

having his bil l s col l ected. ‘Vhat per cent above cost musthe charge for goods, that he may cl ear 1 0%1 6 . A broker purchases stocks at an average of 9%bel ow

par, sel l s them at an average of 7 above par, and makes85300 . What was the par value of the stocks1 7 . 2070 of a l ot of barl ey, original ly 5000 bushel s, was

destroyed by fire, the cost hav ing been 351 } per bushel .What per cent wil l be gained on the l ot by sel l ing the

remainder at $ 2 per bushel1 8 . The rate of a cl ock is 5 fast. How many

seconds does it gain in a week

MISCELLANEOU S EXAMP LES . 367

1 9 . I sel l 4} of a l ot of goods for 9 , and thereby l oseFor what sum must I sel l the remainder to make

on the whol e20 . What woul d be due May 1 , 1 896 , on a note given

for 1 000, at 5%interest, dated March 26 , 1 893, on which200 was paid at the end of each year from the date of

the note2 1 . How many cubic yards of earth must be removed

in di gging a cel l ar 1 0 feet deep which is to measure insidethe wal l s 27 feet l ong and 1 5 feet wide, the wal l to be2 feet 6 inches thick22 . What is the compound interest on a note for $51 000,

dated September 1 4, 1 896, and paid December 2 , 1 897 interest at 6%per annum, payabl e semiannual ly23 . A water tank 3 feet in depth and 4 feet square issuppl ied by rain from the roof of a bui l ding 30 feet l ongand 20 feet wide . What depth of rain must fal l to fil l thetank24 . A man agreed to pay $51 200 with interest at 6%

per annum. Suppose the payment of interest due at the end

of the first and second years was deferred, and that simpl einterest was al l owed upon the deferred payments, whatwoul d be due at the end of the third year ?25 . Mr. Lyon offered to sel l his house for 4500, but,

finding no customer, he was obl iged to keep it five years ;he then sold it for $ 8000 . If he received $ 200 per yearfor rent above what he paid for taxes and repairs on the

house, money being worth 6%a year,how much did he

gain by keeping the property What per cent did he gainon the 4500

26 . In N orth Eastl and, near Spitzbergen, there is said tobe a gl acier from 2000 to 3000 feet deep. At the average

depth of 2500 feet, what average pressure is exerted by thegl acier upon a square foot of underlying earth

,ice being

as heavy as water


2 7 . Suppose coal to be worth 356 a ton,and the net cost

of manufacturing coal gas to be 1 5%of the price of the coal .If a ton of coal yields cubic feet of gas, what is thecost of gas per thousand feet ?28 . August I insured my l ife, paying a premium

of 35 a year . After I had paid my insurance for 1 1years, the company went into bankruptcy, and on February1 4

,1 896 , the sum of was sent me in ful l for al l

demands on the company . Al l owing the use of money to beworth how much money did I l ose by the transaction2 9 . It is a principl e in mechanics that the l engths of

the two arms of a l ever are inversely proportional to theweights at their ends. What weight 6 feet from the ful

crum wi l l bal ance a weight of 80 pounds 5 feet from the

ful crum 2 feet from the ful crum 1 5 feet 6 inches30 . How far from the support of a til t must a boy weigh

ing 1 20 pounds be pl aced to bal ance another boy weighing96 pounds seated 1 0 feet from the support31 . T . Banks Co. imported from France 220 meters of

cl oth costing francs per meter, paying 1 5¢ per yardfor duties. If the cl oth is sol d at 35 per yard, howmuchis gained (1 meter 39%in .)32 . How many square yards of canvas are there in a cir

onl ar tent 200 feet in diameter, the upright wal l being 1 6feet high, and the roof extending from the top of the wal lto a point over the center 50 feet from the ground33 . Find the first cost of an articl e of which 1 00 can be

made with raw material s costing 350, l abor as1 50, and

other fixed charges 55200 ; and find the sel l ing price togain How much woul d this price be affected byraising the wages of al l the l aborers34 . In the Centigrade thermometer the freezing pointof water is marked 0° and the boil ing point In the

Fahrenheit thermometer the freezing point is marked 32°

and the boiling point When the Centi grade ther


42 . A contractor has 20 days in which to do a piece ofwork . He hires 30 men

,who after working 8 days strike

for higher wages and remain idl e 6 days, after which theyreturn to their work . But for the strike the work woul dhave been done at the end of the i 8th day . How manymore men must the contractor hire that he may fin ish hiswork within the time al l owed by the contract43 . A hot-air register in a school room is 2 feet l ong by

1 foot 6 inches wide,and hal f the area is taken up by the

grating. How much air per minute must pass througheach square foot of the opening of this register into theroom to supply each of 42 persons with 4 cubic feet offresh air every minute44 . A schooner beating against the wind sail s SE.

6 mil es, then SWV. 1 2 mil es, then SE. 1 2 mil es, then SW.

1 2 mil es, and final ly SE. 6 mil es. How many mil es is itin a straight course from the point she l eft to the point shearrives at45 . What is the l ength of the edge of the largest cubethat can be cut from a gl obe 9 inches in diameter46 . What wil l a pine log weigh whose l ength is 1 8 feet,

and which measures 3 feet across the l arger end, and 24feet across the smal l er, pine being as heavy as water,which weighs pounds to a cubic foot4 7 . What is the number of square feet in a wal k sur

rounding a circul ar garden which is 25 yards across, thewalk being 4 ; feet wide48 . Find the cost of material s and l abor for 1 00 square

yards of l ath and pl aster work, as fol l ows, and make out abil l for the same . Common l ime, 4 casks lumpl ime

, cask pl aster of P aris, caskl aths, 2000 20 per hundred ; common sand, 7 l oads

white sand,2 7} bu. nail s, 1 3 l b. hair,

4 bu . mason ’s l abor, 31, days laborer, 3

days cartage,

MISCELLANEO U S EXAMP LES. 371

49 . An ancient cypress tree in Lombardy is 1 2 1 feethigh. What must be the l ength of a cord that wil l reachfrom the top to the ground 50 feet in a horizontal l ine fromthe base50 . The trunk of this tree is 23 feet in circumference at

the base. If the trunk were a perfect cone to the top of the

tree (1 2 1 feet) , how many cords of timber would it contain51 . The intensity of l ight from a given object variesinversely as the square of the di stance . If your book is 2 7inches from the l ight and mine is 6 feet 9 inches, howmany times as great is the l ight on your book as on mine 9

52 . P ure iron weighs times as much as water. A

cubic foot of water weighs 1 000 oz. Av . How many cubicinches are there in a cube of iron weighing 1 lb. Av .

53 . Find the diameter of a cast-iron bal l weighing 9

pounds, supposing that the iron weighs 7 .2 times as

“

muchas an equal bulk of water.

54 . The l argest known diamond in the worl d in its

original state weighed 900 carats of 3 31)r Troy grains each.

What was its weight in Troy units55 . The Kohinoor, now owned by Queen Victoria, once

weighed 1 86 1 1 if carats, but l ost in cutting 82 ,5K carats. Whatis its present weight in Troy units What per cent waslost in cutting56 . Among the crown jewel s of Russia is a diamondthat weighs 1 94 carats, which was bought for and

an annuity of What is the yearly expense of the

jewel to Russia if money is worth 4%per annum57 . In the templ e at Baalbec is a stone 66 feet l ong,

1 2 feet wide, and 1 2 feet thick . If its weight is timesthat of water, what is the weight of the stone 958 . The l argest savings bank in the worl d is in Gl asgow .

In 1 893, bel onged to depositors. At

per pound, what was the average in U nited Statesmoney bel onging to each depositor


59 . A building was somewhat damaged by fire. The

owner and the agents of the insurance companies agreed

that $ 4500 woul d cover the damage . There were threepol icies of insurance, one for 5000 in the Royal InsuranceCO .

, one for 7500 in the North American, and one for

2500 in the Farmers’ Mutual . What amount shoul d eachcompany contribute towards making up the 4500

60 . A party of emigrants bought a .township of government l and at an acre . Reserving 1 50 acres for publ ic

purposes and setting aside 200 acres which were worthl ess,they divided hal f of the remainder into farms which weresol d at $ 4 an acre, and the other hal f into farms sold at

$ 5 an acre . After sel l ing al l the farms, and paying forthe township, how much money had they l eft6 1 . James Fiske bought, June 8, 1 894, 1 0 bal es of cl oth,

1 4 pie‘bes in a bal e, 43 yd. in a piece, at 8¢ per yd.

, for

which he gave his note on interest at 6 On the 4th ofNov .

,1 894, he sol d 1 bal e at 1 2 if a yd. , and gave the pro

ceeds in part payment of his note . On the 3d of May, 1 895,he sol d 1 bal e at 1 5 and paid on his note the amount hereceived. On the 1 7 th of Sept , 1 895, he sold the remainderat 1 6 and settl ed the note . How much did he gain62 . A grocer buys coffee at $ 29 per hundred pounds

and chicory at per hundred pounds, and mixesthem in the proportion of 2 lb. of chicory to 5 lb. of coffee.

He sel l s the mixture at 38 a pound. What is his gain percent6 3 . What is gained by sel l ing a hundred powder kegs at

1 632g

apiece, the cost per hundred being for makingand for hooping, eight hoops at per hundredbeing required for each keg, and the value of the otherstock being 8 $3 per keg ?64 . When the shadow of a man 5 ft. 1 1 in . tal l is found

to measure 8 ft . 3 ih .

,what is the height of a tree the

shadow of which at the same moment measures 57 ft. 8 in .

374 MISCELLANEO U S EXAMP LES.

7 1 . A right pyramid, 20 inches high, has a base 1 0 inchessquare. What is the l ength of each edge of the pyramidWhat is the sl ant height of each face What is the area

of each face7 2 . Aright pyramid, 40 inches high, has a regul ar hexagon

for a base, each side of which measures 1 0 inches. Whatis the l ength of each edge of the pyramid ? What is thesl ant height of each face What is the area of each face7 3 . A tennis court is 78 feet l ong and 36 feet w ide . If

a rol l er is 1 5 inches in diameter and 34 inches l ong, howmany times must it revol ve in going the l ength of the court,and what is the l east number of times it must go l engthwiseof the court to cover it74 . What is the weight of a granite rol l er 5 feet in

diameter and 6 feet l ong, granite being times as heavyas water, and a cubic foot of water weighing 625 pounds75 . t at is the weight of a wedge which is 2 inches

square on the head and 1 0 inches in l ength,measured on the

sl ant surface, the iron being times as heavy as water7 6 . Find the weight of 1 5 steel treads of a winding

stairway, each tread being 1} an inch thick and 3 feet l ong,with a width of 1 0 inches at one end and of 5 inches at theother end, the steel being times as heavy as water.

7 7 . If the capital stock of a railway company isthe earnings for six months and the expenses

for the same period how l arge a div idend maybe decl ared, setting aside any fraction of 1%as a reserve

What amount woul d be thus set aside

SU PPLEMENT .

Expl anati on of the U se of the Dri l l Tabl es.

(P ages 29 , 44 , 7 1 , 89 , 1 46 , 1 59 , and

1 . The Dril l Tabl es and Exercises provide for practice additional

to what is furnished by the ordinary exampl es of the book . They

may be used in part, or may be omitted al together, at the discretionof the teacher. The cl ass using them may do the same exampl e, or

each membermay be assigned a separate exampl e by the same dictation . Independence and accuracy can be secured by the l atter pl an .

When al l the members of the class do the same exampl e, the tabl esneed no explanation . In most of the exercises, no copying of figures

wil l be necessary ; the pupil can work directly from the tabl e, writingthe resul t on a sl ip of paper placed beneath the figures used. Other

exercises, as 1 20 to 1 3 1 in Dril l Tabl e No. 1 , w il l need to be copied.

When each member of the cl ass is assigned a separate exampl e bythe same di ctation, the fol l owing suggestions wil l be helpful :

Dri l l Tabl e No. 1 . P age 2 9 .—Exercises 85— 1 04. The teacher

gives out the exercise to the whol e cl ass, and directs the pupil s to

number themselves al oud, the first pupil saying“one ,

” the next

“ two,”and so on to twenty . If there are more than twenty pupil s in

the cl ass, the remaining pupils wil l number themsel ves, beginningwithone again.

At a given signal the pupi l s add, the first taking column 1 ; the

second, column 2 the third, column 3, and so on. At another givensignal they read their answers, the teacher verifying them by the key .

In Exercises 1 05-1 1 4 the pupil s l etter themselves m, n , 0 , etc. , to

v. The first pupil adds the doubl e column m, the second adds n, the

third adds 0 , etc.

In Exercises 1 20— 1 3 1 the pupils l etter themselves a , b, 0 , etc.

The first pupil adds in l ine a , 2324 , 2658, etc . , to zz, the second adds

in l ine b, 1 222 , 1 355, etc . , to zz, and so on. In this exercise the numbers may be either copied or added without copying.

875

376 SU P P LEMENT.

Dril l Tabl e No. 2 . P age 44 . In this table , Exercises 1 78-1 89 , the

pupil s having l ettered themsel ves a , b, 0 , etc . , the first takes

from 1 00 the second takes 1 2 1 4 1 3 from 1 00, and so on .

In Exercises 238— 243 the pupil s l etter themsel ves a , c, e, g , i , k.

The first pupi l takes 5542 from 5848 the second takes 1 735 from5641 , and so on .

Dril l Tabl es Nos . 3 and 4 . P ages 7 1 , 89 .— The teacher who has

used the preceding tables wil l need no explanation of the use of these .

There are 1 2 examples in each exercise, and the pupils wi ll letter

themsel ves from a to l .

Dril l Tabl e No. 5 . P age 1 46 .— In each of these exercises the pupils

wil l number themsel ves from 1 to 1 5.

Dril l Tabl e No. 6 . P age 1 59 . Here the pupil s wi l l number themsel ves 1 to 1 4 . In Exercise 242—255 the first pupil reads the numberswritten in l ine 1 , columns E and G ,

the second in l ine 2 and so on.

Dril l Tabl e No. 7 . P age 1 8 1 . Needs no explanation .

Contracti ons in Mul ti pl icati on. (P age

2 . Some contractions in mul tipl ication and division are given in

Arts. 9 1 , 1 4 1 , 1 72 , 237 , and 238 . Those in mul tipl ication which fol lowmay sometimes be useful .

Rul es .

To mul tiply by any number whose terms are al l 9 ’s

Annex as many zeros to the exp ression for the multip l icand

as there a re 9’s in that of the multip l ier, and from the number

thus exp ressed subtract the multip l icand thus,27 x 27 2673 .

1 . 28 x 99 3 . 4868 x 999992 . 92 x 999 = ? 4 . 247 x 998 = ?

3 . To mul tiply by a composite number : Separate the

multip l ier into convenient factors, multip ly the mul tip l icand byone of the factors, and that product by another factor, and

so on, ti l l a l l the factors have been used ; the last product is

the answer : thus, 41 x 25 4 1 x 5 x 5.

5 . Mul tiply 374 by 45 ; by 1 08.

6 . Mul tiply 369 by 1 44 . 7 . Mul tiply 453 by 1 32 .

378 SUP P LEMENT.

The number 3483 is now separated into two parts, one of which is

divisible by 9, and the other equal s the sum of its digits . Every number can be so separated. Hence a number is divisi bl e by 9 if the sum

of i ts digits is divisibl e by 9 .

3 . D ivisibi l ity of numbers by 3 . Any number that is divisibl e by 9is divisible by 3 . Therefore, when a number has been separated into

two parts, as shown above, the first part is divisible by 3. The otherpart is equal to the sum of its digits . Hence a number is divisible by3 if the sumof its digits is divisibl e by 3.

4 . Divisibi l ity of numbers by 1 1 .

ILLU STRATION.

80000 = 72 72 X 1 1 + 8

6000 : 546 x l 1 — 6

400 36 X l l + 4

20 2 x 1 1 — 2

7 + 7

86427 x — 6 — 2

— sz l l

The number is now separated into two parts, one of which

is divisibl e by 1 1 , and the other equals the sum of one set of al ternate

digits, l ess the sum of the other set . If the sums are equal , or if theirdifference is divisible by 1 1 , the entire number must be divisibl e by 1 1 .

G reatest Common Factor of Two or more

N umbers . (P age

7 . I l lustrativeExamp le. Find the g. c . f. of 52 and 9 1 .

Divide the greater number by the l ess, andWRITTEN WORK then divide the l ess number by the remainder,

52 ) 9 1 (1if there is any . Continue dividing the last

divisor by the l ast remainder until nothing52

remains. The last divisor wil l be the g. c . f.

sought

39 To find the g. c. f . of more than two num

(3 bers, fi nd the g . c. f . of any two of them and

39 then of that common factor and a third num

ber, and so on ti l l a l l the numbers are taken.

This method of finding the greatest . common factor ofnumbers depends on the fol l owing principl es

LEAST COMMON MULTIP LE. 379

I. Any factor common to two numbers is al so a factor oftheir sum and of their diy

'

erence.

Thus 3, which is a common factor of 24 and 1 8, is a factor of

24 1 8) and of 24 Since 24 is equal to a certain

number of 3’s, and 1 8 to a certain other number of 3 ’s, their summust be a number of 3’ s, and their difference must be a number Of 3 ’s.

II. The greatest common factor 0f two numbers is equa l to

the greatest common factor of the sma l ler of them,and the

remainder Obtained by dividing one of them by the other.

Thus , the g. c . f . of 1 43 and 52 is equal to the g. c. f .

52 ) 1 43 (2 of 52 and 39 .

1 04 Since 39 1 43 52 52 , al l factors common to 1 43

39 and 52 are al so factors of 39 , and hence common factorsof 52 and 39 . Therefore the g . c . f . of 1 43 and 52 is a

common factor of 52 and 39 , and cannot be greater than the g. c. f .

of 52 and 39 .

Again , since 1 43 52 52 39 , al l factors common to 52 and 39

are al so factors of 1 43, and hence common factors of 1 43 and 52 .

Therefore the g. c. f . of 52 and 39 is a common factor of 1 43 and

52 , and cannot be greater than the g . c. f. of 1 43 and 52 .

The g. c. f . of 1 43 and 52 and the g . c. f . of 52 and 39 are , then ,

two numbers, neither of which can be greater than the other ; they

are therefore equal .

Leas t Common Mul ti pl e of Tw o or more

N umbers . (P age

8 . I l lustrative Examp le I. What is the l east commonmul tipl e of 6, 9 , and 1 5

WRITTEN WORK.

Exp lanation. The least mul ti

pl e of 6 is 6 , which may be ex.

6 2 X 3pressed in the form 2 x 3 .

9 3 x 3 The l east mul tiple of 9 is 9 ,

1 5 3 x 5 which may be expressed in the

1 . . m. = 2 3 3form 3 x 3 . But in 6 we have

0 X X Xal ready one of the factors (3) of

9 ; hence if we put with the prime factors of 6 the remaining factor

(3) of 9 , we shal l have 2 x 3 x 3 , which are al l the factors necessary

to produce the l . c. m. of 6 and 9 .

380 SU P P LEMEN T.

The least multipl e of 1 5 is 1 5, which may be expressed in the form3 x 5. In the l . c. m. of 6 and 9 we have one of the prime factors (3)of 1 5 ; hence if we put with the prime factors of 6 and 9 the remainingfactor (5) of 1 5, we shal l have 2 x 3 x 3 x 5, which are al l the primefactors necessary to produce the l . c . m. of 6 , 9 , and 1 5.

The product of these factors is 90, which is the l . c . m. sought .

Nora .— In finding the l east common mul tipl e, the factors of the

given numbers sel dom need to be expressed, and the written work

may be greatly reduced. Thus , in this example the written work maybe simply l . c. m. = 2 x 3 x 3 x 5 = 90 .

Rul e .

To find the l east common mul tipl e of two or more numbers : Take the prime factors of one of the numbers ; with

these take such p rime factors Of each Of the other numbers in

succession as are not conta ined in any preceding number, and

find the product Of al l these prime factors.

Additi on of M xed N umbers . (P age

9 . Exampl e 91 (page 25k 67g 1 4575

l . c . m. 75

25}673}

This method of writing the work has

the advantage of brevity, as the nevr

numerators are expressed without thedenominators.

Ans. 1 3,-g

Div is ion of Fracti ons . (P age

1 0 . I l lustra tive Examp le. What is the quotient of iWRITTEN WORK.

EXPLANATION l .-The quotient of 3 divided by

2

3g : divided by which is as large as 2 is 3 times

5 x 2 4 x 3

Ans.

of .g, or5 x 2

,which equal s 1 g. Ans. l i .

EXPLANATION 2 . The quotient of § divided by l is t of di vided

by i (which is gas large as 1 ) is 3 times 4 or4

23; and of i divided

by g is l of 3 times 3 or ; i:g, which equal s l g. Ans. I}.

382 SU P P LEMENT.

in the rema inder. Thus we find that 99 times the given circulate

equal s 63 ; therefore once the given circul ate is 33, or Hence the

To change a circul ating decimal to a common frac tionTake the repetend for thefigures of the numerator, and for the

figures of the denominator as many 9’s as there are figures in

the repetend. C hange the f raction to its smal lest terms.

Change the fol l owing to common fractions in their smal lest terms

25 . 2 7 . 29 . 31 . 0 42857126 . as 30 . 32.

s 9 1N .92 7 =_fr0 " 0 it i f

1 0 1 1 0

35 . 37 . 39 .

36 . as. 40 .

Compound Subtracti on (P age

1 5 . To find the di fference of time betw een tw o dates .

Il lustrative Examp le. What is the time in years, months,and days fromMay 1 1 , 1 897, to August 7 , 1 899

1 899 y. 8 mo. 7 d.

August being the 8th month and May

1 897 5 1 1the 5th, subtract 1 897 y . 5 m. 1 1 d. from

1 899 y . 8 m. 7 d . , a l lowing 30 days for

2 y. 2 mo. 26 d. 1 month, and 1 2 months for 1 year.

Annual Interest. (P age

1 6 . Simpl e interest, taken upon the principal , and uponeach year’s interest of the principal due and unpaid, isannual interest .

Rul e .

To compute annual interest : Compute simp le interest onthe principa l for the time it is on interest. Also, on one year

’s

simp le interest for a period Of time equal to the sum of the

times for which the yearly interests several ly rema in unpa id.

Add the results.

RULES FOR PARTIAL PAYMENTS. 383

What is the annual interest of for 5 yr.

3 mo. 1 8 da. , at 6%

The Vermont Rul e for P artial P ayments .

1 7 . 1 . Compute annua l interest upon thep rincipal to the end

of thefirst year in which anypayments aremade a lso compute

interest upon the payment or payments f rom the time they are

made to the end Of the year.

2 . App ly the amount Of such payment or payments first to

cancel any interests that may have accrued upon the yearly

interests, then to cancel the yearly interests themselves, and

then towards the payment of the p rincipal .

3. P roceed in the same way with succeeding payments, com

puting, however, no interest beyond the time of settlement.

The N ew Hampshire Rul e .

1 8 . This is the same as the VERMONT RU LE, with thefol l owing provision

If at the time of any payment no interest is due except what

is accruing during the year, and the payment or payments are

less than the interest due at the end of the year, deduct such

payment or payments at the end of the year without interest

added.

The Connecti cut Rul e for P artial P ayments .

1 9 . This is the same as the U NITED STATES RU LE exceptas fol l ows1 . When LESS than a year

’s interest has accrued at the time

of a payment, excep t it be the last payment, find the difi'

erence

between the amount of the principa l for an ENTIRE year, andthe amount of the paymentfor the ba lance of a year after it is

made ; this difl'

erence wi l l form the newprincipa l .

2 . If the interest which has arisen at the time Of a payment

exceeds thepayment, compute interest upon thep rincipal only.

384 SUP P LEMEN T.

TABLES OF DENOMINATE NUMBERS.

Measures of Extension.

20 . LONG MEASURE .

1 2 inches ( in . ) 1 foot

3 feet 1 yard

5; yards or 1 65feet 1 rod

320 rods or 5280 feet 1 mil e

The fol l owing measures are al so used

1 hand 4 inches (used to measure the height of horses) .1 span 9 inches (from the end of the thumb to the end of the

l ittl e finger extended) .

1 cubit 1 8 inches (fromthe elbow to the end of themiddle finger) .

NOTE 1 . The standard unit of l ength is the yard, which is identical

with the imperial yard of G reat Britain .

NOTE 2 . The yard is al so divided into hal ves, fourths, eighths, etc.

2 1 . SQUARE MEASURE.

1 44 square inches (sq . in . ) 1 square foot (sq .

9 square feet 1 square yard (sq.

301 square yards, or 272} sq. ft. 1 square rod (sq .

1 60 square rods 1 acre

640 acres 1 square mile (sq.

22 . Come MEASURE.

1 728 cubic inches 1 cubic foot ( cu.

2 7 cubic feet 1 cubic yard (cu.

1 28 cubic feet 1 cord used in measuring wood.

Wood is general ly cut for the market into sticks 4 feet l ong, and

laid in pil es, so that the l ength of the sticks becomes the width of the

pile. A pil e 4 feet wide, 4 feet high, and 8 feet long, contains 1 cord.

One eighth of a cord is cal l ed 1 cord foot. 1 cord foot contains 1 6

cubic feet.

386 SUP P LEMEN T.

Measures of W ei ght.

28. In weighing groceries and most common goods Avoirdupois weight is used. Troy weight is used in weighingsilver, gold, precious stones, and articl es that requi re greataccuracy in weighing.

29 . AvOIRDU P O Is WEIGHT .

1 6 ounces (oz . ) 1 pound

2000 lb. 1 ton

NOTE 1 . In weighing some articl es, as iron and coal at the mines,and goods on which duties are paid at the U nited States custom-houses,the long ton of 2240 lb. is used. In this weight

NOTE 2 .— A cubic foot of water weighs 62 } lh. , or 1 000 oz.

avoirdupois.

28 lb. 1 quarter4 qr. 1 hundredweight

20 cwt . 1 T .

30 . TROY WEIGHT.

24 grains (gr. ) 1 pennyweight

20 pennyweights 1 ounce

1 2 ounces 1 pound

31 . COMPARISON or WEIGHTS.

1 75 lb. Troy 1 44 lb. Av.

1 75 oz. 1 92 oz. Av .

7000 gr. 1 lb. Av.

5760 gr. 1 lb. Troy.

NOTE.— The standard unit of weight is the Troy pound. From

this the other units of weight are derived .

32 . The fol l owing denominations are al so used1 4 pounds 1 stone.

1 00 pounds of grain or flour 1 cental .

1 00 pounds of dried fish 1 quintal .1 00 pounds of na ils 1 keg .

1 96 pounds of flour 1 barrel .

200 pounds of beef or pork 1 barrel .

280 pounds of sal t 1 barrel .

DENOMINATE N UMBERS. 387

APOTHECARIES’ WEIGHT .

3 . In mixing medicine, apothecaries use the Troy pounddivided into ounces (oz. or 3 drams (dr. or scrupl es

(sc. or and grains. They al so use the fluid ounce (f. S) ,fluid dram (f. minims or drops (BL) .

34 . WEIGHT . 35 . FLUID MEASURE .

20 grains l scruple . 60 minims l fluid drachm.

3 scrupl es l dram. 8 fluid drachms l fluid ounce.

8 drams 1 ounce. 1 6 fluid ounces 1 pint (O) .

1 2 ounces 1 pound . 8 pints 1 33 1 10 1 1 (0 0 1 1 3)Apothecaries buy and sel l in large quantities by avoirdupois weight.

Measures of Capacity .

36 . LIQUID MEASURE . 37 . DRY MEASURE .

4 gil ls (gi. ) 1 pint 2 pints 1 quart2 pints 1 quart 8 quarts 1 peck

1 gal lon 4 pecks 1 bushel

NOTE 1 . A pint of water weighs about a pound avoirdupois .

NOTE 2 .— The standard unit for l iquid measure is the gal lon.

NOTE 3 .— The standard unit for dry measure is the bushel .

The gal l on contains 231 cu . in. ; the bushel , 0 1 1 . in .

NOTE 4 .— In buying and sel l ing grain and many other kinds of

produce, the bushel is reckoned at a certain number of pounds. Thus,

potatoes have 60 lb. to a bushel and corn has 56 lb . to a bushel .

NOTE 5. In determin ing the capacity of reservoirs , etc. , 3 1 } gal

lons are considered a barrel and 2 barrels, or 63 gal lons, a hogshead.

38 . Time .

60 seconds (sec. ) 1 minute

60 minutes 1 hour24 hours 1 day

7 days 1 week

365 days l commoor 52 weeks 1 day

1 1 year.

366 days 1 l eap year .1 00 years 1 century .

NOTE . In most business transactions, 30 days considered a

month, and 1 2 months a year.

388 SU P P LEMENT.

39 . LEAP YEAR .

The earth revol ves around the sun in 365 days 5 hours 48 minutes

and 50 seconds nearly, but we cal l 365 days a year. It wil l be seen

that what we cal l a year is nearly 6 hours l ess than the true year, and4 such years nearl y 1 day l ess than 4 true years. To rectify this error,366 days are al lowed to every fourth year. The year of 366 days iscal l ed a l eap year.

The addition of a day in every fourth year is too much by a numberofminutes, which in one hundred years amounts to about three fourthsof a day . To balance this error, onl y 365 days are al lowed to the

final year of a century in three centuries out Of every four.Hence any year is a l eap year, when the number denoting the year

is divisibl e by 4 and not by 1 00 , and when it is divisibl e by 400 .

40 . NUMBERS . 41 . P APER .

1 2 units 1 dozen . 24 sheets 1 quire .

1 2 dozen 1 gross. 20 quires l ream.

1 2 gross 1 great gross. 2 reams l bundle.

20 units 1 score. 5 bu ndl es l bal e.

42 . BOOKS .

book formed of sheets fol ded in

2 leaves is a fol io . 1 2 l eaves is a duodecimo,4 l eaves is a quarto . or 1 2 mo .

8 l eaves is an octavo . 1 6 l eaves is a 1 6 mo.

Measures of Value .

43 . U NITED STATES MONEY .

1 0 mil l s (m.) 1 cent (ct. or r) .1 0 cents 1 dime

1 0 dimes or 1 00 cents 1 dol l ar1 0 dol lars 1 eagl e.

The standard unit of value is the dol lar. Its standard weight in

gold is grains.

390 SU P P LEMEN T.

The Metri c System of W edghts and Measures .

47 . The Metric System of Weights and Measures is a decimalsystembased upon the meter and used in the greater part of Europe

and for scientific calculations in the U nited States.

NOTE 1 . It was intended that the meter should be one ten-mill ionth of the distance from the equator to the poles, but later calculations have shown it to be a l ittle l ess than that.

NOTE 2 . The word meter means a measure. The standard

meter is a certain bar of pl atinum careful ly preserved at P aris. The

standard meter of the U nited States is a copy of this bar kept at

Washington. The meter-sticks made for ordinary use are copies of

the standard meter.

48. The mul tiples of the meter are designated by the G reek

prefixes Does Hecto Kilo and Myriathe decimal parts are designated by the Latin prefixes deci centi

(rte) : and mil l i (whit ) The abbreviations of these prefixes are

their initial letters, those of the multipl es being in capital s, and those

of the decimal parts, in smal l l etters.

49 . The standard un its of the several measures with their valuesare as fol lows

STANDARD ABBREVIAORIG IN

LEG AL U . B. EQU IVA

U N 1 7 . TIONS. LEN“ .

Eq . to P .

m. Xm.

1 0 m. x 1 0 m.

m. Xm. Xm.

1 cu. m.

Capacity 1 cu. dm.

Weight G ram 1 on cm of watergr.

Are, pronounced a ir stere, pronounced stair ; l iter, pronounced

l etter.

METRIC SYSTEM. 39 1

50 . The metric system being a decimal system, units of any

denomination in the system can be changed to those of a higher or

lower denomination by moving the decimal point.In al l but the square and cubic measures, the units increase and

decrease by a scale of tens.

The units of square measure increase and decrease by a scal e of

hundreds, and those of cubic measure by a scal e of thousands.

In the fol l owing tables the standard units are printed in capitals

and the other units usual ly employed in business or science are indi

osted by ful l-faced type.

51 . LO NG MEASURE. 52 . SQUARE MEASURE. 53. LAND.

1 0mm. 1 cm. 1 00 sq. mm. 1 sq . cm.

1 0 cm. 1 dm. 1 00 sq . cm. 1 sq . dm.

1 0 dm. 1 METER. 1 00 sq. dm. 1 SQ . METER l centare

1 0m. 1 Dm. 1 00 sq . m. 1 sq. Dm. 1 ARE (a) .

1 0 Dm. 1 Hm. 1 00 sq. Dm. 1 sq . Hm.

1 0 Hm. 1 Kil ometer. 1 00 sq. Hm. 1 sq . Km.

1 0Km. l Mm. 1 00 sq . Km. 1 sq. Mm.

54 . SOLID MEASURE .

1 000 cu. mm. 1 cu. cm.

1 000 cu. cm. 1 cu . dm 1 l iter.

1 000 cu. dm. 1 CU . METER 1 store.

55. WOOD MEASURE . 57 . WEIGHT .

1 0 da. 1 STERE. ( 1 cu. m. ) 1 0 mg. 1 cg.

1 0 s. 1 decastere . 1 0 cg. 1 dg.

1 0 dg . l GRAM56 . CAPACITY. 1 0 g. 1 Dg.

1 0 ml . 1 cl . 1 0 p g. 1 Hg .

1 0 cl . 1 dl . 1 0 Hg . 1 Kg .

1 0 dl . 1 LITER ( 1 cu. dm. ) 1 0 Kg . 1 Mg.

1 0 1 1 D1 1 0 Mg. 1 quintal1 0 D1 . 1 H1 .

1 0HI. l n , ( 1 cu. m) .1 0 Q 1 Tonneau (T. )

392 SU P P LEMENT.

58. Measure off upon a straight stick ten times the

l ength of the decimeter represented in the margin. You

have now the standard of l ength. Indicate upon the

stick decimeters and centimeters . From these al l the

other metric measures may be obtained .

U pon any conven ient floormeasure Off a square meter ;upon a level plat of ground,measure off the are (ten metersSquare ) ; in the corner of a room, measure off the stere, onemeter each way. A l iter, a cubic centil iter, may be eas ilyformed out of pasteboard or tin, al so a cubic centil iter,

which fil led with water woul d weigh a gram.

Or, for the gram, take a strip of sheet l ead that weighs

exactly a nickel and divide it into five equal parts. One

of these parts weighs a gram.

Exercises .

59 . a . Measure off 1 0 meters on a string, withknots to indicate the meters .b. Find the l ength and breadth of the school

yard,of a garden, etc.

6 . Measure 1 0 decameters and pace it Off. Howmany Of your paces make a decameterd. How many of your paces make a kil ometere. How many kil ometers is it from your home

to school ?

60 . Approximate Equival ents .

The equivalents in U nited States measures here given are accurate

enough formost purposes, and are easy to remember.

A decimeter 4 inches .

‘

A stere

3 ft. 3 ; in . ,Ameteror 1 1

16 yards.

A l iter

A decameter 2 rods. A decal iter

A kilometer 4} of a mil e. A hectol iter

An are4 sq . rods , A gramor 4

15 of an acre. A kilogram

A hectare 2 } acres. IAmetric ton

of a cord .

l iquid qt. ,or of a dry qt.

1 peck and 1 qt.

23bushels .

1 5; grains .

2 ; pounds av.

2200 pounds av.

394 SUP P LEMENT.

ADDITIONAL EXAMPLES FOR OCCASIONAL USE.

62 . Integral Numbers and U nited S tates Money .

1 . If I can buy cl oth at one pl ace for 35 cents a yard and at anotherpl ace for 28 cents a yard, how much money do I save by buying 25

yards at the l atter pl ace2 . When the introduction price for readers is 50 cents , and the

exchange price is 36 cents , what is the cost of a new set of readers in

a town requiring 675 new readers, and giving 250 ol d readers in

exchange

3 . Nine hundred and sixty peopl e struck for higher wages, andwere out of work 6 weeks in consequence . If their average earningswere a day , what was their entire l oss for the time

4 . Of a barrel of beef (200 pounds) which cost 1 7 dol lars, 1 36

pounds were sol d 1 2 cents a pound and the remainder 8 cents

a pound. How much was gained 3

5 . What is the average cost of milk per month for a family whosebil l s for the year are as fol lows :

6 . Aman bought a barrel of beef for Including 55 cents

for freight and 35 cents for carting, howmuch did the beef cost hima pound

7 . Mr. Breck received 1 75 for hismonth’s sal ary ; of this he paidfor rent $ 20 , for help 1 2 , meat bil l grocer’s bil l milk

fuel cl othing and for other expenses.

How much of his sal ary remained unexpended8 . A deal er bought 328 bushel s of Wheat 8755 a bushel , 745

'

bush

el s of cats 56 y a bushel , and gave in part payment $425, drawinga check for the bal ance. For how much was the check drawn

9 . Mr. G oss and Mr. Smith went on a journey, agreeing to share

the expenses equal ly.

Mr. Goss paid Mr . Smith paid

Fares, $ 25, $42 , Fares,Hotel bil ls, $ 1 6 Carriage hire, 3Refreshments, P orter,

Which is in debt to the other, and how much1 0 . Of 745 bushel s of oats which cost 34 11 a bushel , 326 bushels,

being damaged , were sol d 25 ¢ a bushel , and the remainder 45}a bushel . Was there, on the whole, a gain or l oss and howmuch

ADDITIONAL EXAMP LES. 395

1 1 . I started out shopping with and bought with the money2 col l ars apiece, 6 col lars at the rate of 3 for 25 2 pairs ofcufis 20 55, 4 yd. gingham 1 7 51 and 2 yd. 33 ft, 1 4 yd. l inen

33y, one and a hal f yd. lace 1 8 55, two and a hal f yd. edging

22 How much money shoul d I have l eft

1 2 . Mr. P hipps kept 42 bushel s of cranberries through the winter,but found in the spring that a third of them were decayed. He then

sol d the remainder for a bushel and received in payment theprice which he might have received for the 42 bushels in the fal l .

What was the price per bushel in the fal l1 3 . Mr. P l ympton had $ 2 1 45 on deposit in a bank. He drew out

$572 of his deposit at one time, $67 at another, and gave a check for

the balance towards the payment of a debt of 1 700 . What was theamount of the check he gave, and how much of his debt remainedunpaid

In the foll owing exampl es, supply dates, etc. , when wanting.

1 4 . Make out a bil l against yoursel f from the N .Y. Central R .R.

Company for the freight of 1 carl oad household goods, 1 2000 lb. ;

3 boxes goods, 1 450 1b. ; 1 piano, 1 200 1b . ; freight, a hundred .

1 5 . Make out a bil l against Mr. Edwin Train for the fol l owing l ots

of coal sol d him per ton of 2000 lh. : 2 1 00 lh. , 2540 l b. ,

2 1 00 lb. , 2400 l h., 2480 l b . , 2430 lh. , 2460 lb.

1 6 . James Otis sol d to Harrison G rey 57 bushel s of corn a

bushel , and 29 bushel s of turn ips 45;!xa bushel , and received

in part payment. Find how much was stil l due Mr. Otis and make

out the bil l .

1 7 . Two customers ordered the fol lowing at a restaurant : roastbeef 4055, veal pic 3555, 2 tumbl ers milk 5h pudding pie

pickles 551 , sauce 555. Make a statement of the above with amount

due ; al so tel l the proper bi l l s and coins to return if 1 0 should be

given in payment.

64. Fracti ons .

1 8 . To what must you add the difference between 1 1 153 and 53,

that the sum may be 1 6}1 9 . How many jackets , each jacket requiring 1 g yards of cl oth ,

can be made from 43§ yards, and how much cloth wil l be l eft

20 . Divide 52 into two such parts that of one part shal l equal 1 1of the other.

396 SUP P LEMEN T.

2 1 . A train l eaving M at 1 0 o’clock, r .x . , reaches N at 5} o

’cl ock

the next morning . The distance from M to N by rail is 1 755mil es.

What is the average rate of the train per hour2 2 . At 1 0 A.H . Mr. Brice starts at Springfield forWorcester , distant

54 mil es, and travel s at the rate of 8 miles an hour. At the same

time Mr. Smith starts at Worcester for Springfield, and travel s at

the rate of 9 } mi les an hour. How far apart wi l l they be at the end

of an hour In how many hours wil l they meet

2 3 . Two boys are to have for hoe ing a field of corn ; if one

hoes 5 rows while the other hoes 7 rows, till the work is done, howmuch shoul d each boy receive2 4 . If 42 lb. of mapl e-sugar can be made from 1 50 gal . of sap,

how many pounds can be made in three weeks from the sap of 1 28

trees, the average daily yie ld per tree being 7} gal .2 5 . At 7 o

’cl ock, A.H . , A started on a journey, and traveled at the

rate of 7 mi les an hour ; at 9 o’cl ock B started fromthe same place , and

traveled in the same direction at the rate of 9 } mil es an hour . In

how many hours did he overtake A

2 6 . Aman owned of a hotel which cost 1 2000 ; he sold of his

share to a third party at cost. What part of the hotel did he then

own , and how much did he receive for what he sold

2 7 . At 855 per yard for cal ico, and 33¢ per pound for cotton, what

is the cost of material s for a pair of cal ico bed comforters, each beingin l ength yards , and in width equal to 2 ; breadths of cal ico , and

requiring 3 pounds of cotton2 8 . A flagstaff 58 feet long is fastened to a buil ding in such a way

that 136 of what is above the roof equal s 3. of what is below. How

much is above the roof ?

65 . Decimal s .

2 9 . How many bushel s of wheat, each 60 pounds, must be used tomake a barrel of flour, of the wheat being l ost in the making.

30 . January 1 , a gas meter registered cubic feet of gas con

sumed ; April 1 , it registered cubic feet. At per thou

sand cubic feet, what was the cost of gas used in the intervening time

3 1 . How many hours w il l it take a person to travel mil es at

the rate of 52mil es per hour3 2 . A and B are walking the same way al ong a road. At noon A

was mil es ahead of B . When wil l B overtake A, if Awalks

mi l es an hour , and B mil es an hour

398 SU P P LEMEN T.

48 . Before Lake‘ Haarlem was drained it was 1 5 mil es in length

and covered acres. What was its average width49 . I have a rectangul ar fiel d 80 rods long and 6 rods wide. If

this fiel d is div ided into 1 0 equal rectangular house lots, having theirfronts on the longest side of the field, what wil l be the cost of fencing

to inclose and separate the l ots $6 a rod

50 . A street 3 rods wide and 200 rods long is on an average 1 0

inches above grade howmany cubic yards of earth must be removedto bring the street to grade5 1 . What is the weight of the air in a room 25 ft . l ong , 20 ft. wide,

and 1 2 ft . high , water weighing 770 times as much as air, and a cubic

foot of water weighing 1 000 ounces

52 . When you can get 3 gil l s of water by mel ting a quart of snow,

dry measure , how much snow must you mel t to fil l a 1 0—gal lon tubful l of water [Answer in bushels, pecks, and quarts . ]53 . In 1 00 parts by weight of air there are 22 parts of oxygen.

What is the weight of the oxygen in a room 20 ft . l ong, 1 8 ft. wide,

and 1 0 ft. high ?

54 . If, when flour is 5 per barrel , a 4-cent loaf weighs 3 oz. , what

should a 6-cent loaf weigh when flour is per barrel ?

67 . Miscel laneous .

55 . When stock is quoted at how many shares can be bought

for brokerage56 . It takes 4 men 5 days to bui ld 1 6 rods of fence ; how long wil l

it take to buil d 90 rods if 4 boys help, each boy doing hal f as muchwork as a man does

57 . If the weight of a bl ock of sandstone 3 ft. l ong, 2 ft. wide, and1 ft. thick, is 840 lb.

,what is the weight of another block of sand

stone 6 ft. l ong, 2 ft. 6 in . wide, and 2 ft. thick

58 . Suppose al l the water fal l ing upon the surface of New YorkState during a rainfal l of 1 in . should be gathered into one cubical

vat just large enough to hold it ; what would be the dimensions of

the vat, the area of New York State being square miles59 . Two contractors have finished work for which they have been

paid One of them empl oyed 40 l aborers for 1 25 days of

9 hours each ; the other 25 l aborers for 1 1 0 days of 8 hours each.

Divide the money between them in proportion to the l abor eachfurnished.

ADDITIONAL EXAMP LES. 399

60 . The distance around a fiel d containing one acre being 52 rods,what is the distance around a fiel d of similar shape containing 9

acres6 1 . How many pounds of flour must a baker use to make 1 00 lb.

of bread, if the bread weighs 30 more than the flour used6 2 . From a bil l amounting to 1 1 40 , there is thrown off $40 ; that

is equival ent to a discount of what per cent

63 . I bought a lot of wool at 20 bel ow the asking price, and sol d

it for $ 2600 , and by so doing I gained 30%on the cost. What wasthe asking price

64 . I buil t a house costing $ 5000 upon a l ot which cost $800 .

The house being burned, the insurance company paid me of the

cost of the house . I then sol d the land for $ 1 200 did I gain or l ose

by the transactions, and what per cent

6 5 . A marine insurance company took a risk of at

and reinsured of their risk in another company at Should

the property be lost at sea , how much would the first company l ose

66 . Water weighs 62%lb. to the cubic foot, and milk 64g lb. By

what per cent is mi lk heavier than water6 7 . What is the interest of 1 cent for 1 day at 1 a year (360 days) ?68 . Find the amount at 7%of $ 820 from March 1 0 to December

4 of the same year.

6 9 . Find the difference between the accurate interest of

at 6%and the interest by the common method, the time being fromMay 24 to October 1 5 of the same year .7 0 . What per cent on my investment do I save by buying , April 1 ,

coal for $ 60 for which I shoul d pay $ 96 the first of the fol l owing

December, money being worth 6%a year7 1 . A merchant sol d goods amounting to $6500, one hal f on 4

months’ credit, the rest on 6 months, and got the notes discounted

at a bank at 6 How much was real ized7 2 . A person holding three promissory notes dated January 1 , one

for 600 payabl e in 2 months , another for 800 payabl e in 4 months,

and another for 750 payabl e in 6 months, exchanges them for a

singl e note for the sum total . When shoul d this note be made payable

7 3 . If the side of a square inscribed in a circl e is 40 feet , whatis the area of the circl e74 . Find the convex surface of the frustum of a regul ar octagonal

pyramid, the slant height of the faces being 8 inches, the l ength of

each side of the upper base 5 inches, and that of each side of the

lower base 7 inches.

400 SUP P LEMEN T.

7 5 . If a: V0 49 1 2 }I x 75, what is the simplest value of z7 6 . Howmany cubic feet are there in a granite monument 1 5 feet

h igh , 40 inches square at the base , and 24 inches square at the top7 7 . F ind the entire surface of a cube each of whose edges measures

4} inches .

7 8 . How many cubic inches of water can be contained in a gl obe

whose diameter inside is 1 5 inches7 9 . What is the area of a triangle whose sides measure 20 , 28,

and 32 inches What is.the l ength of each side of a similar triangl e

containing one hal f the area

80 . Wishing to find the number of cubic inches in an irregul arpiece of stone , I sunk it in a cyl indrical glass of water and foundit raised the water 2 5} inches. The glass was 5 inches in diameter .How many cubic inches were there in the stone8 1 . A wheel 6 ft. in diameter is turning at the rate of 800 revolu

tions a minute . At the rate of howmany miles an hour is a point onthe rim of the wheel mov ing

8 2 . If a square contains 342} square feet, what is the l ength of one

of its sides

83 . In a rectangular courtyard there are laid paving stones,

the number in the l ength being twice the number in the width. Whatis the number each way84 . Two paral l el sides of a quadri lateral fiel d are 42 chains and

34 chains respectively, the distance between them .is chains .

How many acres are there in the field

85 . The sides in a quadrilateral field taken in order around the

fiel d measure 25 chains , chains, chains, and chains

respectively ; the l ength of a diagonal drawn from the beginning of

the first to the end of the second side is chains. Howmany acresare there in the fie ld8 6 . How many square inches are there in the surface of the largest

sphere that can be put into a cubical box whose dimensions inside

are 1 2 inches each

87 . A father has settled on his daughter 1 500 a year. What sum

must he invest in government 4 per cents at 1 1 33} to produce that

amount of income

88 . If by investing in 5 per cent bonds I get an annual income of43 on my money, at what price were the bonds bought

Typography by j . S. Cashing 8: CO ., Norwood, Mass.

402

80 .

81

82 .

83 .

84.

Am.

333 2549 .

P ag e 2 6 .

338Daisy, 0770 15 .

Jumper, 6894 lb.

both, lb.

sq . mi .3

P ag e 2 7 .

3 9 1 80 .

933

mal es ,femal es .

3

P ag e 2 8 .

3333 6381

3

P ag e 2 9 .

Answers to dril l exercises wil l be foundin the teacher’s Key .

1 .

1 1 .

1 2 .

1 3.

1 4 .

HPo

ms

e

ms

wp

P ag e 3 3 .

81 9 .

532 .

347 .

206 .

408 .

8 1 6 .

624.

51 7 .

1 003 .

709 .

785.

323.

644 .

866 .

ANSWERS.

P ag e 34 .

2092 .

598.

239 1 .

908

0 1 99.

2579 .

988.

6309

4802 .

P a g e 35 .

638.

3808.

8598 .

4651 .

6 1 69 .

1 355.

4079 .

1 479 .

6224 .

1 098.

346 .

6682 .

909 1 .

Am.

1 8 1 1 42 .

P ag e 3 7 .

27 1 0 .

5688.

1 828 .

59 .

1 76 .

894 in.

1 49 1 .

6989 .

249 1 ft .

3 1 6 .

Ten 98’s.

rd.

P ag e 39 .

38 y .

3 789 .

3 31 65.

3550 m.

P ag e 40 .

1 522 m.

bush .

1 240 gr.

333 1 50 .

m.

in .

P ag e 4 1 .

6288 ft.

4465 ft.

1 332 ft .

3587 ft .

2 1 1 7 ft.

AN SWERS. 403

Em. A71 3 .

1 51 . Mt . Blanc

1 663 ft . higher.

1 52 . Mt. Blanc ;6065 ft. higher.

1 53 . Mt. Hercul es ;3724 ft. higher.

1 54. 42 1 4 ft.

1 55. 98 l b.

1 56 . rd .

1 57 . 69 m.

1 58. 1 826 ; 83 y .

1 59 . 1 763 ; 1 3 y.

1 60 .

P a g e 4 2 .

1 6 1 .

1 62 .

1 63.

1 64 . 3 2255 ga in .

1 65. 1 1 1 .

1 66 . inbah .

1 67 .

1 68.

1 69 .

1 70 . Over P aris,Over P hi l . ,

P ag e 4 3 .

1 7 1 .

1 72 .

1 73. lb.

1 74 . Dr.

Cr.

Bal .

1 75. Dr.

Cr.

Bal .

1 76 . Dr.

Cr.

Bal .

1 77 . Dr.

Cr. 3 1 22 005.

Bal .

P ag e 44 .

Answers to dril lexerciseswil l be foundin the teacher’s Key.

406

Ana.

7778 ; 2 rem.

9044 ; 1 rem.

2 rem.

1 rem.

6 rem.

773 packages.

«M oms ;33 cans.

249 cows.

372 cars .

762 cords .

603 squads536 squads .

750 hours250 hours .

392 W . 1 da . rem.

1 60 yd . 2 ft .

1 79 1 yd. 1 ft.

523 yr. 8 mo.

452 it .

385,568,264 f.

290 ,01 3,444g.

AN SWERS.

Ans.

P ag e 79 .

3342 pupil s ;38 pupils .

2688 1b. ; 1 792 l b.

3 32 . 8 1 .

3 8086 .

8

P ag e 80 .

3cu. ia . ;cu. in.

cu. in.

3 3833 i .

P ag e 8 2 .

1 84 .

20791 ,or

2281 1 ,or

6532.or

99%i ,or

26552 9

or

5623.or

AM .

or

or

or

or

201 367 ,

or

or

77233,or

3941 1 .or

1 803217,

or

2641 31 .or

2 70381 .or

43771 ,or

or

58,or

73753or

2985743,or 2985.430

or

7005332,or

40 ,53 1 1 175”or .078

or

32 1 43 3 .

or

1 08 1 79 1 ,or

AN SWERS . 407

Am. Eb. AM .

P ag e 83 . 1 7 1 . 395g ,

1 44 . inches .or

1 72 ° 3T“§ 2 a

or 3 .6 1 4P a“ 84 °

1 73.

58

1

6

8

1

3. 5

23” or

Q

8386 packages

4553 rem.

57 kegs

4 1 lb. rem.

centals.

or

or

2431 338.or

23388.or

1 1 631or 1 1 5. 289 m

1 533337or 1 53 .846 "

P ag e 85 .

37 customers ;20 lb. rem.

28 customers.

56 lb.

P a g e 8 7 .

1 200min . ; 20 h . 1 95. 32 sheets.

bbl . 1 96 . 834

70 lb. rem. or sq . m.

23 tickets ; 1 97 . 720 bbl .

1 7 rem. 1 98. 81 2 trunks ; 1 99 . $ 1 1 75.

3 1 rem. 200 . 1 1 9 bbl .

20 rugs ; 30 rugs . 20 1 . acres .

1 1 bbl . ; rem. 202 . 348fi fig ,or 48 .305 204 . 1 4 3 5 2,

NH, or 1 4 .334

or 1 9 . 696 205 .-o

2083, 206 . 576gg,or or

1 74.

1 75.

1 76 .

1 77 .

1 78.

1 79 .

1 80 .

1 81 .

1 82 .

1 83.

1 84 .

1 85 .

1 86 .

P a g e 86 .

3or 3 mil l s .

3or 4 mil ls .

6624.

1 9 1 7 .

1 1 ; 24.

245.

3 1 47 .

1 05

67 1 33.or

3 76 .

1 87 . 1 6

1 88.

1 89 .

1 90 .

1 9 1 .

1 92 .

1 93.

1 94 .

a3

7 .

or

972 .

5280 .

Ana.

P ag e 88 .

207 .

208. 66go;

highest Mon

day,209 . 89 m. 5080 ft.

2 1 0 . sec.

2 1 1 . mi les.

2 1 2 .

P a g e 89 .

Answers to dril lexerciseswil l be foundin the teacher’s Key.

P ag e 9 1 .

1 . 80 .

2 . 44.

3. 6 .

4 . 1 6 .

5. 36 .

6 . 4 .

7 . 8.

8. 43 .

9 . 3.

1 0 . 34 .

l l . 53.

1 2 . 8883sq . yards.

1 4.

1 5. 1 6 days .

1 6 . A, 1 077 votes ;B

,1 507 votes .

1 7 . 1 600 coins .

1 8. 3 1 70.

P a g e 9 2 .

1 9 . 3 4500 .

20 . $ 1 .

2 1 . $ 279 .

3 285.

23 .

P ag e 9 4 .

24 .

25.

26 . OHdays.

27 . 2 1 6 strips.

l oaves.

29 . 3 3 .40 .

408 AN SWERS.

Ana. AM .

1 25 feet. Pm 1 0 1

83. Balance ,$ 546 . 84. 83 656 .

24,130606 times .

3Pm m

85.

86 . 3P age 9 5 .

87 .

388.

3 7-5089 . 8more .

90 $ 1 0 05G ained 8 540 ; P ag e 9 8 .

per A.

P 1 0 3225 men. 66 .

a”

6 7 . inhab. 93 ,

68. inbah. 94,

69 . inbah. 95,

1 350 seeds. 70 . inbah . 96 .

7 1 . inbah. 9 7 ,

Av. 72 . inhap. 93 ,

sq . mi . 99 .

74 . 1 000 sq . m1 . 1 00 ,

46 1 20 ? years 75. 1 645 sq . mi 1 0 1 .

47 . 500 mo. ; 1 25 mo. ; 76 . The G reat Lakes ; 1 02 ,

1 0 yr. 5 mo. sq . mi. 1 03,

48. in . larger . 1 04 ,

49 . 77 .

50 . 26 cents. 78. in 1 880 ;51 . in 1 890.

52 . 79 . 1 05.

53 . $ 6000 . 80. 1 06 .

54. 1 6 books ; 348. 1 07 . 1 2 .

1 08. 24.

P ‘S’O 9 7 .

P ag e 1 00 .1 09 .

55. mi . 1 1 0.

56 . ft. ; 8 1 . Amt 1 1 1 . 43 .

1 1 mi . ft . 1 1 2 .

1 1 3 . Sum, 6580 ; dif. , 6524 ; prod . , quo. , 234 tota l ,1 1 4 . Sum

, 6825 ; dif ., 6747 prod . , quo . , 1 74 ; total ,1 1 5. Sum, 7943 ; dif. , 7849 ; prod. , quo . , 1 68 ; total ,1 1 6 . Sum, 8960 ; dif. , 8848 ; prod. , quo . , 1 59 ; total ,1 1 7 . Sum

, 851 4 ; dif., 83 1 6 prod ., quo., 85 ; total ,1 1 8 . 8 . 2 , 3, 4 , 6 1 2 . P ag e 1 0 8 ,

1 1 9 8 262-79 9 . 2 , 4, 8. 1 6 . 1 1 6 =2 2 x 29 .

1 0 . 3 . 1 7 . 1 64 X 4 1 .

“ 8 ° 1 1 . 1 1 . 1 8. 1 76 24 x 1 1 .

4 . 5, 3, 9 . 1 2 . 2,4 , 8. 1 9 . 204=22 x 3 x 1 7 .

5. 2, 4 , 7 . 1 3 . 3, 7 . 20. 252=22 x 32x 7 .

6 . 3 , 5. 1 4 . 1 1 . 2 1 . x 33 x 5.

7 . 2, 5, 1 1 . 1 5. 2 , 3, 4 , 6 , 7 , 8, 1 2 . 22 . 342=2 x 32 x 1 9 .

Ans .

sq. mi .8

sq. mi .2950 sq. mi .385 sq. mi.

inbah.

inbah.

inhab.

inbah.

41 0

235.

2m.

1 07rg.

l 4agg.

l ssggg.

1 72g95.

zo7gg5

72fgg .

1 30gg.zosr.

i 44§gg.238fi .

245§gg.

Page 1 20 .

1 9”gal .483k; yd.

1 7 113r rd .

322 113 lb.

5yd.

P ag e 1 2 1 0

- 5i s2 271 3 1

11 .

33333

738

AN SWERS.

Am.

i gg.1 78fya.

347g.

957gg.

26gdegrees.2} yards .

P age 1 2 2 .

fl .

years .67“ yards.

8 1 1 5.

633.

8 1 53.

. M of his

money .

28g5.

1 3g; ; zegg.

1 4g.agg

'

ux

42g95.

m

Am.

425gsq . in . ;

728} sq . in.

8

2 1 1 .

l b .

1 96.

1 96 .

1 97 .

1 98 .

1 99 .

200 .

20 1

202 .

203.

204.

205.

206 .

207 .

208 .

209 .

2 1 0

2 1 1 .

2 1 2 .

2 1 3 .

2 1 4 .

2 1 5.

2 1 6 .

2 1 7 .

2 1 8.

2 1 9 .

220 .

22 1 .

222 .

223 .

224 .

225.

226 .

22 7 .

228.

229 .

230 .

23 1 .

232 .

233.

234 .

235.

371

331 19425

4§i

883.41o2 .

Page 1 2 8 .

1 452A.

1 8 1 8 .

$4 .31 4.

524 bags.8

890 .

sq. ft .

353 ; i t .

3780§ sq. ft. ;

Page 1 30 .

238 1 1 1

AN SWERS. 41 1

1 26 .

29 towels ; § yd .

left.1 6 sheets ; 9; yd .

left.weeks or 7 w.

1 d .

2 1 4widths .

41 2 AN SWERS.

A71 8 .

8 per day.

AreceivedB received0 received

. fi:

Page 1 3 9 .

56} gal .8 1 680 .

38 women ;304 men .

81 0 ? per qt . ; 50 11 ;

607 ;

Page 1 40 .

8$32 .4 l g

a88 23. 75g=s88

. 25

Ana. Ana.

3 lb ; 75 lb. 96 .

sheep.

acres.99 . 3 1 652 ; 8 4 1 72 .

1 8075.

Page 1 45 .

1 00. $ 360 .

1 0 1 . 750 shingles.Pace 1 4 2 1 02 . l st, 8 1 00 ;0 1 1 8 535 3 1 1 8 6 .

2d , 8 1 50 .

2 ,1?ilr yd . 8 Net loss, 8 1 0.

2 7 1, yd.1 03. 0 2 1 6 .

1 4 s a1

8 1 05. 80 min .

353bushels .1 07. 41

47 days .

1 08. 44days.8 1 09 . 5} days.338§ m. 1 1 0. days3 494. 1 1 1 . 8 52 .7 1 .

l st year 2300boxes . P 3 8 0 1 4 6 .

2d year 1 840 Answers to the dril lboxes. exercises will be found

in the teacher’s Key.

600 girls ; Page 1 4 8 .

200 boys.8 1 880 .

1 2 men.

8 6 .

1 8 days23 lb.

81 1 0 1

16 miles.

962; miles.1 44} ft.532 times .1 2a» ; 6 }fi.

P ag e 1 44

8 N . Y. shillings.

3; 2.

6 1 4 ; 44 ir. , 3 ¢rem.

8a1 9443 lb .

576 pupils.

41 4

257 .681 m .

Am.

0 .885m .

0 .-006

81 .2 .

23 .796 m .

257 . 1 42 m .

4690 .

9666 .666 m

0 .235m

0 .6 1 7m .

49 .453m .

7 .042 m .

s3 5892 .

3 0 . 1 88m .

3£ 1 0 ; 3 rem.

1 29 roubles

Page41 yrem.

1 6 6 .

yd .

1 70 lb . 544 lb.

meters.1 6 .406m yd .

1 89 in in .

33

m.

or

0 .026m .

g ; $2550 .

P age 1 6 7 .

a0

ANSWERS.

cu. in.

gal .

84 .1 5m gal .

48 .38m bu.

Page 1 6 2 .

ED.

222 .

223.

224.

225.

226 .

227 .

228 .

229 .

230 .

231 .

232 .

234 .

235.

236 .

237 .

238.

239 .

240 .

24 1 .

7 1 daysdays.

6 miles .

ft.ft. and

times.

8 1 23 ; 89

cu. in.

855553 lb . or5555.555m lb.

Page 1 58 .

8 57 .

83 252 .

338833

Page 1 59 .

Answers to drill exercises .

will be foundin the teacher’s Key .

7 .

8.

9 .

1 5.

1 7 .

albs.Troy5

1 097} 1 b . Av.

1 411, lb . Troy .

Page 1 6 8 .

380

AM .

3£ 20 549420 . 1 68m marks .

roubles.

51 8 . 1 34m francs.

1 08} shil lings .

Page 1 6 7 .

1 24 in.

340 min.

4 1 80 lb.

1 05 oz.

1 272 d.

6 1 5ft.ft.

sq . ft.2 1 ed. i t.

$640 .

3

8 336 .

1 4 1 0 geog. m.

commonm.

Page 1 6 8 .

ft.sq . ft.

284; rd.

320 gr.

pwt.

d.

in .

9 6534 .

1 750 lb.

33'

1 0 cu. ft. 864 cu.

in .

1 7s. 6d.

5 1 28 . 5d.

203 rd. 1 0 ft. 6 in .

37 min . 30 sec.42 sq . rd. 20 sq.

yd. 1 sq . ft . 72 sq .

in .

2 pk. 6 qt. pt .

6 0 2 . 1 5 t .

2 qt. 2 Qt

Am.

22 sq. yd.

sq . in .

5 ft. in .

1 4 oz.4'

Page 1 6 9 .

5 bu. 1 pt.70 gal . 1 qt .26 lb . 1 2 oz .2 oz . 1 9 pwt. 2 gr.

6 reams 1 6 quires22 sheets .1 2 d . 22 h . 42 min .

1 0 h . 45 min . 4 1

sec .1 40 rd. 3 yd . 1 ft.2 1 sq. rd . 1 3 sq .

yd. q . ft. 1 08 sq.

1 8 cu. ft. 624 cu .

1 7 oz . 3 pwt. 1 8gr.

7899 m. 588 ft.£ 6 1 1 3 . 85d.

£ 4 1 1 3 . 1 0d.

ANSWERS.

2° 53'73° 54 !

2 0 26 ’ 1 0 1 /I

1 24° 1 9'

Page 1 76 .

8 cd. 1 cd . ft.6 T. 1 600 lb .

1 1 4 ft. 5 in .

54 gal . 2 qt. 1 pt.2 gi .

1 96 bu . 3 pk. 4 qt.26 cu . yd . 1 05cu . ft.592 sq. ft.

1 08 sq. rd.

1 pk. 3 qt . l pt .

41 5

Ans.

P age 1 7 2 .

1 8 1 4s. 3d.

24 ed . 6 cd. ft.9 bu . 1 pk.

47 A. 93 sq. rd.

1 83} sq. ft .

850 cu . yd . 1 8 cu.

ft. 968 cu. in .

264 qt.3073 ft . , or 1 86rd . 4 ft.

426 lb.

1 yr. 24 days .1 3° 39 ’

£ 6

204 1%acres.25 l 6 oz.48 gal . 2 qt.

6 bu . 1 pk. 1 qt.46 m. 2345rd.

8 h . 44 min .

5 1 2s. 2d.

ft .Page 1 73 .

cu. yd .

1 ° 3'25"

54'

4° 42'3 1 "

23° 34'

93° 45'

2° 1 4'34"

Am.

1 1 mm.

1 8gdays.36 bbl. ; 4 pk.

renn.

1 00 bottles 6 qt.renn.

68 shoes ; 8 oz .renn.

Page 1 7 7 .

92 cups ; 7 oz.rend.

1 3 cases ; 1 0

rings .seconds .

90 times .times .

3006 § revolu

tions .

40 1. days.

Page 1 79 .

2°

93° 1 7’

1 6° 2 1 ’

45° E.

75'W .

1 8°W .

62°E.

1 6 7° 48’ 45" E.

4 h . 53 min .

sec.

7 h . 23 min . 34 1.89 0 .

4 h . 55 min . 37

sec .

8 h . 9min 24 sec .

8 h . 1 7 mm. 1 6 ;88 0 .

1 1 h . 31 min . 3g88 0 .

5 o’clock 1 7 min .

sec . P .M.

5 o’clock 7 min .

48 sec . a m.

l l o’cl ock 1 8min .

52§ sec . a n .

8 o’clock 58min.

41 6

Am.

Page 1 8 1 .

Answers to drillexerciseswill be foundin the teacher’s Key .

Page 1 83 .

l . 1 800 bricks .

2 .

3 . $ 640 .

4 . 0 2 1 78.

5. 9900 sq . ft.6 . A 0 1 40 ; B 3 1 1 2 .

7 . 1 2 1 ft.

8. yd.

9 . 9 sq. ft. sq . in .

1 0 . sq. mi.1 1 . A., or 1 0

A. 7 1 82 sq . ft.

1 2 .

Page 1 85.

1 3. 576 people .1 4. 1 60 acres.1 5. 440A. ; 1

53 ; $920 .

Page 1 86 .

1 6 . 225 sq. ft 1 51 2 ;sq. ft. ; 304 sq . ft.1 7. 25 ft.1 8 . 76} sq . ft.1 9 . 7 sq. ft.

20. 4 O sq2 1 . sq. ft .

22 . sq . ft .Page 1 8 7 .

23 . ft. ;ft.

24. ln . ;

yd.

25. ft.26 . ft. ;

rd.

27 . ft. ;ft.

28 . sq. rd .

29 . 89 sq . ft.sq. in .

30.

3 1 . sq . ft.

ANSWERS.

Ans .

sq . ft.ft .sq. ft .

sq . ft.sq . ft.

480 cu . in . ;lb.

2475 lb.

47 ft . 3 in .

1 33} cu. yd.

Pag e 1 8 8 .

7 ft. 641, in .

Page 1 89 .

36 board feet.1 6 board feet.56 board feet.2 1 1 board feet.60 board feet.60 board feet.50 board feet.56} board feet.

8 1 6 .

8 1 08 .

8

1 82 board feet.board feet.

85825 board feet.Pag e 1 9 0 .

sq . yd .

4 bundles.shingles.

1 48 sq . ft . ;sq. ft.

1 2 bund. ; 81 75 sq. ft.1 20 clapboards.686 clapboards .Page 1 9 1 .

1 6 rolls .9 rolls .8 rolls.

20 gr.

Page 1 9 8 .

27 bricks .900 bricks .581 1

9T perches .

83

8

Page 1 9 4 .

48 bu.

40 bu .

2 T .

1 33 1, cu . ft .625 lb.

2 ft. 3 in.

8 gal .

(1 .8al

lb 8} oz .gal .

Page 1 9 5 .

8 1 04 .

56“ rd.

86 furrows .205; ft.1 1 oz. 0 pwt . 1 5gr.

J uly 4 , 1 826 .

4 yr. 1 m. 1 4 d .

8 7 .42 .

359

i, gal .

b . 1 0 oz.

93 days.yd.

60 perches ;bricks.

41 8

1 2

37

2 1

46 woodland2 7 pasture ; 1 0 00marsh ; 1 7%tiage .

A. 26 1B, 33

0 ,

W illiam,

Francis,Mabel,Caroline,

rye ; 59 .

com .

6337.1 5.

1 2 .

the sumwhichisPage 2 1 4 .

New York,Chicago ,P hiladelphia,Brooklyn,

St. Louis,Boston,Baltimore

,

San Francisco,Cincinnati,Cleveland,Minneapolis,Omaha,St . P aul ,Denver,Page 2 1 6 .

lb.

8 1 88 1 .

AN SWERS.

Am.

81 50 ft.5350 hours.

8 648 .

sq . yd.8 828.

1 82.

20 .

4800 cu. in.

2580 miles.

8 2 1 5.

5 ct. a pound.

8 5000 .

3 600 .

81 6 yr. 01 d.

Page 2 1 7 .

a day.

700 pupils.50 yd.

250 shares.

1 50 lb.

1 488.

88Earned 8 1 500 .

Spent for rent,coal ,

8 familyexpenses, 8 645 ;a n d s a v e d884.

8 4.

8

Pag e 2 1 9 .

3

a1 300 .

a9000 .

8 400 .

1 3

3

3-1275

Wheat at

barley at 7com at 74g} ,oats at

80 °

. 9 1 5 ;

Pag e 2 2 2 .

88000.

82350.

8

Page 2 2 8 .

0 2250 .

3233 .

3

Pag e 2 24 .

Direct discountof 25

28better,

by A.

2 5 .

880 ; 8 1 6 a year.

8 1 6 1 03 3.

To pay 2 a

year.

1 ;

8830 .

Page 2 2 6 .

847 .

86850 more83700 more82600 less .8825, 1 00 .

Page 2 2 7 .

One-third8

Page 2 30 .

R e a l e s ta t e ,per

sonal property ,86240 .

A tax rate of8 makes anactual overlay of830,800 . 1 2 ,whichis less than 5%of the valuation .

ANSWERS. 4 1 9

Ana.

Rate ,Actual overlay,warrant

,

Page 2 3 2 .

82688.

8

859 .

8

Page 2 3 3 .

38888

8 1 92 .

8 5950 .

Page 2 34 .

8 25.

8 4000 .

8 486 . 75.

8 2266 .

42 shares.888

A1 38 .

82050 .

Page 2 35 .

8 2 1 0 .

8 205081 5 1-88882 carrying8 to surplus .

8 2875.

Page 2 3 7.

8888 336 .

8888

Page 2 38 .

8888888

8 50 .

8 240 .

1 80 .

8 320 .

888888 675.

8 1 25.

$5680 .

8 1 00 .

Page 2 39 .

8 0 . 92 .

8

420 AN SWERS.

Page 2 4 1 .

Eb . Ans . Ans.

72 .

73 . 8 874.

75.

77 8 233 878. 8 079 8 828. 080 . 8 685. 75. 08 1 . 8 1 01 .7 1 . 082 . 8 083 . 8 884 . 885. 886 . P 8 8 0 245

3 y 3 mo. 8 d87 . 4 y 3 mo. 1 4 d88 7 y . 7 mo. 24 d

8 2 y 1 1 mo. 2 1 d90 . 2 y 8 mo .

3 y 3 mo . 1 5 d9 1 . 1 y . 3 mo . 2 1 d

6 mo. 1 8 d.

92 . 8 88 8

888 1 008 1 9 .

422 AN SWERS.

Ea . AM . a s. Ans .

250. 8 252 . 88 8

251 . 8 253. 8 54 . 53

8 8

Page 2 7 1 . Ex . 2 54 .

600 . [Dated in America ]Sixty days after sight of this first “ of exchange, second and thirdunpaid, pay to the order of [Engl ish Merchant] Six Hundred P oundsSterling, value received, and charge the same to the account of

[American Banker. ]To [Engl ish Banker] .

Cost, 8 29 1 3 ; 8 2934.

Page 2 7 2 . Ex . 2 5 5 .

1 75. [Dated in England.]Thi rty days after sight of this first of exchange, second and thirdunpaid, pay to the order of [Eng l ish Banker] One Hundred Seventyfive P ounds Sterling , value received, and charge the same to theaccount of

[Engl ish Merchant. ]To [Your name] .

Page 2 7 2 . Ex . 2 56 .

8 4000 . [Dated in Engl and ]Sixty days after sight of this first of exchange, second and thirdunpaid, pay to the order of [your name] Four Thousand Dollars,value received, and charge the same to the account of

[Engl ish

To [American Banker] .

Page 2 7 2 . Ex . 2 5 7 .

8 3675. [Yourp lace of business, and date]Sixty days after sight of this first of exchange, second and thirdunpaid , pay to the order of [American Banker] Three Thousand SixHundred Seventy-five Dollars, value received , and charge the same tothe account of

[Your name. ]To [Engl ishMerchant] .

Ans.

Page 2 74 .

8 9780 89760.

888 4885 .

88

263 . 8264 . 8450 .

258.

259 .

260 .

26 1 .

262 .

Page 2 75 .

265.

Page 2 7 6 .

266 . 8 1 2 ,003.

267 . 834 ,994 .75.

268. 8 1 5,269 . 836 ,

Pag e 2 7 9 .

270 .

Page 2 80 .

866 1 9 .

8On Wabash l sts,4

G en .Elec . , 5N . J . G en .

U . S. 5s,2 75. On U . S. 4%cou

pons,N . H. 6s,Boston 48 ,Minneapolis 4s,4 007

2 76 . On0

Atch . 4s,5.

Duluth 48 ,2 77 . 82937 .50 .

27 1 .

2 72 .

273 .

2 74 .

Page 2 8 1 .

278. above parbelow par.

279 . 7 1.z%.

280. 86000 .

ANSWERS.

Page 2 84 .

6 mo . 1 3 d.

1 8 months afterthe date of thepurchase .J an . 1 1 , 1 895 .

J une 24, 1 896 .

May 1 7 1 896 .

Page 2 8 7 .

297 . May298 . Aug.

299 . Dec . 7 , 1 893 .

300 . J an .

423

Ans.

3 1 3 . 8 666A larger returnfromstock by about

Rs.

the 6%

Ans .

1 66g.

The former,11020 Of 1%

Nov . 2 1 .

francs .

or about9 d . 2 far.Page 2 83 .

4 s . 1 d . l }; qr.Apri l 23 ° francs.

Dec. 1 6 .

Oct . 1 4 ' 1 mark6 mo . 29 d . francs.

1 francmark.

P 8 8 0 2 88 Page 2 9 5 .

301 . Nov. 25, 1 894 . g 1 03367 1 1

302 . 23455101 1

303 . 20 1 11

304 . 8 49 y305.

9306 . J une 2 1 , 1 8 7 .

307 . 8 33333 3 .

Page 2 9 6 ‘

308.

9000 1 10 .

309 . nn

gem

3 1 0°

320 laborers .2 h . 36 1 min .

P ag " 2 89 ° 2 h . 40 min.

3 1 1 .68T h .

3 1 2 .72T

424 AN SWERS.

Am.

Page. 2 9 7 , 29 8 .6 in .

3 in .

2 and 2 1 in .

60 and 63 ft .57 and 76 in .

Hypotenuse_ 38

§63}m.

paral el line33} in.

Page 2 9 9 .

Yes, for the sidesare proportional.66

515

“ft.

Page 800 .

895;r ft . ; ft.20 ft. 84 ft. 1 }inches square .1fluin. {5 x 1 }m.

“in

x 2 } in.

Page 30 1 .

81 } ft .

1 666 2 times .

Page 80 2 .

4 ft . 8 in . long ;2 ft. 4 in . wide .Change width to

1 ; ft. , and thickness to ft .

in .

Edges, 25 in

sides of base, 1 0m.

a . Ans . Am.

P I SO ! 80 4 .

$ 1 899 ; $2532 .

“33650881 .A56

B:C ,D ,

Pag e 3 1 2 .

2 3 4kin .

0

86, 1 863, é;240 acres . 1 8.

28, 35, 42 , 35, 44 .

28 mi . 6 1 .

$7000 , $ 8500 , 29 .

$9500 , 75.

81 .

1 9 .

94.

35625 3 99 .

2 7 ; 1 .

65 ; 75.

1 7 ; 1 1 .

Page 300 1 5 ; 25.

38 ; 6

76 ; 24 .

42 ; 6

1 8 ; 36 .

49 ; 49

8 99 ; 99 .

5528118 76 9337 ; 7 1 .

51 ; 1 9 .

89 ; 79 .

426 AN SWERS.

m Am' Afl‘. AM .

1 76 . About 2 ft. 1 0 in. Circumference pm 35 1 .

square and 1 ft. mi .6

5 in . thick. diameter633 £

1

3

5 1 9 ft.1 77 . in. 7925 mi .

6

0 1 -49 ft

1 78. in.

9 . ft .1 79 . in . 1 NOT3 .

— 'rhe Bide being

represented by w, the apo

Page 84 2 . them is m asor

7} sq. ft.— 2

1 47 sq. ft.sq . ft.

1 200 sq. ft.25§ sq. ft.86 1 48 sq. ft.

sq. yd.

sq. yd.wws

a

e

s

wp

r

Page 344 .

1 1 . ft1 2 . m1 .1 3. ft.1 4 . Inner, ft .outer, ft.

1 5. ft.1 6 . sq . in .

1 7 . sq . in.

1 8. sq . in .

1 9 . sq. ft.

20. ft. ; 7 .98 ft. ;ft .

Page 34 5 .

ft .

Each segmentsq . in .

1 9 11,in . or

1 n.

sq . yd. ;sq. yd.

sq. ft.in .

57° 1 7 ’

Arc in.

1 ’ in .

1 ”

in .

in .

Page 84 8 .

73.

63 .

80.

65.

$85‘

70.

7 11 02 .

80.

72 .

96 .

73 .

74 ,

75.

76 .

in .

77 .

Page 34 9 78.

24 ft. 79

1 94 yd.

80

1 06 mi. 8 1

75 mi .30 ft. ; 34 ft . 83

29 in.

84

30 ft. ; 3 1 20 sq. ft.

37 in .

in . ; sq. in . 85

Page 3 50 .

86 °

28 ft . in . 87 .

8 .66 1n . ; 433 z5oo; 88.

sq . in .

Each triangle , 89 .

sq. in . ; 9 1 .

each segment, 92 .

sq . in .

in .

sq . in . 94 .

in .

in .

in .

cu. in .

96 ; sq. in .

f of 1 cu. ft.Sq . ft.

Page 35 8 .

sq. ft.cu. ft.

cu. in .

sq. in. ;cu . in.

sq. ft .

sq . ft.cu . in .

cu. in .

gal .

sq. ft.cu. ft .lb .

cu. in .

lb.

Page 359 .

cu. in

D iameter,depth , in .

1 5783 cu . in.

2 1 5 cu. ft.lb.

qt.sq. in.

cu. in.

sq. mi .2 1 6 cu . in .

cu . in .

cu. in .

cu. in.

Ans .

Volume ofcylinder=g1rR3sphere r R8

cone t Rs.

Page 3 60 .

in.

in.

cu . in .

sq. in .

The area of theconvex surface isfour times thatofthe base .Height,D iam. of base ,

Page 36 1 .

30 in.

20, 24 , 28 in .

1 1 00 ft.As 9 to 1 6 .

One square isdouble the otherin area .

ih . ; as 1

to V}.

Page 36 2 .

1 200 sq . ft .Its area will be3g of its originalarea, and willbe diminishedby9

in .

in .

31 25 gal .

1 0 in .

7 .07 in .

in .

Page 3 6 3 .

793%cu. in.

cu. ft.lb .

oz .ozoz.

ANSWERS.

Am.

Page 36 4 .

in . in.

64 tons.49 and a.fraction.

The diameter ofSaturn is about

and that ofJ upiter about

times thatof the Earth .

in .

in .

Silver, in. ;copper, in .

Lower frustum,

in . ; upper

small cone,m.

Page 36 5 .

l st,2d,3d , 34th, 31 05.

mi.8

westeast .

{5768.

mi .T.

The latter,greater.Page 36 6 .

8 1 0 .

or 1 07 .

33

33‘s

3 160116

sec .Page 36 7 .

3 30 .

a

427

Am.

237 cu. yd.

0(L96 in .

0 32 70 ; 72m.

lb . , or70 T. 625 1 b.

Page 36 8 .

966 2 lh . ; 26§ lh. ;2033 lb.

8 i t .

sq. yd .

$ 7 ; wouldb e ra i s e d to

F . C .

Page 36 9 .

3ft.

32 ft.mi .on $ 1 ;

3 more .

Page 3 70 .

20 more men.

1 1 2 cu . ft .

mi .in .

2 T. lb.

sq. ft.

Page 3 7 1 .

ft .cd .

9 times .cu m.

in .

5 oz . 1 8 pwt. 1 8 gr.

1 3 pwt. gr ;44 2

7v

428 AN SWERS.

Ana. Am.

l b. , or772 T. 400 l b.

8

Pag e 3 7 2 .

Royal , 8 1 500 ;N .A. , 82250 ;Farmers’ , 0 750.

8 left.

0

4 1 ft. 4 in.

Page 3 73 .

3 1 81

155332, l n ., or 0 .

1 37 1 . 77 y .

y .

2 1 ft .

1 8331 4 y . {g m2d , ( 1 . 59 .7573

”

72

68

(5 . 69

75.25%3d, 69 64%Page 3 74 .

Edge ,in . ; slant height,

in . ; area,sq. in .

Edge ,in . ; slant height ,

cu

Page 882 .

sq. in .

times ; 1 3

times .or

9 T. 1 29 1 lb .

O! .

5 lb . oz.l b.

Page 3 76 .

1 . 2772 ,

Am.

Page 888 .

Page 3 9 3 .

649 . 1 1 5m

300 times.

or

1 2m length8393 1 .

81 35 Sum , or 4 oz.7 pwt . gr.

Tonncaul’ 0 1

‘

5 T.

7200 Kno

Page 3 94 .

8888 2 , or88 or888Mr. Smith ,99 gain .

Page 3 9 5 .

88 3.

Check, 8 1 506

unpaid, 8 1 94 .

35. 1 6 .

9

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