Ma Teema Tics F or Comm on Schools Part III Higher Arithmetic

MATEEMATICS FOR COMMON SCHOOLS

PART III

HIG HER A RITHMETICINCLUDING

EASY ALGEBRAIC EQUATIONS AND SIMPLEGEOMETRICAL PROBLEMS

JOHN H. WALSH

MM TBNDBNT Ol' PUBLIC INSTRUCTION,

BROOKLYN , N .Y.

REVISED

BOSTON, U.S.A.

D. c. HEATH co.,PUBLISHERS

1898

PREFACE.

Marnm rros roa Connor Sonoors is a one-book arithmetic inthree

parts . Part I., anEwnsuraar Anrrnumrc

, contains those portions of

the subjectneeded by allpupils of the commonschools : addition, subtraction, multiplication, and divisionofwhole numbers ; sh ple fractions ; andthe most commonly used denominations of compound numbed .

Part II.,anIsrsausnrarnAsr'rnu srro, intended for pupils of the fifth

and sixth school years, gives a sufficiently fulltreatment of commonanddecimal fractions, and of compound numbers, and takes up the simpler

and more practicalparts of percentage and interest. A shbrt chapter on

easy algebraic equat ions of one unknownquantity is added , to be used

whenever it is deemed advisable.

Part III. , a HIGHER Aar'rnxs 'rm,

completes the ordinary grammar

school course inthis subject, and contains, inadditionthereto , a chapter

onalgebraic equations and one onelementary constructive geometry , with

applications. These two chapters are placed , for convenience, at the end of

thework ; but it is intended that su itable parts of each be taught from time

to time throughout the last two years of the grammar school.

The special features of the work are its divisionof the arithmetical

portioninto half-yearly chapters, instead of the ordinary arrangement bytopics ; the omission. as far as possible, of rules and definitions ; the very

greatnumber and variety of the examples ; the use of the equationinthe

solution of arithmetical problems, especially inthose of percentage and

interest ; and the introductionof the elements of algebra and geometry.

Believing that there is some foundationfor the complaints frequentlymade by business menand high -schoolteachers that grammar-schoolgradu

ates are too oftenslowand inaccurate inordinary compu tations, the author

8 5965

iv PREFACE.

has furnished throughout the entire work systematic drills and rev iews in

the addition, subtraction, multiplication, and divisionof ordinary numbers

and of fractions.

Inth is endeavor to enrich the grammar school course inmathematicsthe attempt has not beenmade to shortenit so much as some may desire.

The intelligent teacher canand should do the remainder for h imself, byrigorously omitting allsuch t0pics as he finds unnecessary .

J . H. W.

Baooxnrn, N.Y.

,December, 1893.

CONTENTS. PART III.

CHAPTER X.

Anon-snare Equarroxs

ONE UNKNOWN QUANTITY

Clearing of FractionsTransposing

CHAPTER XI.

Panoram a: Isrssnsr Drsooum Soar-acne sunVonuxssPnsomwraon

To find the Base or the RateProfit and Loss

Msasuanu sn'rs

Inrssasr

Interest-bearing NotesSpecialDrills

ApproximationsShortMethods

Basx Drsoous 'r

Discount of Interest-bearing NotesEnglishMoney

Com macmr. Drsoousr

Suaracss u p Vowxss

CHAPTER XII.

Stuns u rnCOMPOUNDInrasssr Drscomrr Gauss ANDErrso'r

Paars sasmr Bonus AND S'rocxs Excnas on Louorrunn

anti) Tm s— Suaracns arm Vommss

v

vi CONTENTS. PART III.

SIMPLE INTEREST .

To find Principal, Rate, orTime

Interest by Aliquot Parts

COMMERCIAL DISCOUNT .

BANK DISCOUNTTo find Face of Note ,

Rate ofDiscount, Time

SpecialDrills

ShortMethodsMEASUREMENTS

CAUSE AND EFFECTPARTNERSHIP

Approx imations

BONDS AND STOCKS

COMPOUND INTERESTEXCHANGE

Domestic Sight Exchange

CircularMeasureTime Drafts

LONOITUDE AND TIMEBills ofExchange (Foreign)

CHAPTER XIII.

PARTIAL PAYMENTS—RATIO AND PROPORTION— SOUARESURFACES AND VOLUMES .

PARTIAL PAYMENTS—U. S. RULEPresentWorth and True Discount

SURFACES AND VOLUMESSQUAREROOTRATIO

SpecialDrills

PROPORTIONApplications of Square Root

CONTENTS . PART III.

MEASUREMENTSExact Interest

PARTIAL PAYMENTS MEROKANTS’ RULE

CHAPTER XIV.

EQUATION or PAYMENTS MENSURATION OF SURFACES AND VOLUMm BOARD MEASURE ANNUAL INTEREST GOVERNMENTLANDS—METRIC SYSTEM

EQUATION OF PAYMENTSMENSURATION or PLANE SURFACES

SpecialDrills

SURFACES OF SOLIDSPrisms and Cylinders

Pyramids and Cones

VOLUMES .

LumberMeasureSurface of Sphere

CURE ROOT

Volume of Sphere

ANNUAL INTERESTGovernment Lands

METRIC SYSTEM

CHAPTER XV.

ALSERRAIC EQUATIONS— ’

l‘wo UNKNOWN QUANTITIES THREE UN

KNOWN QUANTITIm PURE QUADRATICS —AFFECIED QUAD

RATIOSADDITION OF ALOERRAIC QUANTITIES

SURTRACTION or ALOERRAIO QUANTITIES

Remov ing Parentheses

Two UNKNOWN QUANTITIES

TRREE UNKNOWN QUANTITIES

viii CONTENTS . PART III.

MULTIPLICATION OF ALCERRA IC QUANTITIESPURE QUADRATICSAFFECTED QUADRATICS

CHAPTER XVI.

ELEMENTARY GEOMETRY— PRORLEMS IN CONSTRUCTION—PRACTICALAPPLICATIONS— CALCULATION or HEIGHTS AND DISTANCESMENSURATION

ELEMENTARY GEOMETRYExercises inConstructionProblems inConstructionEqualTriangles Equ ivalent TrianglesSimilarTriangles

CALCULATION OF HEIGHTS AND DISTANCES .

MENSURATION OF SURFACESPrisms

, Cylinders, Pyramids, Cones

Frustum of Pyramid orCone

Sphere

VOLUMESPrisms and Cylinders

Pyramids and Cones

Frustums of Pyramids and Cones

Oblique Prisms

Sphere

APPENDIX.

TARLES TIME BETWEEN DATES —DAYS OF GRACE PARTIAL PAYMENTS : CONNECTICUT, NEW HAMPSHIRE, AND VERMONTRULES

INDEx

HIGHER ARITHMETIC.

CHAPTER X .

ALGEBBAIOEQUATIONS.

ONE UNKNOWN QUANTITY.

OralExercises.

1 . What number increased by 12 equals 16?

2 . 9 added to a number equals 14 . Find the number.

3 . What number diminished by 7 equals 8 ?

4 . 18 diminished by what number has 10 for remainder ?

5 . Eight times what number equals 64?

What number multiplied by 9 gives 63 for product ?

7 . Three times a certain number added to twice the samenumber equals 40. What is the number ?

8 . 36 is equalto 10 times What number added to 8 times thesame number ?

G ive values of x , y, z, etc

1 . 4 . 18

2 . 5 .

3 . 6 . 9y= 63

448 ARITHMETIC.

7 . 3 x + 2 z = 40 10y + 8y— 4y

= 42

a. 10z + 8 z = 36 2E+ 4 z = 52 — 12

9 . 4 v — 3 v = 72 3 z + 4 z = 30— 9

10. 11w— 2w= 27 12y— 5y

= 25+ 10

1 1 . 3x — zz + 5z = 54 6w+ 6w= 16+ 8

850. Slate Problems.

1 . A horse and a wagoncost together $600. What is theprice of each , if the wagon costs twice as much as the horse ?

Let a: cost of home

then 2 :c cost ofwagon.

Cost of both 2x a: a : 600

3 a: 600

a: 200

2x 400

Am . Cost of horse, $200 ; ofwagon, $400

2 . Divide 100 into two parts, one of which shallbe four timesas large as the other.Let a: a one part

then 4 3: the other.

a: 4 4: 100.

3 . is divided among three children , the second of

whom receives twice as much as the first, and the third of whomreceives Six times as much as the first. Requ ired the Share of

each .

as, 2x,6 x.

4 . Ina class of 54 pupils, there are twice as many boys as

girls. How many are there of each ?

5 . The sum of two numbers is 78. One is five times as largeas the other. What are the numbers ?

6 . 156 is equalto seventimes a number added to five timesthe same number . Find the number.

ALCERRAIC EQUATIONS. 449

7 . The difference between three times a certain number andnine times the same number is 66. What is the number ?


whom receives twice as much as the first, and the third ofwhomreceives three times as much as the second . What is the Shareof each ?

9 . The sum of two numbers is 72, and the greater is 5 timesthe other. What are the numbers ?

10. John,Henry , and James have 54 marbles . Henry has

twice as many as John , and James has as many as the othertwo. How many has each ?

11 . The sum of the ages of mother and daughter is 42 years.

What is the age of each,i f the mother’s age is Six times that of

her daughter ?

12 . A man paid $ 96 for an equal number of hats and coats,paying $ 2 apiece for the former and 10 apiece for the latter.How many of each did he buy ?

13 . Divide 41 into four parts, the first being twice the second,the second three times the third , and the third four times the

(Let E the fourth .)

14 . The sum of three numbers is 180. The first is double the

second, and the third is three times as large as the sum of the

other two . What are the numbers ?

15 , Mr. Smith paid 81 cents for sugar and flour, the samequantity Of each . For the sugar he gave 5¢ per pound , and forthe flour 4¢ per pound. Howmany pounds of each did he buy ?

16 . The length of a rectangular field is 24 rods, its breadth is

a: rods, its area is 456 square rods. Find the value of z .

17 . It takes 340 feet of fence to enclose a square lot. What

are the dimensions of the lot ?

450 ARITHMETIC.

18 . Mrs . B. divides $ 120 betweenher sonand her daughter.She gives the latter twice as much as she gives the former.What is the Share of each ?

19 . The earnings of a man and his son during Januaryamounted to $ 175, both hav ing Worked the same number of

days. The father ’s wages were $4 per day ,and the son ’

s wageswere $ 3 per day . How many days did they work ?

20. The sum Of $ 240 is divided among four children , twoboys and two girls. Find the share of each

,if each girl

’

s shareis double that of each boy .

2 1 . A man worked twice as many days as his son. Theircombined earnings amounted to $ 165. Find the number of dayseach worked

,if the father earned $4 per day and the sonthree

fourths as much per day .

22 . A boy 's bank contains indimes, nickels, and cents.

There are twice as many nickels as there are dimes,and three

times as many cents as there are nickels. Howmany are thereof each ?

23 . Ipaid 75¢ more for a rollof 15-cent ribbon than I didfor a roll of 12-cent ribbonof the same length . How manyyards did each rollcontain ?

24 . A rectangular field whose length is four times its breadthrequires 250 rods of fence to enclose it . What are the dimensionsof the field ? (Make diagram.)

25 . A girlpaid 60 cents for a speller and a reader,the cost

of the former being one-third that of the latter . Find the costof each .

26 . The sum of two numbers is 72, and the smaller is onefifth of the other . What are the numbers ?

27 . Mary,Susan, and Jane have 54 hickory nu ts. Susan has

one-half as many asMary , and Jane has as many as the othertwo . Howmany has each ?

https://www.forgottenbooks.com/join

452 ARITHMETIC.

853 . Sight Exercises.

G ive values of x, y,2 , etc

2 . 6 .— 5

Z 7 §+g= 10

854 . Slate Exercises.

Find the value of the unknown quantity (x).

Ineach of the following equations, multiply both members by the leastcommondenominator of the fractions.

a: a: a:lo 26

Multiplying by 12, we have 6 40 4a: 3 a: 312

a: a:2 0 x 44

Multiply by 6. 6 3 3 3: 2 zc==264

3 .

4

;2

3

”4s 15 . 2x +

27 x: g :

3 4x 156 16 x +

524

a: 2 3: 75x 33 x5 . 28

2+3 100 50

81

18 . S}E—2{-x= 45

2x 3 x 8x 2x19 ,

3 4 3 5136

s. zgx= 115 so. 3gx = 116

23

22

am

ALGEBRAIC EQUATIONS. 453

z a: x _

a: x

45 -23

5 9 4

1 . The sum of two numbers is 90,and the smaller number is

one-fifth of the larger one . What are the numbers ?a:-90.(SH -

5

2 . Divide 100 into two parts, one of which shallbe 2} timesthe other.

3 . After losing of his money,a manhas $ 714. How many

dollars had he at first ?

(a:98

4 . A horse was sold for $240, the seller thereby gaining onethird of what he originally paid for it . How much did he payfor it ?

a:

5 . One-half of a number added to one-fourth Of the samenumber equals 66{ What is the number ?

6 . The difference between 4 of a number and 4} of the samenumber is 15. Find the number .

7 . One number is {L of another . Their sum is 55. Whatare the numbers?

8 . Find a fraction equivalent to t » the sum of its numeratorand its denominator being 60.

(Let 7 a: numerator and 8x denominator.)

9 . Find a fraction equ ivalent to the difference betweenitsnumerator and its denominator being 24 .

454 ARITHMETIC.

10. The sum of two numbers is 480, and the quotient obtainedby dividing the greater by the less is 7. What are the numbers?

1 1 . Find two numbers whose difference is 522 and whosequotient is 30.

12 . A boy buys apples at pears at and oranges atthe same number of each . How many of each does he buy , if

he pays 81If for all?

13 . A girl bought 70 cents’ worth of peaches and plums.

She paid 3 ¢ each for the peaches and each for the plums,buying four times as many of the former as of the latter . Howmany Of each did she buy?

14 . is divided among three persons, the second of

whom receives three times as much as the first , and the thirdthree and one-half times as much as the first . Find the shareof each .

15 . A farmer paid for a cow three-sevenths as much as hepaid for a horse. HOW much did he pay for each , if the lattercost $80more than the former ?

16 . Three times a man ’

s money increased by two-thirds of hismoney is equalto How much money has he ?

17 . After giving away 43 of his marbles and losing 4 of them,

Joseph has 24 left . How many had he at first ?

18 . Bought a coat,a hat , and an umbrella for $ 15, paying

for the hat 1} times as much as for the umbrella , and for thecoat 3} times as much as for the hat. Find the price of each .

19 . A merchant purchased two pieces of cloth for$ 240, paying for one piece twice as much per yard as for the other. The

former contains 36 yards and the latter 48 yards . How muchdoes he pay per yard for each ?

20. A farmer sold 4 times as many cows as horses, receivingfor all$840, at the rate of $40 for a cow and $120 for a horse .How many of each did he sell?


TRANSPOSING .

856. Sight Exercises.

G ive values of x , y,2 , etc

1 . x + 15= 21

2 . 2y+ l5= 2l

3 . 2

4 . 4w

v5 . 8

3: = 12 1=2

3 11

857 . what ?

Whenz x = 21+ what ?

If inthe equation 2x + 15 21,we take away 15 from the

first member , what must we do to the second member to preservethe equali ty ?By transp osing we mean bringing the unknown quantities

(x, y, z, etc .) to one side of the equation , and the known quantities to the other .

NOTE. Inbringing a quantity from one side of the equationto the other,the Signof the quantity is changed.

Find values of the unknownquantities.

NOTE. Clear of fractionswhennecessary ; thentranspose.

1 . x + 37 = 56 5 . x + 3x = 25+ 11

2 . 4 z 6 . 5x = x + 40

3 . 3E 7 . 3x — 20= x —8

4 . 7E+ 13 = 111 8 . 12 — 3 x = 45— 4 z

456 ARITHMETIC.

9 . SE— 6= 48 + x

10. 3 x + 6= 9 — 2x + 12

11 . 2x — 2 — 16= x + 10

x3

8 24

13 .

27 2714 0 510+


1 . The sum of three numbers is 51. The second is 5 less thanthe first, and the third is 10 less than the first . What are thenumbers ?Let a: a firstnumber,

a: 5 second number,a: 10 th ird number ;

z + x—5 + z —10= 51.

Transposing~

3 : 66,

a: 22,first number,

a: 5 a 17, second number,a: 10 12

,third number.

2 . Add 45 to four times a number , and you willhave seventimes that number . What is the number ?

(7 z = 45 4 x .)

3 . Nine times a number less 27 equals six times the number .Find the number .

4 . Two boys have together 48 marbles . One has 18 morethan the other. How many has each ?

(as, a:

7x — 5x = 20+ x + 4

6 x 14 16+ 9:

2 x—11+6x— 60=5z+25

x x = 105

z2” +

2 3

3 x x _ 3 x

5 2 4+ 27

ALGEBRAIC EQUATI 457

5 . The length Of a rectangular lot is 75 feet more thanthebreadth . The distance around it is 250 feet . What are itsdimensions?

6 . A piece of land containing 86 acres is to be divided intotwo fields

,one of which shall be 8 acres larger thanthe other.

How many acres ineach field ?

7 . At a certainelection votes were cast for two can

didates, the successful one receiving 318 more votes than hisopponent . How many votes did each receive ?

8 . A man, being asked his age, replied that if he were halfas Old again and 7 years more he would be 100. What was hisage ?

”

9 . The sum of two numbers is 96, and their difference is 72 .

Find the numbers .

(Let a: less,to 72 greater.)

10. After paying 4 and i of my debts, I stillowe $45. Howmuch did I owe originally ?

1 1 . Divide 45 into two parts, one of which shall be 6 lessthan twice the other .

12 . William has $5 more than John , and three timesWilliam’

s

money added to five times JObu ’

s would be $ 103 . How manydollars has each ?

13 . I bought 3 cows and 4 horses for $635, paying$80apieceless for the cows thanfor the horses. How many dollars apiecedid Ipay for each ?

14 . Mary has a dollar indimes and five-cent pieces. She

has 11 more of the latter than of the former . Find the numberof pieces of each denomination .

15 . Divide 100 into two parts whose difference shallbe 48 .

16 . Ina class of 54 pupils , the girls outnumber the boys by12. Howmany are there of each ?

458 ARITHMETIC.


Whom receives more thanthe first,and the third of whom

receives more than the second . Find the share of each .

18 . The greater of two numbers is 11 more than3 times theless. Their difference is 33 . What are the numbers ?

19 . A boy spent a dollar for postal cards,2-cent stamps

, and

5-cent stamps. He bought 15 more 2-cent stamps than 5-centstamps, and 15 more postal cards than 2-cent stamps. Howmany of each did he buy ?

20. A farmer has 88 head of stock— horses, cows, and sheep.

He has 17 more cows than horses,and the number of sheep is 22

greater than that of the cows and horses together. Howmanyare there of each ?

460 ARITHMETIC.

865. OralExercises.

1 . Find of 1 gal.

2 . 37570 of $ 24

3 . of 81 cows

6% of 150lb.

5 . 4% of 125 bu .

6 . 62570 of 1 pk .

7 . of $200

8 . 9970 of 200 gal.

9 . of $640

10. {70 of 800yd .

866. Slate Exercises.

1 . Find 63195 of

2 . 1270 of

3 . of $28 .80

4 . of

5 . of $92 .40

6 . 450% of

7 . 2070 of $51.60

8 . of $89.70

9 . of $ 73 .28

10. of

1 1 . of $25.6o

12 . of

13 . of

14 . 860% of $ 38

66396 of 66 horses

of lyd .

81% of $300

2470 of 80 sheep

40% of

20% of 65 rd .

10% of 15 lb.

of $60

of $ 72

1 of $96

of $24

25% of

60% of

6% of

of

12% of

of

4070 of

16070 of

s4% of $ 23 .75

66mg of $82575% of

saw. of $ 392

933% of

PERCENTAGE. 461

TO FIND THE BASE OR THE

29 . Find 257. of 280.

280 is called the base, 25 is called the rate .

Multiplying 280 by 12056 , we get 70, which is called the p er

centage; that is,dt

868 . Base x1

18

3: Percentage.

30. Find a: per cent of 65.

3 1 . If a: per cent of 65 is 26, find the value of z .

32 . Find 25% of x .

3 3 . If 25 per cent of a: is 42 , what is the value of x ?

34 . Increase a: by 20% of itself.

35 . If a: increased by 20 per cent of 3: equals 132, what is thevalue of z ?

3 6 . Diminish a: by of itself.

37 . Find the value of x,when x diminished by 33} per cent

of i tself equals 78 .

38 . Find x% of .3,

39 . What is the value of x ,if x% of {L

40. Find of x .

4 1 . Find the value of x ,when of x 23 .

42 . What (x) per cent of 65 is 26?

4 3 . 24 is 18 per cent of what number (x)?

44 . 250% of what number (x) 180?

45 . What number (as) increased by 25% of itself equals 85?

46 . g what per cent of i » ?

462 ARITHMETIC.

47 . is what per cent of 4?

3 xf 5 i .c.

,3 g

. Clear of fractions and solve.

4 100 6’

125 4

48 . Find the interest on 2: dollars for 1 year at

49 . What sum at gives a yearly interest of $44 ?

50. What is of

51 . What per cent of 88 is 33 ?

52 . Find the difference between of 800and per cent of 800.

53 . Find the selling price of a horse that cost $ 175 and was

sold at 2570 profit.

54 . How much insu rance does a man receive for whenthe rate is

55 . What is 16% of

56 . 3} is what per cent of it ?

57 . What per cent of is

58 . is 95% of what sum of money ?

59 . What number increased by 16% of itself equals

60. 984 is 133496 of what number ?

61 . 470 of a number is 81. What is the number ?

400 400

62 . Animporter paid duties amounting to If theduty was 25% of the value of the goods

,what was their value ?

63 . A collector deducts commission , and returns to hisemployer How much did he collect ?

a:

1—20ioz or, a:

4or, 333

35

PERCENTAGE. 463

64 . A commission merchant receives 2596 commissionforbuying grain for a customer.

’

The cost of the grainand his commission amount to How much does the grain cost ?

a: cost of the grain 5° commission.

40

65 . A capitalist sends a commission merchant $8 ,670 to investincotton and to include commission at How much doesthe commission amount to ?

66 . A commission merchant receives to invest inoatsafter deducting 2% commission . Howmany bushels of oats at30 per busheldoes he purchase ?

Should the commissionmerchant deduct 2% of or 2% of the cost

of the oats ?

67 . A house is insured forg of its value at The annualcost (premium) is What is the value of the house ?

Let a: value. Then xL 15”

68 . What willbe the taxes on a house worth andassessed at g of its value, the tax rate being perof assessed value ?

Oral

3 is what part of 6 ?

3 is what decimalof 6 ?

3 is how many hundredths of 6

3 is what per cent of 6 ?

6 is what per cent of 3 ?

What number is 50% of 6 ?

3 is 50% of what number ?

2 is what of 100?

2 is what of 200?

464 ARITHMETIC.

10. What number is 5% of 100?

1 1 . What of 20 is 1 ?

12 . 4 is What of 200?

13 . 3 is 570 of what number ?

14 . What per cent of 9 is

15 . What number, increased by 1 of itself, equals 10?

16 . What number,increased by 25% of itself

,equals 20?

17 . 65 diminished by 5% of itself, equals what ?

18 . Buying price $ 100, selling price Gain

19 . Cost $80, profit Selling price ?

20. What principalwillgive $ 30 yearly interest at

21 . At i of how much will I pay for insurance on

22 . If insurance costs 370 ,howmuch canI get for$60?

23 . Bought sugar at 4 ¢ per 1h . ; sold it for 5 [5 Profit

NOTE.—Per cent of profit is based uponthe cost.

24 . is what of g?

(Reduce both to a commondenominator.)

25 . I} is what of

PROFIT AND LOSS.


Find the profit or loss,and the selling price

1 . Cost gain

2 . Cost loss

3 . gain

4 . Cost $875 ; loss

5 . Cost gain12570 .

PROFIT AND LOSS . 465

Find the profit or loss per cent .

N. B.

— Profit or loss per cent is based uponthe cost.

6 . Cost $600 ; selling price $618 .

Gainorloss x% of cost. 18 x% of 600.

7 . Cost selling price $802.

8 . Cost selling price

9 . Cost selling price

10. Cost $84 ; selling price

11 . Selling price 1088

Gainor loss m% of cost.

12 . Selling price $150 ; gain$ 25.

13 . Selling price 1088

14 . Selling price gain

15 . Selling price loss

Find the cost, and the profit or loss


Let a cost.

17 Selling price 1088


19 . Selling price $204 ; loss

20. Selling gain

21 . A manbuys a horse for $275, and sells it at a profit of20 per cent . How much does he gain?

22 . A cow is sold for $ 75, onwhich the profit is $ 15. Whatis the gain per cent ?

23 . A lot is sold for $960, which is 20 per cent more than itcost. Find the cost of the lot .

466 ARITHMETIC.

24 . Tea that costs 32 ? per pound is sold for 48 What is

the gain per cent ?

25 . A man buys a horse for 175 and sells it for$200. Whatper cent does he gain ?

26 . What per cent is lost on a horse costing $200, and sold ata loss of 25 ?

27 . What is the selling price of dress goods costing 334 2 peryard

,on which a profit of 124} per cent is made ?

28 . Sold a coat for thereby losing 16 per cent . Whatwas i ts cost ?

29 . How much did I gain on a house forwhich Ipaidmy profit being 2} per cent ?

30. A man after spending 10 per cent of his money for

clothing,25 per cent for board , and 30 per cent for incidentals,

has $ 70left. What d id he have at first ?

3 1 . The population of a city was in1880. In1890 it

was What was the gain per cent inten years ?

(Wh ich is the base ?)

3 2 . The population of a city was in1890, a gain of

over the previous census. What was the increase per cent ?

(Have the correct base.)

3 3 . 16 shares of bank stock , face (par) value $ 100 each , weresold at 2§ per cent above the face value . How much wasreceived for the stock ?

34 . What will be the cost of 84 shares of stock , face (par)value $50 each , at 3} per cent below par?

V/35 . Find the interest on at 4 per cent

,for 3 years

7m m15 days.

468 ARITHMETIC.

6 . How many sheets of tin 6 inches by 4 inches will berequ ired to cover a roof 20 feet by 30feet, making no allowancefor overlapping ?

7. The owner of a lot 25 feet by 100 wishes to bu ild aroundit a tight board fence 6 feet high . How many running feet offence willthere be ? Howmany square feet ?

(Drawthe development of the fence, marking onit the dimensions.)

8 . How many boards 12 feet long, 6 inches wide , willit taketo bu ild such a fence ?

9 . A plot of ground 300 feet by 200 feet is bought atper acre . It is sold inlots 100 feet square at $ 160 each . Whatis the profit ?

10. How many cakes of ice 4 feet long , 2 feet wide , can be cutfrom a rectangular pond 320feet long , 160 feet wide,no allowancebeing made for waste ? (Cancel.)

1 1 . If the ice is 1; feet thick , how many cubic feet of ice willthere be ? Find its weight intons

,at pounds per cubic foot .

12 . G ive the length of a shed 15 feet high,32 feet wide

,that

willexactly hold the ice cut from the above pond . (Cancel.)

13 . The owner of a plot of ground 640 feet long , 440 feetwide , cu ts two streets, each 40 feet wide , through the middle of

the plot,one running north and south , the other east and west .

How many square feet of land had he originally ? How manysquare feet has he left for building purposes ? (Make diagram .)

14 . What will it cost him,at 602 per square yard, to make

the above streets ?

15 . If the remaining ground ; is divided into building lots 25by 100, howmany lots willthere be ?

16 . How many square yards of plastering will be requiredfor the walls and the ceiling of a room 21 feet wide , 30 feet long ,9 feet high

,deducting for 2 windows, each 6 feet by 3 feet, and

a door 8 feet by 4} feet ?

PERCENTAG

PROFIT AND LOSS.

Bought for $ 36 ; sold for$40. Gain per cent ?’

Bought for $40; sold for $ 36. Loss per cent ?

Cost 36¢ selli ng price 40 Gain per cent?

Cost $24 ; gain Selling price ?

Selling price 70$5; 1088 Cost?

Buying price 702 gain Selling price ?

Cost $20; selling pri ce $29. Gain

Cost $20; selling price $290. Gain

Cost $20; selling price Gain

Cost $20; selling price Gain

Selling price $ 300; loss $ 100. Loss

Cost $ 300; gain $ 100. Gain

Selling price $ 175 cost $ 150. Gain

Selling price $ 375 ; gain Profit ?

Cost selling price Loss

Selling price 1088 Cost?

Cost loss Selling price ?

Selling price gain Cost ?

Selling price $ 64 ; profit $ 16. Gain

Cost selling price $42. Gain

Selling pri ce 1088 Loss?

Cost $24 ; gain Selling price ?


Selling price loss Loss?

69

470 ARITHMETIC.

Cost 1088 Selling price ?

Cost $84 ; selling price $184 . Gain

Selling price $700 ; gain Cost ?

Cost gain Selling price ?

Selling price loss 13470 . Loss?

Cost selling price Loss

Selling price 1088 Cost ?

Cost gain Profit?

Selling pri ce gain Profit ?

Cost 1088 $70 . Loss ?

Selling price profit Gain

Cost gain Gain

Selling price 1088 Loss

Cost gain Selling price ?

Selling price 1088 4370 . Loss?

Profit $ 36 ; gain Cost?

Loss 1088 Selling price ?

Cost 1088 Loss ?

Selling price profit Gain

Cost $ 84; 1088 Loss


Profit gain Selling price ?

Loss 1088 Cost ?

Cost $ 75.52 ; gain Profit ?

Selling price 1088 Loss

Cost loss Loss?

PERCENTAGE. 471

873 . In calculating interest, take 30 days to a month , 12months to a year .

Rate

100

NOTE. Change giventime to years .

2 yr. 6 mo .- 2} yr.

- § yr.

lyr. 7 mo .- 19 mo .

n Hyr.

44 mo . 10 da.

=-4} mo .=igyn

Time (inyears) Interest.

1 yr. 5 mo. 15 da. 17} mo .

12

5 mo . 17 da. 167 da. m yr.

875.

Find the interest on

1 . $ 750, for 24years, at 670 .

2 . 34 months, at

3 . from Oct. 1 to Oct. 21, at

4 . 8 mo . 12 da ., at

5 . $ 360, 33 da .,at

6 . 2 mo . 3 da .

,at

7 . $ 400, 1 yr . 1 mo . 1 da . , at 4470 .

8 . $ 720, 21 da . ,

9 . 8 da .,

10. $630, from April1, 1890, to Jan . 16,1892

,

1892—1 — 16

1890— 4 1

11 . 2 yr . 1 mo . 7 da at

12 . 3 yr . 4 mo . 9 da. ,at 4470 .

472 ARITHMETIC.

13 . $960, 11 mo . 24 da .,

14 . 1 yr . 9 mo . 20 da.,

15 . 4 yr. 7 mo . 10 da .,

876. Amount Princip al Interest.

Find the amount

16 . $813 , from April19, 1889, toMarch 4, 1894 , at17 . $960, from J an1, 1893 , to Dec . 21, 1894 , at 470 .

18 . for 3 yr . 6 mo . 9 da ., at

19 . for 17 da . ,at

20. $144, for2 yr. 5 da .,at

2 1 . for 15 mo . 27 da .,at

22 . for 1 yr . 5 mo . 29 da . , at

23 . for 7 yr . 13 da .,at

24 . for 3 yr . 5 mo .,at

25 . for 1 yr . 2 mo . 3 da . , at

877 . Interest-bearing Demand Notes.

26 . SAN FRANCISCO, J an. 7 , 1893 .

Ondemand , I promise to pay William Britt,or order

,Seven

Hundred Sixty-five 11096 Dollars, value received , with interest

at 6 per cent

ARTHUR TOWNSEND.

How much money will be requ ired to pay the above note,

with interest,July 15

,1894 ?

27 . A demand note,dated Sept . 25, 1892 , with interest at

8% from date,is paid J an. 2

,1895. How mu ch was due, the

face Of the note being $ 750?

28 . Find the amount dueMarch 4 ,1894 , on a note for

datedMay 20, 1892 , with interest from date at

INTEREST. 473

29 . Find the amount necessary,Oct. 16

, 1896, to pay a noteof with interest at 6% from Aug. 15

, 1892.

30. An interest-bearing note for is dated April 3,

1886. Howmuch is due on it for principaland interest Jan . 2,

1894 ? Rate

878. OralProblems.

1 . Find the interest on $300, for 1 year 7 months, at

$ 12 per year is howmuch for 7 months ?

2 . On$60, for 33 days, at

for 360 days is howmuch for 33 days ?

3 . Ou‘

$l20, from Jan . 1, 1893 , to July 1, 1894 , at

4 . How long will it take $ 100 to produce $ 15 interest at6%

5 . A t what rate per cent will$50 produce $6 in2 years ?

6 . What is the interest on $ 300, at from Feb. 1 toFeb . 21 ?

7 . What part of a year is 72 days ?

8 . Find the interest at for 90 days, on $ 150.

9 . On$ 240, for 36 days, at

10. What is the amount of $200,for 3 years 1 month , at

11 . How long willit take $ 1 to make 1 interest at 5%

12 . How long willit take any sum to double itself at

13 . How long willit take to double itself at

14 . At 4 per cent per month , find the interest on $90 for 16months.

15 . 5% per year is 1% for how many days ?

16 . 4470 per year is'

1% for how many days ?

17 . Find the interest on $ 75, at for 72 days.

474 ARITHMETIC.

18 . Find the interest on $63 , at 4470 ,for 80 days.

19 . Find the interest on $570, at for 60 days.

20. Find the interest on $840, at 670 ,for 30 days .

21 . Find the interest on $ 150,at for 120 days.

22 . Find the interest on $275, at for 90 days.

23 . Find the interest on $ 360, at for 144 days .

879 . Questions from Examinations forN.Y. State Certificate.

1 . What fractional part of 30 rd . 5 yd . 1 ft . is 8 rd . 4 yd .

2 ft . ?

2 . A coalbinis 6 feet long and 4 feet wide . How deep mustit be to contain 5 tons of stove coal, if one tonoccupies 36 cubicfeet of space ?

3 . A manwalking at the rate of 3 mi . 96 rd . per hour willwalk how far in3 hr . 16 min. ?

5 . If a merchant pays per yard formuslin , and sells thesame for 74¢ per yard , what is his gain per cent ?

6 . Find the cost of 96 feet of pine lumber at $ 25 perM.,

and laths at $ 3 perM.

7 . Divide by and express the result inthe formof a fraction .

8 . In the written number the number expressed bythe first two (left-hand) figures is how many times the valueexpressed by the second two figures ?

9 . Anagent buying wheat is ofi'

ered a commission of perbushel, or one of 45per cent, and he chooses the former . The

average price paid per bushelis 9142 Does he gain or lose byhis choice , and how much per bushel?

476 ARITHMETIC.

880. Questions from U. S. CivilServiceExaminations.

1 . The Navy Department had square yards of bunting, which it ordered to be made into flags as follows

10measuring 6 X 9} feet,20measuring 7 x 144 feet,5measuring 14 x 334 feet,

and the remainder measuring 34x 67} feet. How many of thelatter were made ?

2 . A ship has provisions to last her crew of 450 men 5months . How many men must be discharged to make thesupply hold out 9 months?

3 . The government purchased 96 acres of land at peracre

,185 acres at per acre

, and 36 acres at peracre . What was the area of the land

, what its cost, and whatthe average price per acre ?

4 . A clerk has anincome of per annum . He pays20 per cent of it for board , 14 per cent for washing , 2 per centfor incidentals

,15 per cent for clothing

,9 per cent for other

expenses, and loses invarious ways 50 per cent of the amountthen remaining. What sum does he have left ?

5 . Add the following

Two thousand (units) and three thousandths .

Two hundred-thousandths.

Nine millionths.

Sevenmillions seventy-nine .

Tenhundred-thousandths.

6 . Multiply 9 846358 by7 . Divide by 3. of g of87? and give the result indecimals.

8 . What is the interest on for 93 days at 5} percent ?

EXAMINATION QUESTIONS. 477

9 . A regiment of 940men , during the war lost 532 of theirnumber by death and 125 by desertion . What was the percentage of loss ineach case , and what per cent remained forservice ?

10. Find the cost of each of the following items of a soldier ’s

12 oz . Of pork at per bbl. (200

1 lb . 6 oz . of flour at 6 .O2.lJrper bbl. (196

oz . of beans at per bushel(60

oz . of rice at per 100lb .

oz . of potatoes at per bushel(60

oz . of coffee at per pound .

oz . of sugar at per pound .

1 1 . Express in words the following amounts :

7seg .

12 . Subtract from

13 . Divide by

14 . What is the product of ‘i‘ of 3 of 3 of State theresult indecimals.

15 . Anarmy fought two battles. Inthe first it lost 15 percent

,and inthe second 20 per cent of the original number , after

which it mustered men . What was the original strengthof the army ?

16 . A owned 4 of a Sh ip and sold 4 of his share to B,who

sold 4 of what he bought to C, who sold 1? of what he bought toD. What part of the whole vesseldid D buy ?

17 If a letter-carrier indelivering letters takes stepsina day , each step averaging 20 inches, how many miles doeshe walk ?

478 ARITHMETIC.

18 . A farmer exchanged 340 bushels of corn worth 752 perbushel, for barley worth $ 1 per bushel, and oats worth 50¢ perbushel. How many bushels of each did he receive , the quantityof barley and oats being equal?

19 . Thirty-two clerks are to distribute letters on a

certain day . Half of the Clerks are experienced menand half ofthem new men . If each experienced mandoes twice as much as

a newman , how many letters will be distributed by one of eachkind

20. A merchant sold a lot of damaged sugar at a loss of 25

per cent,receiving Howmuch did the sugar cost

him ?

881 . U. S. CivilService. Clerkships inDepartment of Interior.


Two thousand (units) and two thousandths.

Two hundred-thousandths.

Nine millionths.

Sevenmillionseventy-nine .

Four hundred thousand seven hundred ninety-seven .

One hundred sixty-seven million,eight hundred seventy-one

thousand , two hundred forty-nine .

Nine hundred eighty-seven million,Six hundred fifty

-fourthousand , three hundred ninety-one .

Tenhundred-thousandths .

One hundred eighty-nine thousand,one hundred sixty-seven .

2 . Multiply by

3 . Divide by and divide by 63 .

4 . What is a pile of wood 15 feet long,10} feet high , and

12 feet wide worth, at $4} per cord ?

(1 cord a 128 cu . ft.)


5 . Find the interest on for 1 year 8 months and 15days, at 670 .

6 . Find a common divisor of 72 and 90.

7 . is what per cent of

8 . 10f is what per cent of

9 . If seven men canmow a field in19 days,inhow many

days will20men mow it ?

10. When a gold dollar is worth inpaper money , whatwillbe the value ingold of a dollar incu rrency ?

1 1 . Reduce 74 to a decimal.

12 . Add {3 to Q .

13 . Subtract {t from .97 .

14 . Divide of 11,by 4 of 131 .

882. WorcesterPublic Schools. ExaminationQuestions. Oral.

1 . How many hours ing of a day ?

2 . What is the interest of $300 for3 years 4 months, at

3 . A manis 60 years old ; of his age is 3 of his son’

s

age . How old is his son?

4 . Buy cloth at 45¢ a yard,and sell it at 60¢ a yard .

Gain per cent what ?

5 . What is the difference between 4 of 6} and ‘i of

6 . Five bushels is 5% of what number of bushels ?

7 . There is a pole standing so that 4} of it is inwater and 4as much is inthe sand . What part of the pole is inthe sand ?

8 . What is of

9 . A manbought a watch for $64 ,and sold it at a gain of

124} per cent . What was the selling price ?

10. How many feet ina quarter of a mile ?

480 ARITHMETIC.

11 . How many oranges in4, 4, 4, and 110 of an orange ?


13 . What is the cost of 74 pounds of sugar at 11 if a pound ?

14 . In24 peeks how many quarts ?

883 . BrooklynPublic Schools. ExaminationQuestions.

1 . Add the greatest and the least of the three fractions 44,

4, 44 ; and divide the sum by the remaining fraction .

2 . Multiply 82 ten-thousandths by 7 and 5 hundredths, anddivide the product by 705 millionths .

3 . To make 15 per cent profit , what must goods be markedthat cost per yard ?

4 . The dividend is the quotient is 87 , the remainder is45. Find the divisor .

5 . What per cent of a school is boys, and what per centgirls

,there being 640 of the former and 560 of the latter ?

6 . 43 gallons 3 quarts 1 pint alcohol are sold forWhat is the price per gallon

7 . How many planks 16 feet long and 6 inches wide willbeneeded for a floor 48 feet long and 32 feet wide ?

8 . A horse costing $ 160 is sold for $ 180. What is the gainper cent ? What is the loss per cent when a horse costing 180

is sold for$ 160

9 . Find the interest on for 1 year 5months 14 days,

at 34 per cent ?

10. A merchant sold 600 barrels of flour for at a lossof 44per cent . What did the flour cost him per barrel?

11 . What per cent of is

REVIEW .

SPECIAL DRILLS.

884. G ive sums

425 99 999 425

99 576 576 999

685 99 999 685

99 599 599 999

885. G ive differences

565 99 999

488 99 999

794 99 999

898 99 999

886. G ive products

24 x 21 z1 x 31

33 x 21 32 x 31

42 x 21 43 x 31

51 x 21 s1 x 31

887 . G ive quotients

16 .25 228 19

18 .25 234 18

24 .75 238 17

36 .75 336 16

888 . G ive answers

154x 6 214x 4

5i + t 74 4 41444 - 14 64 4 - 4

481

482 ARITHMETIC.

889 . OralProblems.

1 . What will be the cost of 48 yards of cloth at per

yard ?

2 . How many square yards ina piece of carpet 48 yards long ,27 inches wide ?

3 . How many yards of carpet 27 inches wide willbe neededto cover a floor containing 48 square yards ?

4 . Paid for groceries,

for dry goods, and 99¢for sundries. What is the total?

5 . From a chest contain ing 254 pounds of tea ,84 pounds

were sold . How many pounds remain?

6 . At 374¢ per peck , what willI receive for 4 bushels of

potatoes ?

7 . 834 yards of cloth are divided into 9 pieces. How manyyards are there ineach piece ?

8 . Ibuy hardware to the amount of I give the storekeeper two $5-bills. Howmuch change should I receive ?

9 . What willbe the cost of 24 yards of calico at 44¢ per yd .

10. What willIhave to pay for 19 base-balls at each ?

1 1 . At per yard,what willbe the cost of 120 yards of

silk

12 . For $ 120, how many yards of silk can I buy at $ 18 74per yard ?

13 . What willbe the cost of a tonof hay at 974¢ per cwt . ?

14 . A square field requ ires 320 rods of fence . How manysquare rods are there inthe field ?

15 . How many acres are sq . rd.

16 . At 44¢ per yd . ,how many yards of calico can Ibuy for

17 . If slate pencils cost 2 mills each,how many willI receive

for $ 4 ?

484 ARITHMETIC.

SHORTMETHODS.

Slate Exercises.

x 4 x 4Deduct Deduct 4Ans .

Ans .

Multiply by 94.

-10 times number .

4 number (Deduct).

Ans .

892 . Find products :

1 . 176 x 44 x 94

2 . 273 x 44 x 84

3 . x g x 74

4 . x 44 x 114

5 . x 4 658 x 994

6 . x 4 658 x 994

7 . x 4 725 x 1194

8 . x 4 347 x 794

9 . x 496

418 x 894

10. x 44 543 x 494

893 . Multiply2 1 . 418 x 99 724 x 36 x 45

22 . 674 x 874 484 x 44 x 94

23 . 36 x 999 x 25 576 x 944x 124

24 . 48 x 125 x 71 95 x 36 x 194

25 . 64 x 7§ x 334 74 x 31 x 13 x 94

AgriculturalimplementsAnimals

Breadstufl’s

Carriages and cars

Copper and manufacturesCottonand manufacturesFertilizersFru it and nutsFurs and skins

HOps

Ironand steel,and manu

Leather and manufac

MusicalinstrumentsNavalstoresOils

ProvisionsSpiritsSugar, refinedTobacco and manufac

turesWood and manufacturesAllother articles

REVIEW . 485

Supply the value of Allother articles , and give Increaseor Decrease of the value of each item ,

and the totalincrease.

486 ARITHMETIC.

Divide . Do not write products

Write answers (Art . 385)

14 .

MEASUREMEN1719.Slate Problems.

6 .

7 .

8 .

9 .

10.

1 . How many boards 16 ft . long, 8 in. wide , willbe requ ired

for a tight fence 8 ft . high , around a piece of ground 240 ft . long ,180 ft . wide ? How many posts

,6 ft . apart ,willbe needed ?

2 . A room is 18 ft . long , 15 ft. wide . The walls and the ceiling contain 930 sq . ft . What is the height of the room ?

3 . What will it cost to cover a table 6 ft . long, 24 ft .

wide,

with baize 4 yd . wide , at 75 per yd . ?

(Arts . 385, 616)

MEASUREMENTS. 487

4 . 160 square rods make 1 acre . Howmany square yardsare there inanacre ? About how many yards long is a squarefield containing 1 acre ?

5 . A 40-acre field is 160 rods long . How many rods of

fence are needed to enclose the field ?

6 . A room 30 ft . long , 24 ft. wide , 15 ft . high,contains 40

persons. How many square feet of floor space are there for eachoccupant ? Howmany cubic feet of air space are there for each ?

7 . How many yards of carpet, 27 inches wide , would beneeded for the floor of such a room ?

8 . How many bundles of laths, each bundle covering 3 squareyards, would be needed for the walls and ceiling of the aboveroom

,no allowance being made for doors and windows ?

9 . A farmer owned a recta ngular piece of ground 30 rodslong

,27 rods wide . He sold three lots 18 x 8 rods, 12 x 3 rods,

15 X 9 rods.

Find the number of square rods inthe original piece . Markinthe diagram the area of each lot sold , and the area of the partremaining.

10. Howmany rods of fence will be needed to enclose thepart remaining?

488 ARITHMETIC.

1 1 . Ina box containing 50 square feet of glass, howmanypanes willthere be 6 inches long , 4 inches wide ?

1 gallona 231 cubic inches.

1 bushel cubic inches.

1 cord 128 cubic feet.

12 . How many gallons willfilla tank 22 feet long, 14 feetwide

,9 feet deep ?

Indicate Operations. Cancelwhere possible.

13 . How many cords of wood are there ina pile 48 feet long,16 feet wide , 12 feet high ?

14 . A binmeasures 49 feet long, 32 feet wide , 10 feet 8 inchesdeep . How many bushels of grain willit hold ?

15 . Inbu ilding a sewer one thousand yards long , anexcavation 6 feet wide was made to an average depth of 104 feet. Howmany cubic yards were dug out ?

16 . If the sewer pipe occupies a space equalto a box 44 feetwide

,44 feet deep , and yards long , how many cubic yards

of earth willbe left after the excavation is filled in

899 . St. PaulPublic Schools. ExaminationQuestions.

1 . Iborrowed at and kept it 3 yr . 8 mo . 18 da.

What was due the lender at the expiration of that time ?

2 . Made a loan of $ 12040 for 1 yr . 9 mo . at Whatinterest did I receive ?

3 . Mr. B’

s ground is 14 rods by 10 rods, which is 40%larger than A ’

s. Find the value of A ’

s ground at $5 a sq . rd .

4 . What per cent of a sum of money can be earned in1 dayi f 72% of it canbe earned in9 days ?

5 . Write the analysis of the second .

6 . At $ 34 a cord , what is the value of the wood that can bepiled under a shed 50 ft . long

,25 ft . wide

,and 12 ft . high ?

BANK DISCOUNT. 489

7 . Inbu ilding a house,350 joists 12 in. by 4 in. were used ,

which together amounted to cu . ft . Find length of each .

8 . If I buy 94 bu . of chestnuts at $44 a bu . and retail themat $ 124 a pint , what is my whole gain

9 . If 14 acres of meadow yield 324 tons of hay ,how much

do 54 acres produce ? Analyze .

10. What is the value of

BANK DISCOUNT.

900. Wm . Brown and Sons receive the following note inset

tlement of their account with Thomas TierneyBos'rorr,May 30, 1893 .

Thirty days after date , Ipromise to pay to the order ofWm .

Brown and Sons, Three Hundred Fifty-Fou r “

17055Dollars, value

received,at the Park NationalBank .

5535441556 . THOMAS TIERNEY .

901 . Th is money is payable 33 days afterMay 30,3 days Of grace being

allowed . IfWm. Brownand Sons desire to use the money at once, theymay have thenote d iscounted at a bank . In th is case

,the bank deducts

from the face of the note the interest thereonfor 33 days, and

pays over the difl'

erence (proceeds).

Face Ofnote

Discount (Int. for 33 da ) (atProceeds


Find the discount at 6% onthe following , allowing 3grace ineach case . (See Appendix .)

1 . A 30-days note .for $75.

2 . 15-days note for

3 . 60-days note for

4 . 20-days note for $ 96.

5 . 4-months note for $336.

490 ARITHMETIC.

Find the proceeds, at on

6 . A 6-months note for 180.

7 . A 3-months note for

8 . A 24-days note for

9 . A 90-days note for$ 180.

10. A 72-days note for

903 . Incomputing the discount onanote, the banks ascertainthe exactnumber of days.

A 3-months note, dated February 1, is payable three days afterMay 1,wh ich isMay 4 . The discount is takenfor27 3 1 30 4 92days. A 90

days note of the same date is payable 93 days after February 1, wh ich 18

May 5.

The year, however, is considered to contain360 days ; the interest inthefirst case being takenfor {324 of a year, and inthe latter for 1125’s Of a year.

Find the discount , at on

11 . A 1-month note for $600, dated Feb . 6; 1894 . (48614)

12 . A 2-months note for $240, dated July 17, 1894. (46554 )

13 . A 3-months note for $ 360, dated April8 , 1894 .

14 . A 4-months note for $ 84 , dated Dec . 24 , 1894 .

15 . A 6-months note for datedMarch 4 ,1895.

16 . A 60-days note for$ 240, dated July 17, 1894 .

17 . A 90-days note for $ 360, dated April8 , 1894 .

904 . Ineach Of the preceding examples, it has beenassumed that thenote has beenpresented for discount the day onwh ich it was made.

Insome of the following examples, the notes are discounted at a later

date, and the term of discount is to be ascertained ; that is, the time betweenthe date Of discount and that of maturity (including the days of grace).The term of discount of a 30-days note datedMay 1, and discountedMay 19, is 15 days.

492 ARITHMETIC.

907 . AnInvoice (German).ELBERFELD,

Dec . 16, 1892.

INVOICE.

ForMESSRS . BROWN J ONES , NewYork ,Mom HAUSELMANN SCHMIDT.

We send you ,throughMESSRS . VON DEB. OHE MEYER, at

Antwerp ,1 case , marked H . S . containing

12 pieces Rhaetia

4 pieces Caledonia

brown

111. means meters . M. stands formarks.)

Find the total number of yards of dress goods inthe abovecase (1 meter inches), and the totalcost inU. S. money

(1 mark cents).

REVIEW . 493

SHORTMETHODS.

908 . Whenmultiplying mixed numbers, many accountants preferreduce them to improper fractions.

909. Multiply without reducing1 . 43§ x 8§ 4 .

350 5 . 42; x 12121% 6 . x 8;

2 . s7i x 94 7 . 569W };

516 s . 83§ x 31I“ 9 . 721 x 10;

3 . 17; x 7} 10. 953 x 11}

910. 76} x 95 x

765 product by 10 709} product by 11

123.less product by 4 913;product by

752} Am . 9; 36H product by 47551} Ans.

841X 56421; product by 516%product byproduct by i

4553 Ans. 5§

Multiply11 . 19gx 4; 15 . 51} x 6}

16 . 63; x 10g12 24; x 3;

(4 - 1) 17 . 24

13 . 35; x 71} 18 . 31} X 123

19 . 42gx 912,14 . 40§ x 85

20. 56%x 7§

494 ARITHMETIC.

912 . BostonExaminationQuestions. Mental.1 . If 5} yd . of cloth cost 66 if, what cost 12} yd . ?

2 . What part of a cu . yd . is 3 cu . ft . ?

3 . What is the difference between a floor 40 ft . square andtwo others each 20 ft . square ?

4 . What is the interest of $ 12, for 1 yr. 4 mo .,at

5 . Sold anarticle for $60, thereby losing Cost ?

6 . If Ibuy a dozen pencils at 2 if each , and sellat 3 ¢ each ,What is the gain per cent ?

7 . How many sq . yd . inthe surface of a cube that containsone cu . yd . ?

8 . If a yard of ribbon contains 144 sq . in. , how wide is it ?

9 . A t 75¢ a yard , what willbe the cost of carpeting a floor15 ft . long , 4 yd . wide ?

10. How long willit take a boy to pay for a su it worthand a pair of boots worth if he earns a day ?

1 1 . 180 is 10% less thanwhat number ?

12 . In1§ yd ., howmany inches ?

13 . Sold a saddle for $ 18, which was more thanthecost . Cost ?

14 . What is gained on goods sold at double the cost ?

15 . What is the interest of for 60 days, at

16 . Sold flour at a profit of and gained 25 Whatwas the cost per barrel?

17 Bought 4 a ton of sugar at per lb . How muchdid Ipay for it ?

18 . If 2} pk . berries cost one dollar , what would 3 qt . cost atthe same rate ?

EXAMINATION QUESTIONS . 495

19 . Bought 5 bu . nuts at a dollar a pk . , and got 5% OR forcash . How much did Ipayfor the nuts ?

20. In a certain school 40 pupils are present and 10 are

absent. What are absent ?

2 1 . Ina certain class 1} of the pupils are under 10 years , 4 of

them are between10 and 12,and the rest are over 12 . What

are over 12 years ?

22 . Mr. Eastmancollects bills for me , and Ipay himHe pays over to me $ 56. How much d id he collect ?

23 . With of his money,Robert bought 12 oranges.

How many oranges could he have bought with allhis money ?

24 . Mrs. Allen bought 7 chairs at $4 apiece,2 tables at $9

apiece , and a carpet for $30. She paid two $50 bills. Howmuch change was due her ?

25 . If a boy earns a week and saves 2% of it,how

much willhe save in10weeks ?

26 . What is the of gain,when boots which cost a

pair are sold for

27 . 30% ofMr. Brown ’

s money is inthe bank , and 50% in

realestate ; the remainder, is inrailroad stocks. Howmuch has he inall?

28 . How much difference between4of 20 and 20divided by 4?

29 . What is the premium for insuring on my houseat 470 ?

30. 20% of 115 is 33470 Of what number ?

913 . BostonExaminationQuestions. Slate.

1 . Change .03125 to a commonfraction insmallest terms.

What is the product Of one hundred one thousandths by ten

thousand one hundred one hundred-thousandths ?

Divide .01001 by 001.

496 ARITHMETIC.

2 . If I am compelled to lose on damaged goods, howmust I sellthose that cost me

3 . What is the interest of for 2 yr . 6 mo . 24 da .,at

6%What is interest ? The principal? The amount ? The rate ?

4 . Find the interest ona note for $250, dated J an. 21, 1890,

and paidMay 30,1890, at

Find the amount of fromMay 15, 1887, to July 21,1890, at

5 . What is of $832 ?

$832 is of what sum ?

6 . How large a sale must a merchant make , at a profit ofthat h is gainmay be

7 . By the census of 1880,the population of a certain city

was By the census Of 1890, its populationwasFind the per cent of increase .

8 . Each Of two boys bought 100apples for a dollar . The firs tboy sold his , 4 apples for the second sold his, 5 apples for

Which boy gains the more per cent ? How much more ?

9 . From 4 A . Of land I sold one piece 20 rods square , andanother piece conta ining 16 square rods How much did Ihaveremaining

10. Multiply one hundred eleven millionths by five and six

tenths.

11 . A quantity of coalwas bought for $900. For what mustit be sold to gainBy selling a house for a mangained on the cost

What was the cost ?Bought a horse for $800, and sold it for $600. What did

Ilose

12 . If a manpaid $ 189for a load of hay weighing 11» tons,what would he pay at the same rate forg of a ton ?


13 . Sold my house and farm of 944» acres for Al

lowing for the house , what did I receive per acre for theland ?

14 . What number is that which,diminished by will

leave 2916 ?How long will200 lb. flour last 18 persons if each person is

allowed 14 lb . per day ?If 4 of 4 of a ship cost what is 4 Of it worth ?

15 . The dividend was the quotient .027 . Whatwas the divisor ?

x

Write your rule inpointing Off indivision of decimals.

16 . Change to other methods Of expression , 4, i , g, .164.

17 . A note of dated July 5, 1888 , was paid June 7,1890

,with interest at Whatwas the amount paid ?

18 . Maine had , by census of 1880, sq . mi . andinhabitants ; NewHampshire had sq . mi . and in

habitants ; Vermont had sq . mi . and inhabitants ;Massachusetts had sq . mi . and inhabitants ; Connecticut had sq . mi . and inhabitants ; RhodeIsland had sq . mi . and inhabitants ; Texas had

sq . mi . and inhabitants . Texas is how manytimes as large inarea as NewEngland ? Carry the answer totwo decimalplaces. Whatwas the average population per squaremile inNewEngland ? InTexas ? If the land inRhode Islandis worth onthe average 4 Of a millper square foot , what is thevalue Of the entire land inthe state ?

19 . A flock Of sheep has beenincreased by 25070 Of its num

ber,and now numbers What was the originalnumber ?

20. Bought a house for and sold it so as to gainWhat di d I sellit for?

498 ARITHMETIC.

21 . If this room is 35 ft . long , 23 ft . w ide , and 13 ft . high ,how many square yards of plastering will it requ ire , making noallowance for doors, etc . ?

22 . A commissionof was charged for sellingworth of goods. What was the rate Of commission ?Sold goods at a loss Of anactualloss Of Whatwas the prime cost ?

23 . Harry Hedge earns $ 12 a week . He pays for

board,

for car fare , $03 75 for library fees, andfor other expenses. Inhow many weeks would he save

24 . The milk from a herd Of 15 Jersey cows, sold at 6¢ a qt . ,

amounted inone summer to How many quarts weresold , and what was the average quantity from each cow ?

25 . A woman has three children . She pays for each $ 15 a

year for having his clothes m.ade , a month for h is mending,

and a week for his washing. How much could she save ina year if she knew how to wash , make clothes, and mend ?

26 . A coaldealer bought tons of coalat $5 a ton . Hesold 40% of it at $ 7 , 20 %Of i t at and the remainder at

How much did he gain ?

27 A man insured his house for his store forand°his goods for at What did his insurance come to ?

28 . A pole stands 4 inthe mud , inthe water, and 32 ft . inthe air. How long is the pole ?

29 . What is the amount Of the following : 23 yd . at

17 yd . at 16 yd . at 10 yd . at 14 yd . at20 yd . at 12 yd . at

30. Bought flour for and sold it for $9. What is theper cent Of gain?Bought for $9 and sold it for What is the per cent of

loss ?What per cent is gained by selling goods for 2 times their cost ?

500 ARITHMETIC.

4 . G old coin contains 90 per cent gold , 9 per cent silver , 1per cent copper . Find the quantity Of each metalin50 doubleeagles each containing 516 grains .

5 . How many spoons, each weighing 2 oz . 18 pwt . , can bemade from 5 lb. 9 oz . 12 pwt . silver ?

6 . How much money insilver dollars, 412} grains each , willweigh 165 lb . avoirdupois grains)?

7 . What fraction Of a pound avoirdupois is a pound troy ?What per cent of anounce avoirdupois is a troy ounce ?

8 . What is the value , at per oz . troy , of a silver pitcherweighing 4 lb . 8 oz . avoirdupois ?

9 . At per ounce,what is the value of the silver con

tained ina half-dollar, which weighs grains, 195 being puresilver ?

10. What is the capacity , ingallons (231 cu . of a barrelthat willcontain2} bu . cu .

11 . A tank 16 ft . long,14 ft . wide

,8 ft . deep , is lined with

lead onthe bottom and sides . How many square feet of sheetlead willbe required ? (Draw the

“ development ” Of the tank .)

12 . How many cu . ft . of lead willbe needed for the above , ifthe sheet lead is 31, in. thick ?

13 . A cubic foot of water weighs oz . Lead is timesas heavy . How many pounds of lead willbe needed to line thetank ? (Cancel.)

14 . How many gallons willthe above tank hold ? How manybushels

15 . 8 lb. 3 oz . troy is what per cent of 4 lb. 7 oz . ?

16 . What per cent Of a lb . avoirdupois is a troy pound ?

17 . If 3 T. 740lb. hay cost how much will17 T. 380

lb. cost ?


18 . A man sold 18 bbl. sugar , each containing 306 lb. ; 21

bbl. , each containing 297 lh . ; 5 bbl. , each containing 291 lb .

What is the average weight per barrel?

19 . A , B,and 0 buy a farm . A pays B pays

0 pays What per cent Of the purchase money does eachfurnish ?

20. If 11 weavers in9 days weave yards,what will 1

mando in1 day ? 6 men in7 days?

21 . The tax rate of a certain city is l%% uponthe assessedvalue of property . If this value is 7570 Of the actualvalue , howmuch taxes doesMr. Smith pay upon a house and lot, the actualvalue of which is

916. Philadelphia Public Schools. OralWork.

1 . 96 17)

2 . What decimalis equalto

3 . What part Of 100 is


6 . g Of 40 yards gOf 15 yards

7 . At $ 12 per ounce , what is 3 of a pound of gold worth ?

8 . peeks are how many quarts ?

9 . How many square feet ina board 10 feet long and L}feet wide ?

10. If 9 pounds of sugar cost 36 what will12 pounds cost ?

11 . What fractionalpart of a minute is 12470 Of it ?

12 . 25 is 2070 Of what number ?

13 . What per cent is lost inbuying potatoes at a bushel,

and selling them at 6015 a bushel?

502 ARITHMETIC.

14 . If Ibuy butter at 30if a pound , how much per cent do Igain by selling it at a pound ?

917 . Philadelphia Public Schools. Slate Work.

1 . Multiply eight hundred nine and one hundred twenty-fivethousandths by six and seventy-eight hundredths . Divide theproduct by fifty-four.

2 . Find the quotient of 5205} g of 91.

3 . Find the greatest common divisor of 216 and 360.

4 . Find the least common multiple of 13 , 15, 26, and 39.

5 . If two-thirds of a yard Of silk canbe bought for $4, howmany yards canbe bought for $34?

6 . A drover sold 250 sheep for whichwas 15% morethan they cost. What was the cost Of each sheep ?

7 . A man shipped 600 barrels Of flour, and lost of it

by storm he sold 75% of the remainder. What per cent of thewhole remained ?

8 . A owns 65 shares (par value , $50 each) Of aninsurancecompany , which declares a dividend of 2} per cent . Requ iredthe amount Of A '

s dividend .


1 . How many rolls of paper,each 8 yd . long and 18 in.

wide , willpaper the sides of a room 16 ft . by 14 ft . ,and 10 ft.

high,deducting 174 sq . ft . for doors and windows ? Illustrate .

2 . Find the sum Of 356 , 3, 43, 11

11 , 3

1355 , indecimals, correct to

fourth place .

3 . Of 200 bu . is 2470 of how many bushels ?

4 . If a grocer’s gallonmeasure is tOO small by l gi . , whatdoes he make dishonestly inselling 2 hhd . of molasses , averaging58 gal. 2 qt . 1 pt . each , worth $ 8 0 a gallon ? (Art .

BANK DISCOUNT. 503

5 . What will it cost to cement a cellar bottom 48 feet 6inches long and 25 feet wide

, at $ 4 5 a square yard ?

6 . If corn selling for 21¢ a bushelmore thancost gives a

profit of 3070 ,what did it cost ?

7 . Bought a hhd . Of sugar containing 8 cwt . 48 lb. for

and paid freight and cartage . At what price perpound must it be sold to gain

8 . How many tablespoons, each weighing 2 ounces, can bemade from 2 pounds 10 ounces Of silver ? Analyze .

9 . After taking out 15% Of the grain ina bin,there re

mained 40 bu . pk . How many bu . were there at first ?

10. Write the analysis Of the fifth .

DISCOUNT OF INTEREST-BEARING NOTES.

919 . Slate Problems.

BROOKLYN ,N.Y.

,Oct. 15

,1894 .

Sixty days after date Ipromise to pay to the order of HarmanP . Payne

,Forty-eight 15695 Dollars, value received , with interest

at

354815096 . GEORGE P . POST.

1 . Find the amount due on the above note at maturity .

If the above note is discounted at a bank , the discount is takenonthe

amount due at maturity .

Find the proceeds of the above note if discounted Dec . 1 , 1894 ,

at

2 . Find the proceeds Of a 90-days note for $ 175, bearingInterest at discounted 33 days after date , at

3 . Find the proceeds Of a 60-days note for $ 350, bearinginterest at discounted at 10 days after date.

504 ARITHMETIC.

4 . Find the proceeds of a three-months note for $840, bearing interest at discounted at bank 47 days before maturity

,

at

5 . A four-months note for $ 720, datedMarch 17 , 1894, bearing interest at is discounted at May 10. What are theproceeds ?

REVIEW.


1 . Change inches to rods, etc .

2 . Change feet to rods,etc .

3 . Change 964 yards to rods, etc .

12 pence (d .) 1 sh illing (a)

20 sh illings 1 pound (46)A farthing is one-fourth Of a penny , and is generallywrittenas a fraction.

4 . Reduce 4 ,000d . to d .

5 . How many pence in£ 87 178 . 6d . ?

6 . What willbe the cost of 150 yd . silk at per yd . ?

3 6 38 . 6d . , read three and Sixpence.

If £ 1 what will be the cost inU . S . moneyof 75 books at 18 pence each ?

8 . A merchant sells 37 coats at £ 3 58 . each,less

What is the amount of his billinEnglish°

money ?

9 . Find 25% of £ 183 148 . 8d .

10. A silver dollar weighs 412} grains. How many ouncesof pure silver are there in silver dollars i f the coin is 1

95

pure silver ?

1 1 . The one-cent pieces weigh 48 grains. How many dollarswould weigh 120 pounds avoirdupois grains to pound)?

REVIEW . 505

12 . A coaldealer buys 150 tons of coal,

pounds each,at

per ton . He sells it at per ton,giving lb .

tothe ton . What is his profit ?

13 . Three workmen receive fordoing a piece Of work .

One worked 32 days, the second worked 53 days, the thirdworked 41 days. What is the share of each ?

14 . Three men engage ina business venture . One furnishesanother furnishes a third furnishes They

gain What is each one ’s share of the profit ?

15 . A garrison of menhas rations for 40 days. Tendays later it receives a reinforcement of 300 men . How manymore days willthe rations last ?

16 . Two trains start at 9 O'clock from towns A and B

,120

miles apart . The train leaving A travels 20 miles per hour , theother 30miles per hour . What time do they meet

, and at whatdistance from A ?

17 . Divide $900 among four persons, so that the second will

have three times as much as the first ; the third , twice as muchas the second ; the fourth , as much as the three others.

18 . Divide $540 among three persons, so that the first willhave $48 .more than the second , and the second 75 more thanthe third .

19 . Four menworking 7 hou rs a day need 15 days (105hours) for a piece Of work . How many days would it take 6men

,working 10 hours per day , to do the same work ?

20. A grocer buys some eggs at 15 if per dozen. He breaks

15, and then finds that by selling the rest at 16¢ per dozen , he

willneither gain nor lose . How many eggs did he buy ?

21 . If a person lends me $ 250 for 8 months, for how longought I to lend him $400 as anequ ivalent ?

22 . A fort had‘

a garrisonOf men, and provisions for 18weeks. If men were sent away , how long would the pro

visions last ?

506 ARITHMETIC.

23 . At per ounce , what is the value of 12 bars of puregold

,each weighing 12 lb. 5 oz . 16 pwt . 5 gr. ?

24 . A plank that is 20 feet long and 12 inches wide hasthe same area as another plank 24 feet long . What is the widthof the latter plank ?

25 . Mr. Jones owes and has only to pay hisdebts. How mu ch should he payMr. Smi th , to whom he owes

$576?

922 . U. S. CivilService. Assistant Weighers, CustomHouse.


132 873 63

Express infigures

2 . Twenty-five thousand one hundred one .

3 . Forty-four million , one hundred one thousand one , andone ten-thousandth .

Express inwords

4 .

5 .

6 . A merchant bought 51 tons 17 cwt. 3 qr . 25 lb. of wool,

and sold 27 tons 4 cwt. 2 qr . 27 lb . Of the remainder , one-halfwas lost by fire . How much had he left?

28 lb. 1 quarter ; 4 quarters 1 cwt.

508 ARITHMETIC.

10. The product of three factors is two of the factors

are 23 and 11. What is the third factor ?

11 . Solve by cancellationHow many pieces Of cotton cloth , each piece containing 42 yd .

at 9} per yd . , canbe bought for 14 firkins of butter, each containing 56 lb. , at 1915per lh. ?

12 . How many minutes are inNovember ?

13 . What must be the depth of a binwhich is 4 ft . wide and6 ft . long

,to contain 40 bu . oats ?

14 . If a locomotive runs 25 mi . 48 rd . in50 m in. , how far

willit runin12 hours ?

(G ive answer inmiles and decimalOf a mile.)

15 . A farmer sold lb. hay at $ 12} per ton , and took inpart payment ft . of boards at $ 11 per thousand . Howmuch remained due him ?

16 . Name and define five terms used inpercentage .

17 . A store rents for $500, which is 10} per cent of its value .What is its value

18 . Bought 80 barrels of flour at 6 per barrel, paying forfreight 30. At what pri ce must I sellit per bbl. to gain 30%on the totalcost ?

19 . What is the amount of for 3 yr . 5 mo . 19 da. ,at

6 per cent ?

20. Three men buy a house for$ 2,500. A pays $500, B pays

$ 900, C pays They rent it for $ 250. What is eachone ’s share of the rent ?

2 1 . If acres of land cost what will acrescost

22 . What willbe the duty,at 55 cents per square yard ,

on 6pieces of cloth , each containing 54 yards, 32 inches wide ?

PERCENTAGE. 509

COMMERCIAL DISCOUNT.


1 . Ona bill of goods amounting to a discount of5% is given for cash . What is the amount paid ?

2 . What will be the cost of 16 gross of Roman candles at

per gross , less

3 . Sept. 1, 1894, I bought tea amounting to If

5% is deducted for payment within ten days, how much would Ihave to pay if Ipaid the billSept . 9 ?

4 . What willbe the cost of 15 cases cocoa each,

less

5 . Bought 5 gross of essence of lemon at per doz .,less

What is the amount Of my bill?

6 . Find the cost of 15 cases of chloride of lime , 50 lb. percase , at per lh .

,less

7 . What willbe the net cost of a billOf plated ware amounting to on which a discount of 33} and 10% is allowed ?

1ess i 28 25Th is means 33}% discount onthe gross amount.

and 10% di scount onthe remainder.

less 110Ans . net.

8 . Find the difference between$390 less discount ,and $390less 33} and 10% discount.

9 . Thurber,Whyland CO. sold the following goods . Make

out the bill, less 50 and 10 and 10 and 10% discount.

500 -pound bags at perM-poundl-poundl}-pound2-pound

510 ARITHMETIC.

10. Find the net cost Of bags at perM. less 60and 10 and

11 . Which is the better discount for the buyer , 40 and 10%or 30 and 20% What willbe the difference on a billOf $ 100?

12 . $ 100less 33} and 10% discount is equalto what ? Whatper cent discount is 33} and 10% equal to ? TO what per centnet is it equ ivalent ?

13 . What per cent discount is equalto 40 and 20% discount ?

14 . Inprice-lists the gross price is generally given. Whatis the gross

,

” or “ list,

” price of plated ware costing $60 net, onwhich a discount of 33} and 10% was allowed ?

15 . What per cent of the list price is the net price Of goods onwhich 20 and 10% discount is allowed ?

16 . What per cent Of the gross price is the net price Of goodson which 10 and 20% discount is allowed ?

1 . Find 12} per cent of 648 .


3 . What per cent Of 50 dollars is 5 cents ?

4 . is 10 per cent of what ?

5 . Write three-fourths Of one per cent , first as a pure decimal,and again as a common fraction .

6 . 3» of a number is what per cent of it ?

7 . 30 and 6 per cent Off the pri ce of an article markedreduces the price to what ?

8 . The selling price was $ 30,and the gain 25 per cent .

What was the cost ?

9 . What number dim inished by 15 per cent of itself is 510?

COMMERCIAL DISCOUNT. 511

10. If a piece Of property is taxed at a tax rate Of

Of one per cent, what is the assessed value of the property ?

11 . Mr. Wilsonbought 96 shares of stock , par 50,for

What was the rate of discount ?

12 . A boy deposited $ 15 inbank . This was 30 per cent Ofwhat he had inbank before making this deposit . What had hethere after this deposit ?

13 . What is the interest of 500, for 3 years 4 months and 18

days, at 6 per cent ?

14 . What is the interest of for 2 years and 6 months,

at 5 per cent ?

15 . What is the exact interest Of $500, for 100 days, at 8 percent ? (Take 365 days to the year .)

What is the interest Of 700, fromMarch 10to November3 , at 5 per cent ? (360 days.)


1 . A piano , marked $ 800, is sold at a discount of 25 andWhat is the selling price ?

2 . Bought goods amounting to 600, less 5% for cash . Whatis the net cost of the goods ?

3 . What single discount is 50 and 10% equalto ?

4 . What single discount is 30 and 30% equalto ?

5 . Paid $ 729 for goods, on which 10% was allowed . What

was the gross price ?

6 . How much willI have to pay for 12 doz . bottles flavor

ing extract , at per doz ., less 10%

7 . What is the “list price Of anarti cle forwhich Ipaid $48 ,

after a discount of 25% was deducted ?

512 ARITHMETIC.

8 . Which is the larger discount , 40 per cent or 20 and

Why ?

9 . What is the difference between a discount of 50 and

and one Of 10 and 5070

10. Find the net cost Of a $ 700 billOf fire works, on which a

discount Of 6070 is allowed .

927 . BrooklynPublic Schools. ExaminationQuestions.

1 .

‘

A man received per day , exclusive Of Sundays, during 1891. He paid $ 73 for clothing for himself and family , $ 15per month rent , per day for provisions, $8 per month forfueland light , and 2515 per day for other expenses . How muchhad he left at the end Of the year ?

2 . Divide the product Of 8} and 11} by their difference.

3 . If a meter measures inches,how many yards will

be equalto meters?

4 . A merchant bought 340 bushels Of potatoes at 80£5 perbushel; 20 per cent of them proved worthless, and were thrownaway . He sold the remainder at per bushel. How muchdid he gain or lose ?

5 . What would it cost, at per square yard , to paint theoutside of a fence , 6 feet high , that encloses a lot 100 feet longand 25 feet wide ?

6 . Divide eighty-four and eighty-four hundredths by fortyeight thousandths.

7 . What is the interest on at 4 per cent , for 197days, taking 360 days to the year ?

8 . A mancando a certain wdrk in18} days . What percent of it canhe do in6} days ?

9 . What is the amount of at 4 per cent,for 7 months

24 days ?


A m anspent 30 per cent of his money for clothes, 20 percent for rent, and had $ 75 left . What rent did he pay ?

11 . What is the value of Of 8 Of 3} (10} 7}

12 . From 16 ten-thousandths take 27 millionths,multiply the

difference by and divide the product by 410.

1 3 . How many bushels Of grain willfill a bin feet long,

feet wide,3} feet deep ?

14 . Iwish to put 111 bu . 2 pk . 4 qt. of grain into 47 bags.

What quantity must each contain ?

15 . What is the difference between per cent of and50 per cent of

16 . A manbought silverware for $ 120, and sold it for $250less 33} and 10 per cent . What was his profit per cent ?

17 . What is the interest on at 4 per cent, for 3 yr .

5 mo . 14 da. ?

18 . A note for $600, drawn January 16, payable 4 monthsafter date

,is discounted March 25 at a bank

,at 6 per cent .

What are the proceeds?

19 . There are 350 pupils on the third floor Of a school,450

on the second floor , and 600 on the first floor. What per cent ofthe pupils are on each floor ?

20. A dry-goods merchant sells goods per yard morethan their cost, and realizes a profit of 8 per cent . What is thecost per yard ?

21 . A manbought 396 acres of land for He sold150 acres at $ 120 per acre , 134 acres at $80 per acre, and theremainder at cost . Did he gain or lose , and how much ?

22 . If 44} yards of calico cost how much must be paidfor 80 yards?

23 . Divide the sum of 75 thousandths and 75 ten-thousandthsby the difference between 75 hundredths and 75 tenths.

514 ARITHMETIC.

24 . What number divided by 320 gives 47 for quotient and

163 for remainder ?

25 . What is the cost Of a pane Of glass 30 inches wide and 50

inches long , at 40¢ per square foot ?

26 . If a river current carries a raft Of lumber at the rate Of 4miles 4 furlongs 20 rods (4 miles 180 rods) per hour , how longwilli t take the raft to float 365 miles ?

27 . Ina schoolroom there are 35 pupils and a teacher . The

room is 30 feet long , 20 feet W ide , and 15 feet high . Howmanycubic feet Of air space has each person?

28 . What is the interest of for 8 months 19 days, at4 per cent ?

29 . A merchant sold a quantity of flour for $ 282 , losing 6

per cent . How much money did he lose ?

30. Ibought bushels of wheat at 80¢ per bushel, andsold it for 84 per bushel

,on a note for 60 days , which I had

discounted immediately at a bank , at How much did Igain ?

INTEREST AND BANK DISCOUNT.


1 . Find the interest on 2:dollars,for 2 years 3 months

,at

a: X 7 87 ; X

2 . What principalw illproduce $ 180 interest in2 years 3months

,at

Let a: the principal.

3 . Find the interest on for 51: years, at

4 . A manloaned at He received $ 765 interest .Find the time .

5 . What sum , at w illproduce $ 240 in3 years ?

516 ARITHMETIC.

2 . Find the area Of each of the following triangles insquarefeet ; base of each 100 feet, altitude 60 feet.

3 . Mark ineach of the following eleven triangles its area insquare feet ; altitude of each , 60 ft .

4 . Find the area Of each of the following four triangles insquare feet ; altitude Of each , 60 ft .

5 . Find the area of each Of the following fou r parallelograms ;altitude Of each , 60 ft.

6 . Find the area of each Of the following trapelzoids ; altitudeof each , 60 ft . The number of square feet ineach is equalto 60multiplied by what

SURFACES AND VOLUMES. 517

7 . One diagonalof each Of the following quadrilaterals measures 100 feet . The perpendiculars let fallonthis diagonalfromthe opposite corners measure 30ft. and 40ft . ,

respectively . Findthe area Of each insquare feet.

8 . Cu t from a strip Of paper,AB , two inches wide, a rectangle,

a rhombus, and a rhomboid, as given inthe accompanying dia

gram. Showthat the three parallelograms are equalinsurface.

9 . Cut from a strip of paper,two inches wide , three trapezoids.

Make one parallel side Of each 2 inches long , and the otherparallel side 3 inches long . Divide up each trapezoid insucha way as to show that its surface is equal to that Of a rectangle2} in. by 2 in.

SURFACES AND VOLUMES.


1 . How many cubic feet are there ina block of granite whosebase is 4} feet square , and whose height is 6 feet ?

2 . Find the value of the above block at 60 per cubic foot,and the cost Of polishing its six faces at 60£5per square foot.

518 ARITHMETIC.

3 . A contractor charged 30¢ per cubic yard for digging acellar 21 feet wide , 60 feet long , 9 feet deep . What was theamount Of his hill?

4 . What willbe the cost of cementing the floor of the abovecellar at per square yard ? Deduct from the givendimensions the thi ckness Of the four walls , eighteen inches each . Makea diagram .

5 . A cubic foot Of water weighs ounces . Marble istimes as heavy . Calculate the weight, intons, of a marble shaft4 feet square and 12 feet high . (Cancel.)

6 . A carpenter is making a cubicalbox whose inside measurement is 1 foot . Each side consists of a single piece Of wood 1inch thick . G ive the dimensions, ininches, Of each of the six

pieces used .

7 . How many pounds avoirdupois would there be ina brickOf pure gold 8 x 4 x 2 inches

,gold being times as heavy as

water ?

8 . Find the weight inpounds Of a cord of pine wood,which

is .66 times as heavy as water .

9 . A cube of marble , 12 inches on a side,is enclosed for

transportation ina tightly fitting wooden box made Of materialone inch thick . What are the outside dimensions of the box ?How many cubic inches inthe wood and the marble together ?How many cubic inches are there Of each ?

10. Aniron cube 2 feet long weighs 2 tons. Howmany timesas heavy as water is iron ? Calculate the weight of an iron cube1 foot long . Of one 3 feet long .

CHAPTER XII.

SIMPLE AND COMPOUND INTEREST. DISCOUNT. CAUSE

AND EI‘

PEOT. PARTNERSHIP.—BONDS AND STOCKS.

EXCHANGE. LONGITUDE AND TIME. SURFACES

AND VOLUMES.

TO FIND PRINCIPAL, RATE, OR TIME.

931 . At what rate per cent will amount toin1 yr . 1 mo . 10 da . ?

Let a: rate.

Then x166xm interest.

amount

Transposing,Clearing of decimals, 804 2:

a: 4} Am . 4} per cent.

932. Inwhat time will produce interest, at 4per cent ?

Let at : time inyears.

Then x f t} x a: interest.z

Clearing of decimals, 342 4: 817

a: H f : time inyears.

2 yr. 4 mo . 20 da .

yr.

133 yr. remainder.

12

1596 mo . newdividend .

2 8 mo . remainder.

30

6840 da. newdiv idend .

0 Am . 2 yr. 4 mo . 20 da .

520 ARITHMETIC.

933 . SlateExercises.

Find rate, time , etc .

Principal,

time , 3 yr. interest, $ 300. Rate ?

Principal,

rate , 4% interest, $ 144. Time ?

Time,8 mo . ; rate , interest, Principal?

Principal, $38 ; time, 2 yr. ; amount , Rate ?

Principal, $ 140 ; rate , time,3 mO. 15 da . Interest ?

Amount , rate,

time , 2 yr . 6 mo . Principal?

7 . Amount,

rate,

principal, $460. Time ?

8 . P rincipal, rate,470 ; time , 1 yr. 7 mo . 15 da.

Amount ?

9 . Time,8 yr . rate , 3% amount

,Principal?

10. Principal, time,3 mo . 5 da . ; interest,

Rate ?

1 1 . Principal,

rate , 4% interest, $632. Time ?

12 . Time,2 yr. 3 mo . 20 da . ; rate , amount ,

Principal?

1 3 . Principal, $ 756 ; rate , time , 3 yr . 4 mo . 20 da .

Interest

14 . Principal, $ 120 ; time , 1 yr . 2 mo . 15 da. interest ,

Rate ?

15 . Amount,

rate , interest , Time ?

16 . Time , 3 yr. 8 mo. 19 da ; rate , amount ,Principal?

17 . Principal, rate,

time,4 yr . 9 mo . 25 da.

Amount ?

18 . Rate , time, 4 yr . 6 mo . 23 da . ; interest ,

Principal?

INTEREST. 521

19 . Rate,3% amount , interest, Time ?

20. Principal, $ 60 ; amount, time,3 yr . 10mO. 11 da.

Rate

21 . Amount, rate , time,2 yr. 4 mo . 6 da.

Interest ?

22 . Interest , rate , time,1 yr . 5 mo . 25 da.

Amount ?

23 . Principal, rate,

interest , $ 37. Time ?

24 . Principal, time, 3 yr . 9 mo . 20da . interest

,

Rate

25 . Amount,

rate,

interest,

Time ?

Inwhat time will$ 100 amount to 109, at 6% interest ?

At what rate will$200 produce $ 16 interest in2 years ?

What principalwillproduce 12 interest in3 years, at

Inwhat time will$300, at produce $ 29 interest ?

Inwhat time will$ 170 produce interest , at

Inwhat time will$360 produce interest , at

Inwhat time will$ 725 produce interest, at

Inwhat time will$45 produce 45¢ interest , at

Inwhat time will$ 72 produce interest, at

Find the interest on $84 for 144 days, at

Find the interest on 125, at for 2 months 12 days.

At what rate will$64 produce 64 ¢ interest in80 days ?

At what rate will$40produce interest in6 months ?

A certain principalproduces 120 interest, at Whathe the interest if the ratewere

522 ARITHMETIC.

A man borrows at and loans it at

18 his profit in2} years ?

What is the interest on 240,for 3 years, at 570

Find the interest on at for 6 months ?

What is the interest on $60, at for 11 months 29

Find the amount Of $250,at for 2 years.

What is the interest on 90,at for 99 days?

1 . Add following, placing the sum bottom

2 . Express the following infigures : One billion ten millioneleven thousand twenty ,

and (decimal) one thousand twentyfive ten -millionths.

3 . Express the following in signand figures : Ten billionone hundred ten m illionone hundred thousand five dollars six

cents and three and three-tenths mills.

524 ARITHMETIC.

2 . Find the interest on at 670 , for 1 yr. 7 mo. 12 da.

x .06.

interest for 1 yr.

6 me .a yr. interest for 6 mo .

1 me . (of 6 mo .) interest for 1 mo.

10 da . (of 1 mo .) interest for 10 da.

2 da . (of 10 da.) interest for 2 da.

interest for 1 yr. 7 mo . 12 da.

3 . Find the amount Of at for 1 yr. 9 mo . 4 da.

principal.5% interest for 1 yr.

6 mo . yr. interest for 6 mo .

3 mo . (of 6 mo .) interest for 3 mo .

3 da. (of 3 mo .) interest for 3 da.

1 da. (of 3 da.) interest for 1 da .

amount for 1 yr. 9 mo. 4 da.

What is the amount of for 3 yr. 6 mo . 20 da. ,

principal.

(10% a )116 interest for 2 yr.

1 yr. (of 2 yr.) interest for 1 yr.

6 mo. yr. interest for 6 mo .

20 da. of 6 mo. interest for 20 da .

amount for 3 yr. 6 mo . 20da.

5 . Interest Of at for 1 yr. 7 mo . 25 da.

1 yr. , 6 mo. ,1 mo . ,

15 da 5 da. ,5 da.

6 . Interest for 10mo . 29 da .,at on

x .04.

interest for1 year;interest for 1 month deduct from interestinterest for 1 day for 1 year.

interest for 10mo. 29 da.

INTEREST. 525

7 . Amount, at Of for 2 yr . 10mo . 4 da .

8 . Interest on at for 1 yr . 4 mo . 12 da.

9 . Interest, at for 4 yr . 2 mo . 7 da .,on $250.

10. Amount Of at for 33 days.

937 . Whenthe time is less than a year, the followingshould be remembered

670 for a year is 1 per cent for 60 days.

570 for a year is 1 per cent for 72 days.

for a year is 1 per cent for days.

4% for a year is lper cent for days.

11 . Find the interest for 81 days, at on

72 days 1%9 days (of 72 da.) a

interest for 81 days.

i

12 . Amount of $954 , at for 4 mo . 10 da.

Principal$954.

3 months’ interest 1%1 mo . (of 3 mo .)10 da. (of 1 mo.)

amount for4 mo. 10 da.

Interest Of at for 93 days.

80 days

10 days

2 days

1 day

Interest Of at for 158 days.

72 da . , 72 da .,12 da .

,2 da .

Amount of for 193 days, at

60 da. ,120 da . , 12 da . ,

1 da.

526 ARITHMETIC.

16 . Find the proceeds Of a 90-days note , for at

Face

60 da .

30 da . Deduct.3 da.

proceeds.

17 Find the discount ona 3-months note , for at

18 . What are the proceeds of a 6-months note , for $ 785,discounted at 670 .

19 . Find the interest on for 1 yr . 3 mo . 17 da . ,at 770 .

20. Find the amount of $683 , for 3 yr . 4 mo . 11 da . , at

938 . N.B.— DOnot use unnecessary figures.

2 1 . Principal, $ 360; 3 yr. 7 mo . 18 da . Interest ?

22 . Principal, $ 613 ; 157 da . Amount ?

23 . Principal, 17 mo. 23 da. Interest ?

24 . Principal, $875 ; 2 yr . 3 mo . 1 da. Amount ?

25 . Principal, $976 ; 325 da. Interest ?

939 . By the time of a note is meant the numberOf days, etc forwh ichi t is drawn. Find the discount for three additionaldays.

26 . Face of note , $254 ; time , 30 days ; Proceeds ?

27 Face of note , $515 ; time , 6 months ; Discount ?

28 . Face Of note , $493 ; time , 60 days ; Proceeds?

29 . Face Of note , $ 717 ; time , 15 days ; Discount ?

30. Face Of note, time,90 days ; Proceeds?

940. Find the exactnumber Of days . Take 360 days to year.

3 1 . Principal, J an. 2 to Dec . 1. Amount ?

3 2 . March 8 to April9. Interest ?

INTEREST. 527

3 3 . 5% Feb . 13 to Sept . 4. Amount ?

34 . Principal, Oct . 1 to Dec . 17 . Interest ?

3 5 . Principal, May 9 to July 3 . Amount ?

By the term of a note is meant the number of days it has to runafter it has beendiscounted , including days of grace.

36 . Face Of note , $ 100 ; term , 63 days ; Discount ?

37 Face ofnote, $200; term,

93 days ; Proceeds ?

38 . Face of note, $300; term,

24 days ; Discount ?

39 . Face of note , $400 ; term ,117 days ; Proceeds ?

40. Face Ofnote , $500 ; term , 88 days ; Discount ?

942 . In examples 41- 45

,inclusive, find the time by compound sub

trac ti on.

4 1 . Principal, J an. 14,1892

,to Sept . 5, 1894 .

Interest ?

42 . Principal,

Feb . 6, 1893 , to Aug. 1, 1896.

Amount ?

43 . Principal, March 25,1894 ,

to J an. 13 , 1897 .

Interest ?

44 . Principal, April15, 1888 , to Feb . 6,1895.

Amount?

45 . Principal, May 6, 1893 , to Dec .

Interest ?

46 . Face of note , $600; time , 60 days ; dated June 13 , discounted June 19. Proceeds ?

47 . Face of note , $ 700 ; time , 90 days ; dated July 2 , dis

counted July 3 . Discount ?

48 . Face of note , $800; time , 3 months ; dated Aug. 14 ,dis

counted Oct . 1 . Proceeds ?

528 ARITHMETIC.

49 . Face of note , $900 ; time , 4 months ; dated January 5,discountedMay 5. Discount ?

50. Face Of note , time , 6 months ; datedMarch 17 ,discountedMarch 17 . Proceeds ?

943 . U. 8. CivilService. Day Inspectors, CustomHouse.

1 . Add these across, placing the totals in the spaces pro

vided , and find grand total.

Totals.

2 . Express the following inwords :

3 . Express the following inwords :

4 . Express inSignand figures one million one thousand tendollars and twenty cents .

COMMERCIAL DISCOUNT. 529

5 . What is the sum of

.66625 ?

6 . Divide Of 4} by {L ofReduce fraction to its lowest terms.

7 . A merchant purchased 3 hhd . wine at 502 per gal. Helost by leakage , from one hhd . from another and fromthe other 4% Of the quantity . He sold the remainder at 292

p er qt. Did he gain or lose by the transaction , and how much ?

( 1 hhd . 63 gal.)

8 . If a laborer received for his services per day, ex

clusive Of (53) Sundays during the year 1887, and paid $ 175 per

year for house rent , per day for provisions, $8 per monthfor fuel and lights, per month for clothing for self andfamily , and per day for incidentals, how much had he leftat the close of the year ?

9 . Five lots Of coal, containing, respectively , 43} tons, 38}tons, 45} tons, 33} tons, and 66} tons , were divided equallyamong five dealers. How much did each receive ? (Tonlb .)Express answer intons, hundredweights, and quarters.

10. What willit cost to carpet a . room , 13 feet long and 18feet wide , with carpeting yd . wide, at perlinealyard ?


1 . Ona billOf goods amounting to a discount Of 5%is allowed. Howmuch must be paid ?

9,s Ans.

Divide by 2, placing the quotient figure one place to the right of the cor

responding figure of the dividend .

530 ARITHMETIC.

2 . Find the cost Of a wagon , the catalogue price of which is

$ 750, the discount being

750x . 70 $525. Ans.

Thenet cost is .70 Of the catalogue price.

3 . What willbe the cost of goods amounting toonwhich there is allowed a discount of

5% (of 10 Deduct.(of

Am .

For rewrite the originalamount, plac ing first figure one place to theright. 5% is Of 10% . is of

4 . less

} Of Am .

5 . less

Deduct 10% 11} of

6 . Find the net cost Of yd . silk,invoiced at per

yard , less 16% discount .

7 . What is the cost,infrancs, of meters Silk

,at

francs per meter , less

8 . What is the net cost of a lot of musical instrumentsamounting to on which a discount of 10, 5, andis allowed ? (Art . 924

,Ex .

9 . What would be the net cost of the same articles,if the

discount were 5,and

10. Find the net cost of the same,at discount .

11 . Goods catalogued at 2: dollars are sold at a discount of 20and What is the net price ?

532 ARITHMETIC.

4 . Express inwords the following figures, signs, and abbre

v iations :

(7 mi . 3 fur. 24 rd . 4 yd . 2 ft . 9 in . + 2 mi . 2 yd. 1

mi . 1 fur. 8 rd . 2 yd . 1 ft . 3 in.

(fur.a furlong.)

5 . Express inwords the following

CCCLVI.

£ 13 158 . 8d .

6 . Add 026, 0137 , and .4 . From the sum subtract 3998 ,

and divide the difference by 21. (Express answer indecimalfraction .)

7 . A merchant bought 84 yd . of linen at 55 per yd .,and

105 yd . Of muslin at 20¢ per yd . He sold allthe linen at 40¢per yd . What must he charge per yd . for the muslin inorderto make up exactly his loss onthe linen ?

8 . A fru it dealer bought a lot of oranges for $240. Hesold of them for of the entire cost ; of the remainder forgof the entire cost ; Of what then remained for of the entirecost ; and the finalremainder for of the entire cost . What washis gainor loss

9 . The owner of 165 shares of gas sto ck sold them at $ 25per share , and with the proceeds purchased two lots , 32 ft . by115 ft . ,

and 30 ft . by 105 ft .,respectively , and had just $ 27 left .

What was the pri ce per square foot of the lots ?

10. At 18 ¢ a square yard,what will i t cost to plaster the

walls and ceiling Of a room 16 ft . long , 12 ft . wide , 14 ft. high ,deducting for two doors, each 8 ft . by 4 ft .

,and for three win

dows, each 7 ft . by 3 ft . ?

REVIEW . 533

1 . A can do a piece of work in5 hours,B in7 hours. HOWlong willit take both working together ?

2 . Anagent collected a bill, and sent to his employer theamount, less commission . If his commission wash owmuch did he remit to his employer ?

3 . My house , worth is insured for of its value,at

What premium do Ipay ?

4 . A floor 6 yards long , 4 yards wide , needs 32 yards Of

carpet to cover it . What is the width of the carpet ?

5 . What willbe the interest on $87 , at 5 per cent, for 144days ?

6 . Find the discount, at 8 per cent, on a note for $ 176,wh ich has 90 days to run.

7 . An agent receives to invest after deducting‘

his

commission Of 11} of the amount invested . What is the agent’scommission

8 . By selling a house for Ilose $ 500. What is myloss per cent ?

9 . A lot is sold for at a loss of 20 per cent . Whatpart of is the loss ?

10. A merchant’

s receipts are his gain is 20 per cent.W hat part of his receipts is profit ?

11 . If 3 menearn $ 72 in8 days, how many dollars will5

menearn in11 days ?

12. If a dealer loses 25% by selling a horse for $ 225, whatper cent would he ‘

gain or lose by selling the horse for $325 ?

13 . Find the cost Of 4 yd . 1 ft . of ribbon , when 2 yd . 2 ft.

cost 40 cents.

534 ARITHMETIC.

TO FIND FACE OF NOTE, RATE OF DISCOUNT,AND TIME.

947 . Iwish to Obtain from a bank . What must bethe face Of a 30-days note , which willgive the above proceeds, ifit is discounted at

Let 2: face of thenote.

_6_x All : discount,100 360 2000

e dsproc e

Clearing of fractions, 11 a:

a:

.531989

Low

Ans. face ofnote.

Face ofnote,

30 days’

discount3 days

’

discount .5027 115 of 30 days.

Proceeds

Deduct

948 . A note for was discounted at The proceedswere How many days had the note to run?

Let a: term indays.

x100x£6 $5: d iscount

i33= proceedsClearing Of fractions,

33 2:

a: 63

Am . 63 days.

536 ARITHMETIC.

2 . Express in words : (a) (6) £ 1788 . 3d .

3 . Find the value Of 2} 4} g.

4 . Divide by .012 .

5 . The regular salary Of a special examiner was $350 perquarter , with $ 3 additional for each day actually employed .

During the first quarter Of 1888 he was employed anaverage of

20 days per month . His expenses when employed were $ 3 a

day,and when unemployed $ 2 a day . Whatwas his net income

for that quarter ? (Leap year .)

6 . A pension is granted to a widow from April 16, 1879, at

$8 per month , and $2 per month for each of two childrenup tothe age Of 16 ; widow

'

s pension increased to $ 12 per month fromMarch 19, 1886. One child attains the age Of 16 onSeptember1,1880, and the other on November 22, 1882. She receives her

first payment onMarch 4 , 1887 . HOW much should this be ?

7 . Calculate the interest on for 1 year 5 months 12days, at 7% per annum .

8 . Anexaminer purchased a house for At the endof five years he found that after paying interest on h is money andalltaxes, he had received a net income from rent amounting to

$ 10 a month . He sold the house for one-fifth more than itsoriginalcost. How much did he make by the transaction

9 . Special examiner A reports 16 cases per month , B 14

casesper month , and C 12 cases per month . A receivesper year . If paid inproportionto work done

,what should be

the salaries Of B and C?

10. Owing to a deficiency inthe appropriation bill, the salariesOf the clerks ina bureau were reduced 18 per cent for the lastquarter Of the fiscalyear . How mu ch did a clerk who was paid

$287 for the last quarter receive during the whole fiscalyear ?

REVIEW . 537

SPECIAL DRILLS.

951 . Find sums

28 87 48

44 66+ 19

75 42 87

18 71 82

952 . Find remainders :

654 90 34 89

988 94 27 88

777 85 42 87

684 79 18 12

953 . Find products

91 x 19 51 x 29

82 x 19 42 x 29

73 x 19 32 x 29

64 x 19 23 x 29

954. Find quotients

378 —z 18 256 16

462 14 289 17

475 19 493 17

448 16 465 15

955 . Find answers

l36 x }290x 1

95

315 x i-g3 78 x };

538 ARITHMETIC.

956 . OralProblems.

1 . Paid 23 ¢ for calico , 27f for ribbon , and 48 ¢ for collars.

What was the amount of my hill?

2 . A farmer had 95 sheep . He sold 39, and 17 died . How

many had he left ?

3 . What willbe the cost Of 16 base balls, at 49 each ?

4 .~How much paint willthere be in27 casks , each contain

ing 75 lb. ?

5 . A man divided a 429 acre farm into plots Of 13 acreseach . How many such plots were there ?

6 . There are 900 men ina certain regiment . How manycompanies of 75 meneach are inthe regiment ?

7 . Find the cost Of 136 lb. sal-soda,at per lb .

8 . At per yd., what will I have to pay for 64 yd .

gingham ?

9 . How many square inches ina sheet of paper 10} incheslong by 4} inches wide ?

10. If 2} yards of cloth are needed for a jacket,how many

jackets canbe made from 18} yd . ?

1 1 . How many yards around a field 96 yards long , 75 yardswide ?

12 . What will be the area , insquare rods, of a triangle 33

rods base , altitude 42 rods ?

13 . How many acres in square rods ?

14 . How many feet ina mile ?

15 . I paid for cloth at per yard . How manyyards did Ibuy ?

16 . What willbe the cost Of 3 lb . 7 oz . of tea, at 64f per lb .

540 ARITHMETIC.

4 . The circulation inG reat Britain at the close of 1882 was

estimated to be , of gold of silverand of paper The value of the pound sterlingbeing what would be the equ ivalent of this circulationinthe currency of the United States ?

5 . Express infigures

Five hundred two million one thousand one and one tenthousandth .

6 . Express inwords


8 . The consular fees collected by the U. S . ConsulatLiverpoolduring the year ending June 30, 1882 , amounted toThe salary was What percentage of the fees wasrequ ired to pay the salary ?

9 . The silver coinage in France,in 1882 , amounted to

francs. The value of the franc is Whatwas the value of this coinage expressed inthe money of the

United States ?

10. Ifasof a pound of z inc costs $3 of a dollar, how manycents willH of a pound cost ?


11 . What must be the annualsales of a merchant that he mayrealize after paying expenses, when h is rate of

profit is 35 per cent ?

12 . The duty onwoolen bunting being 20¢ per square yard ,and 35 per cent ad valorem , give the total duty on 45 pieces,each containing 38 linealyards, the width being 42 inches, andcosting 55¢ per linealyard.

13 . Express infigures the following numbersTen million seventeen hundred seven dollars thirty-seven

cents five mills.

14 . Express inwords the following sign and figures17

15 . Add the following columns of figures so as to show (1)totalexpenses, and (2) totalreceipts ; and subtract the expensesfrom the receipts so as to show (3) the profits .

Expenses . Receipts .

Ofi ce rentFeed and care of horse

Wages of servants

Purchase and repair of instruments

Repairs of carriage and

R. R. fare

MedicinesStationery and printing

Total

Less expenses

Profits

16 . Reduce the following decimal fractions to common frac

t ions, and reduce to lowest terms

.75, . 125, .1875, .4 .

Day v isitsNight v isitsSurgicalOperationsOfi ce prescriptionsMileageMedicines

542 ARITHMETIC.

17 G ive the value Of two and one-half pounds of sulphate Ofeserine , at 33 ¢ per grain . gr . per pound .)

18 . There were 984 patients ina certainhospital, classified as

follows : 369, pulmonary diseases ; 246, nervous diseases ; 123 ,d iseases of heart and 246, various other diseases . G ive the percent Of each class.

19 . What willit cost to carpet an office room measuring 21feet inlength

,and 194 feet inwidth , the carpeting being yard

wide,and costing per linealyard ?

20. A physician accepts , inpayment Of a bill, a note fordue inone year and three months, without grace, at 7

per cent . What amount willbe due at maturity ?

SHORTMETHODS.

Slate Exercises.

X 427

47 236 Multiply by 7 . Multiply this product by 6. Why ?

2 834 16

958 . Find products

1 . x 328

2 . x 273

3 . x 287

4 . x 142

5 . x 189

832

2 631 2

105 248

Multiply by 8 . Multiply th is product by 4. Where isthe second product placed ? Why ?

544 ARITHMETIC.

44 . £ 7 16s. 7d . 45 . 8 yd . 2 ft. 6 in. 46 . 6 bu . 3 pk . 5 q t .

9 yd . 1 ft . 10in.

4 188 . 11d . 5 bu . 2 pk . 7 qt .

£ 20 18 . 3d . 30yd . 0ft . 3 in. 25 bu . 1 pk . 1 qt .

961 . Divide . 2 decimalplaces (Arts . 385,616)

47 . 50.

48 . 51 .,318

49 . 52 .

962. Write answers (Art. 385)

53 .

TABLE OF EXPORTS.

963 . The following table contains the values Of the goodsexported in1891 by the United States to the various countriesOf the world .

Find the totalvalue of the goods exported , and the per cent of this valuefor each section. Carry out to two places of decimals.

Countries to which Exported . Per cent .

1 . Europe

2 . Asia and Oceanica3 . Bri tish North AmericanPossessions4 . West Indies

5. South America

6.Mex ico , CentralAmerica , and British

8. Allother

INTEREST.

1 . 3 of a number is 48. What is the number ?

2 . A base ballclub won 17 games, and lost 13 games. Whatper cent of its games did it win?


4 . 2§ is what per cent of

5 . How many acres ina rectangular farm 1 mile long, 4m ile wide

6 . What per cent of the list price is paid by a buyer whoreceives a discount of 20 and 10 per cent ?

7 . A tank is filled by two pipes, one Of which canfill it in

6 hours, and the other in8. How long willit take both togetherto fillthe tank ?

8 . Find the interest on $80, for 72 days, at 670 .

9 . A man sold a wagon for 420, which was 16% less thani t cost. Howmuch did he lose ?

10. A kilo is lb. Howmany pounds in kilos ?

In the first four examples, carry out to two places of decimals, as

1 . The population OfMontana was in1880,in1890. Find the per cent Of increase .

2 . The population Of South Dakota in1890 was a

gain Of Over the population in1880. What was the

gain per cent ?

3 . The enrollment inthe South Dakota schools was in

1880, in1890. Find the gain per cent.

546 ARITHMETIC.

4 . What was the gain per cent inthe population of a statethat had inhabitants in1880, and in1890?

5 . A man marks an article and sells it at a discountOf 2570 from the marked price . If the article cost him 90

what is his gain per cent ?

6 . G oods costing $8 are sold at an advance of 20 per cent.The marked price is $ 12 . What per cent reductionis made onthe marked price ?

7 . A rug costs $20. It is sold at a profit Of The

selling price is 20% below the marked price . How mu ch isreceived for the rug? What is the marked price ?

8 . What price must cloth , which costs $ 2 per yard , bemarked so

.

that a profit Of 2070 willbe made whenthe cloth issold at 20% less than the marked price ?

9 . Find the net cost inEnglish money of a billof goodsamounting to £83 2s. 6d . less

10% 11, Of £ 83 23 . 6d.

2% t of 10%.

10. A can do a piece of work in12 days,B in15 days. How

long willit take both working together


1 . What is a promissory note ? Anindorser ? Purpose Of

anindorser ? Write a promissory note .

2 . Find the interest, by any Of the ordinary methods, Offor 1 yr . 4 mo . 10 da.

,at

3 . At what rate will amount to in2 yr .18 da .

4 . The net profits of a mill intwo years were andthe profits the second year were 5% greater than the profits thefirst year . What are the profits each year ?

548 ARITHMETIC.

10. How many days from July 1 to Sept. 6 ?

11 . At 1¢ a sq . ft . ,what will a piece Of wire netting 7 ft .

long and 6 ft. wide cost ?

12 . What willit cost to insure a house worth at 5%premium ?

13 . How many eggs, at the rate of 2 for5 cents, canbe boughtfor 60

14 . 75 is what per cent of 300?

15 . What is of 600?

968. WorcesterPublic Schools. ExaminationQuestions. Writteh.1 . What is the interest of from June 6, 1886, to

Feb . 16,1891 , at

2 . Inwhat time will$ 750 gain $ 195 interest , at

3 . What is the rate per cent when the amount of $ 500 isfor 2 yr . 6 mo . ?

4 . What principalwillgain $ 360 in5 yr . 4 9%5 . What is the duty , at 6 ¢ a gallon

,on 250 hogsheads Of

molasses, 60 gallons in a hogshead, leakage (Deductleakage .)

6 . A house valued at was insured for two-thirds Ofits value , at What is the insurance , including for thepolicy

7 . Anagent collected 20% Of an account of $ 750, charging4% commission . What was his commission , and what sum

should he have paid over ?

8 . Bought 480 barrels Of flour , at a barrel,and sold i t

for Find the gainper cent.

9 . By selling a house for a man gained 15% on thecost . What was the cost ?

10. What sum of money put on interest J une 1, 1885, w illamount to $522 Dec. 1, 1892, at 670

FRACTIONS. 549

1 1 . The tax assessed ina town is The property isv alued at There are 500 polls

,each taxed at $2 .

F ind the rate of taxation , and a man’

s tax whose property isv alued at and who pays one polltax . is raisedonproperty , being raised by polltax .)

12 . Find the difference between the greatest common divisorOf 480 and 520

,and the least common multiple Of 5, 6, 15, and 20.

13 . Find the value Of a pile of wood 40 ft . long,8 ft . wide ,

and 4 ft . 6 in. high , at a cord .

14 . A cargo Of flour was bought for 690. For what must itbe sold to gain

15 . Reduce to a simple fraction : (3 4)x

REVIEW FRACTIONS


NOTE.— DOnot use too many figures.

1 . Add 4, 3, g.

2 . Divide each of the following fractions by 6

7.H. 43» 13355

3 . Reduce 5 of 151 Of 15, Of to a simple fraction .

4 . 38g— 21fi .

5 . What fraction Of £ 1 18s . 9d . is 58 . 6d .

6 . Multiply 24%by f of 3.

7 . What is the greatest common di visor of 657.

and

The least common multiple Of 12, 16, 20, 30?

8 . What must be taken from 8155 to leave

9 . Reducem andm to their lowest terms.

10. Which is the greatest and which is the least, Of 4, i of g,and 2§ Of f f ?

550 ARITHMETIC.

11 . What must be added to 3141 to make

12 . Add g of a week , Of anhour, 11, Of a minute .

13 . How much is 9 times each Of the following fractions ?

3. fr. 417 . fi

14 . 303—4 of 7.

15 . Of fi + § Of fit .

16 . What part Of a ten-acre field is 4 A . 100 sq . rods ?

17 . What is the least number that will containeach Of thenumbers 6, 15, 18 , and 20?

18 . What must be multiplied by 4} to produce

19 . What is the value of ill—E} ?

20. What quantity must be divided by 43 to produce 83?

2 1 . Find the value3

How much is iii of 3 da . 15 hr . 32 min.

Reduce 155 mile to rods .

Add 3L, 4, 54. Subtract 4157 from the sum.

Multiply gOf 5%by Divide the result by

MEASUREMENTS.

Slate Problems.

1 . A tank 18 ft . long , 15 ft . wide , requ ires 63 sq . yd . Of

lead to line its sides and bottom . How many feet deep is it ?

(Make diagram .)

2 . A farmer has a 3-acre field inthe form Of a right-angledtriangle . If one perpendicular side measures 242 yards

,what is

the length of the other ?

552 ARITHMETIC.

13 . The area Of a floor is 135} sq . ft .,and its length is 12 ft .

8 in. What is its width ?

14 . How many bushels will a bin contain , its dimensionsbeing 10 ft . 8 in. X 12 ft . 8 in. x 8 ft . 9 in. ? (Cancel.)

15 . Find the capacity ingallons Of a tank 6 ft . 5 ih . ,by 3 ft .

9 in. ,by 4 ft. 6 in.

16 . A farmer wishes to construct a post and railfence arounda square field containing 40 acres. He pays 15¢ each for theposts

,which are placed rod apart . The rails cost each .

If the fence is 5 rails high,how much willthe materialcost ?

17 . A farm 1 mile square is divided into square fields eachcontaining 40 acres . Make a diagram Of the farm

,and say how

many miles of fence willbe needed to enclose allthe fields.

(640A . 1 sq . mi .)

18 . How many cords Of wood ina pile 164 ft. long, 16 ft .

wide , 30ft . high ?

19 . A cubic foot Of water weighs oz . What willbe theweight Of a gallon of water ? (G ive answer in pounds andfraction .)

20. Find the weight Of a quart of mercury , considering mercury 13 times as heavy as water .

21. A plot Of ground 320 ft . long,210 ft . wide , is enclosed by

a tight board fence 6 ft . high . Howmany square yards inthesurface of the fence ?

22 . Find the numkr Of square yardsina sidewalk , six feet wide , on the out

side Of the above plot .

The inner rectangle measures 320x 210 ft .What are the dimensions of the ou ter rectangle ?Find the difference betweenthe area of the ou ter rectangle and that of theinner one.

CAUSE AND EFFECT. 553

23 . Find the area of a gravelwalk , 6 ft . wide,just inside a

fence surrounding a plot 320 ft . long , 210 ft . wide .

Make a diagram. Find difi'

erence of areas, as above.

0A USE AND EFFECT.

1 . If 4 books cost what willa dozencost?

2 . If 3 lb . sugar cost 1342 what willbe the cost of 50 lb.

3 . If 48 lb. tea cost $20, what will12 lb. cost ?

4 . Bought 17 yards Of cloth for $30. How many yardscould Ihave bought for $90?

5 . If 36 men do a piece Of work in105 days,how long will

it take 72 men to do it ?

6 . If 7 railway trucks weigh 14 tons, how much would 29trucks weigh ?

7 . How long willit take 8 horses to plow a field , if 3 horsescan do it in8 days?

8 . What is the height Of a steeple that casts a shadow Of

300 ft . ,if an8 ft. pole casts a shadow of 12 ft .

9 . If 18 men mow 90 acres Of grass in5 days, how manyacres will36 men mow in5 days ? In10 days?

10. If 60 yd . carpet i yd . wide will cover a floor , how manyyards {1 yd . wide willbe required ?


1 . A pi ece of cloth , measured with a yard measure that Is 1

inch too short, appears to be 25 yd . long . What is its tru e

554 ARITHMETIC.

3 . If 3 men or 6 women cando a piece Of work in56 days,inwhat time will1 manand 2 women working together do it ?

4 . If 5 mencando as much ina day as 8 boys , how -long will

it take 32 boys to finish a piece Of work which 15”men cando in

12 days ?

5 . If $ 100 gain $4 in1 year , what will$ 350 gain in34years ?

6 . If 48 horses in10 days consume 180 bu . oats , how manybushels will32 horses consume in10 days ? In12 days? In15

days ?

7 . If 5 men mow 45 acres Of grass in6 days, inhow manydays will12 menmow 90 acres ?

If 5 menmow45 acres in6 days,1 manwillmow45 acres in6 x 5 days.

1 manwillmow 1 acre in5days.

6 x 5

45 x 12

6 x 5 x 90days.

45 x 12

12 menwillmow 1 acre in days.

12 menwillmow90 acres in

Z

Canceling,w 5,

45 x 122

974 . Inpractice, the work is somewhat shortened . Since the numberofdays is requ ired , we write the givennumber Of days los t, with a line

underneath .

Am . 5 days.

5 menmow 45 acres “5"

6 x 5 x 901 manmows 1 acre

12 menmow 90 acres45 X 12

If 5 mendo thework ina certaintime, 1 manwillrequ ire 5 times as

many days. We place 5 inthe numerator (as a multiplier). To qut 1 acre,

he willtake 115 Of the time requ ired to cut 45 acres . Place 45 in the

denominator (as a div isor).12 menwilltake 1

1, of the time 1 manrequ ires. Place 12 inthe de

nominator. To cut 90 acres willrequ ire 90 times as long. Place 90 inthe numerator.

556 ARITHMETIC.

18 . If a ship ’

s crew Of 500men have provisions to serve for48 days

, at the rate of 27 ounces a day for each man , how manymen will the same provisions serve for 60 days, allowing eachman 30 ounces a day ?

19 . How many hours a day must 9 men work so that theymay do as much in16 days as 12 men can do in15 days of 8

hours each ?

20. If 30! is paid for 6 lb . 14 oz . of bread , when wheat isper bu . , what should be paid for 23 lb. 12 oz . ,

when wheatis per bu . ?

NOTE. Reduce weights to ounces,or to pounds and fractions.

976. Admissionto N. Y. State NormalSchools.

1 . Express inwords :

(a) (b) (c)(J ) per cent ;MDCCCLXXXVIII.

2 . Find the sum Of allthe prime numbers to 50.

3 . If A and B can mow a field inseven days ; and A ,B

, andCmow it infive days, for $25, what ought C to receive ?

4 . With 5 bushels of wheat , at per bushel, and 5bushels Of rye

,at 60¢ per bushel, how many bushels of cats , at

30¢ per bushel, must be added to make a mixture worth 50}per bushel?

a: number. Mixture contains (a: 10)bu .

5 . TO g of a score add 1 Of a dozen , and from the sum sub

tract 3 of a hundred . What is the remainder ?

6 . What must be the length Of a load Of wood that is fourfeet wide, and five and one-third feet high

,to contain two cords?

z length .

7 . Bought a hogshead Of molasses containing 128 gallons,at

659‘a gallon ; paid 80¢ for cartage , and lost 16 gallons by

leakage . At what price per gallon must the remainder be soldto gain one-fifth of the entire cost ?

ExAMINATION QUESTIONS. 557

8 . What is the least number that will exactly contain 48,

20,21

, 24 ?

9 . Find the least common multiple of 24 , 44, 65, 100.

10. Sold 50 sofas for 25 Of them were sold at a gainof 20 per cent

,and 25 at a loss Of 20 per cent . What was the

gain or loss on the transaction ?

1 1 . Bought a number Of eggs , and sold 11 of them for what18 cost me . What was my gainper cent ?

12 . A bookseller wishes to mark up the price of a book whichhe is now selling for$2, so that he candeduct 15 per cent, andyet receive the present price . What must be the marked price

13 . What is the difference between.75 divided by 75, and 75divided by .75 ?

14 . A pole extended into the mud 54 feet ; i Of its lengthwasinthe river , and i of it inthe air. What was the length Of the018P

a: length .

15 . Find the exact amount Of $ 68 , fromMarch 13 to June 15,1888 , interest 6 per cent.

Take 366 days to the year.

16 . How many acres ina field 210 ch . long and 50 ch . wide ?

1 acre 10 sq . chains.

17 It took 9 horses to move a stick of timber weighinglb . How many pounds would a stick weigh that would

take 7 horses to move ?

18 . Mr. A sold a horse for $275, which was 5 per cent lessthan he asked for it , and his asking price was 5 per cent morethan the horse cost him . What was the cost of the horse ?

19 . Write inRoman numerals 1889.

558 ARITHMETIC.

20. A man purchased a rectangular field 18 rodswide , containing 10 acres 146 square rods Of land . The land cost him

$ 80 per acre , and he surrounded it by a fence costing 80f perrod . How much did the land cost, and how much the fence ?

21 . By selling a carriage for $ 117 , a dealer lost 10 per centof its cost . For how much should he have sold it to gain 10 percent of its cost ?

22 . A man , dying, left of his estate to h is wife , 3 Of theremainder to his son, and the remainder to h is daughter , whoreceived What was the value of the estate , and whatwas the son

’

s share ?a: value Of estate.

23 . What is the interest of $ 10, for 10 years 10 months and10 days, at 10 per cent ?

24 . If it takes one man 7} days to do a piece of work , howlong willit take 3 men to do 2§ times as much ?

PARTNERSHIP.


1 . B and C gain by trade $ 182. What is the gain Of each,

B having put in 300, and C $400?

The gainOf $ 700 is 182. What should 300gain? What should $400gain

2 . A , B, and C invest $ 720, $ 340, and $ 960, respectively .

The profits are $ 101. What is each one ’s share ?

Howmany dollars of capitalproduce $ 101 profits ?

3 . Two menhire a pastu re for $45. One puts in 15 cows,the other puts in12 cows. What should each pay ?

4 . A and B hire a boat for 50 days, paying $ 30. A uses it27 days, B uses it 23 days. Howmuch should each pay ?

5 . Our standard gold coin consists Of 900 parts gold , 90

parts silver , 10 parts copper . What is the quantity Of eachmetalin50 pounds of coin ?

560 ARITHMETIC.

6 . The smaller of two numbers is and their sum is

Find the larger number.

7 . Express .39, .00036, and as common frac

tions (or mixed numbers).

8 . Divide .826 by 100 ; by and fifty-nine tenthousandths by one thousand .

9 . Find the difference between and and the continued product Of .734 , and

10. Reduce 6 shillings 9 pence to the decimal of a poundsterling .

11 . Express as decimals seven hundredths, forty-three tenthousandths, and ninety-one millionths .

12 . Change 115 , 8316 , Th ,andm , into decimals. Find their

sum .

13 . Express .42796 as a common fraction , and the sum of 315 ,

7-3-5 , and T3531 as a decimal.

14 . X .07 x .0907 .

15 . Divide .0075 by .15,and .00044408 by .0112.

16 . Divisor , quotient,

Dividend ?

.035 x .0056th f17 What 18 e value O

07

18 . Change 69 rods to the decimalOf a mile .

19 . Change .4285 month (30 days) to days, hours, etc .

20. How many meters, each inches, in3 miles 220 rods?

2 1 . Change .1875 bu . to quarts .

22 . What decimalOf a pound is 13 oz . ?

23 . Reduce 4 ft . 1} in. to the decimalOf a rod .

24 . How many links Of in. each ina 4-rod chain ?

25 . A chain is 66 ft . What decimalof an acre is 1 sq . chain ?

561

979. AnEnglish Invoice.

Invoice Of Brass Wire Cloth , per steamship Furnessia , for entryat NewYork , shipped to Arthur Jacksonby Robt.MacDonalddt BrO. ,

ofLeiuth ,the 2lst day Of April

, 1893 .

6 cases marked A . J . 1- 6 .

2 pieces,

4 in. £24 7 8

41 “x 5 ft G m 451

42 x 7 ft . 2 in.

43 “x 7 ft. 8 in.

10

discount

Find cost inEnglish money .

NOTE.

— 1 2 1 shilling 2 pence, the price persquare foot. The remaining pi eces cost 10 pence per square foot.

960. A GermanBill.

MB . C. G . SCHMIDT,Bought Qf

Scarfs

Find cost inmarks. What is the duty inU. S . money at 35%onthe value ?1 mark f .

DRESDEN,Sept. 4,

MEYER SON .

M.

9.

562 ARITHMETIC.

APPROXIMA TIONS.

981 . G ive approximate answers at sight (Art . 890)

1 . 4871 is what per cent of 960?

2 . If 17 bu . 37 lb. of corn cost what will52 bu . cost ?

3 . About how many cords of wood ina pile 25 ft. long, 4 ft .

wide, 5 ft . high ?

4 . How many bushels (14 cu . ft .) can be placed ina bin6 ft .

long, 5 ft . wide , 4 ft . high ?

5 . Howmany acres ina field 50 rods long , 30 rods wide ?

6 . About how many yards are there inthe side Of a squarefield containing 1 acre sq .

7 . At 74 gal. to cu . ft .,about how many gallons will a tank

hold 6 ft . long , 4 ft . wide,3 ft . high ?

8 . .0987

9 . About how many dollars are equalto £ 199 178 . 6d . ?

10. A mark How many marks in$ 100?

BONDS AND STOCKS


1 . The people of a certain town wish to bu ild a street railroad . A company is formed . Five hundred shares Of stock

,Of

the par value Of $ 100 each , are sold .

At the end Of 6 months it is found that the profits are $2,000.

How much should the owner Of 10 shares receive ?

2 . Profits thus distributed are termed dividends . Whatsemi-annualdividend is declared on the stock Of the above railroad ? TO what per cent interest per year is it equal?

3 . Mr. H . has in the savings bank, onwhi ch he

receives a low rate of interest . Hearing Of the success Of the

564 ARITHMETIC.

13 . Canyou state a difference betweenstocks and bonds as tothe rate of income received from each ?

Bonds are redeemed at maturity . Howabout stocks ?If a railroad prove unsuccessful,which claims are first met, those of the

stockholders or those Of the bondholders ?

14 . Why is it necessary sometimes to employ a broker to purchase stOcks or bonds ? What is his fee called ?

Mention some other persons, not owners, through whombuyers regularly make purchases.

16 . What is the base inthe following ?

(0) Insurance ; (5) taxes ; (0) brokerage ; (d ) commission ;(8) interest ; (f ) discount ; (9) stocks ; bonds.

17 . At per thousand , what willhave to be paid intaxes by the owner Of property assessed at

18 . Mr. Cartwright owns a house and lot worthThe tax rate is 2470 , and his tax bill is $540. What is theassessed value of the property ? What per cent Of the actualvalue is the assessed value ?

19 . If the property inthe last problem were assessed at its

real value, what should be the rate to makeMr. Cartwright’s

tax billthe same ?

20. For insuring his property ,Mr. Cartwright pays a yearlypremium Of $ 135. If the rate is 470 ,

for how much is hisproperty insured ?

2 1 . Reduce feet to rods,etc .

22 . A manpaid for bank stock at 245. How manyshares, par value 100, did he buy ? If a quarterly di vidend Of

2470 is declared , how much should he receive

23 . Redu ce inches to rods, etc .

24 . A woman deposited $ 100 ina savings bank Jan . 1, 1892.

On the first of July , interest at the rate Of 470 per annum was

COMPOUND INTEREST. 565

calculated , and entered on the depositor’

s bank book . Jan . 1,

1893 , interest on the new principalwas placed to the credit Of thedepositor. The same was done July 1, 1893 . How much wasthere to the woman '

s credit at the date last mentioned ?

25 . Reduce feet to rods, etc .

COMPOUND INTEREST.

983 . Find the amount of $375, for 1 year , at Considering this as a new principal, find the amount for a year , samerate . Find the amount Of this last principalfor 3 months.

26 . What is the amount Of $ 375, for 2 years 3 months, atcompound interest ?

27 What is the amount of $ 375, for 2 years 3 months, atthe interest compounded semi—annually ?

Principal, $375.

3% 6 months Interest.Amount 6 months.

3 6 months’ interest.Amount 1 year.

3% 6 months’

interest.Amount IQyears.

etc etc .,etc .

Find the compound interest” on $375, for 2 years 3 months,

at 6 per cent, compounded semi-annually .

28 . What is the amount Of 100,at compound interest, for 3

years, interest at compounded annually ?

29 . Find the compound interest Of at for 2 years,interest compounded quarterly .

1

1

etc., etc . , etc.

566 ARITHMETIC.

30. Find the difi'

erence between the simple interest of $ 100,

for 2 yr . 3 mo .,at and the compound interest for the

same time , interest compounded semi-annually .

{a Divide by 4, and put first quoti ent figureone place to the righ t.

(four places of decimals are sufi cient .)

984 . Compound interest is allowed by savings banks . It is

not collectible on notes or other debts.

1 . A capitalist wishes to realize 5% onmoney invested instock . What must be the annualdividend on stock costing 300,inorder to produce this rate ?

2 . What willbe the taxes on property assessed at

the rate being $ 16 per

3 . Find the compound interest on for two years,at

five per cent, interest compounded annually .

4 . What will be the net cost Of anarticle marked $8 , onwhich a discount Of 50, 25, and 10% is allowed?

5 . Find the “list ” price of an article sold for $ 10 after adiscount of 50 and 50 per cent had been deducted .

6 . Paid 90f for anarticle . The discount is 25 and 25 percent . What is the list price ?

7 . One boy cando a certain piece of work in2 hours,a

second boy requ ires 3 hours,a third needs 6 hours. How long

willi t take the three working together ?

8 . Sold a cow for $ 60, losing What was the loss ?

9 . Sold a cow for $60, gaining What was the gain ?

568 ARITHMETIC.

money at 6 per cent per annum , how much per gallon must hecharge inorder that he may realize the full amount of the cost,including the estimated interest .

8 . The owner Of inEnglish consols (3 per cents)sellsthem at 96, and invests the proceeds in6 per cent $ 100 bondswhich he buys at 108. What is the difference. indollars andcents betweenhis income from the console and from the bonds ?

(36

9 . What is the weight (intons, cwt. , etc .)Of the water whichfills a cistern9 feet 8 inches long, 9 feet 4 inches wide , and 6 feet9 inches deep , a cubic foot Of water weighing ounces ? (1ton lb .)

10. A grocer pays 18 per pound for coffee,and roasts it ,

losing 10% Of the weight inthe process. What must he chargeper pound for the roasted coffee inorder to make a profit Of 2070

987 . BostonPublic Schools. Examination Questions. Mental

1 . What would be the cost of 2 bu . blueberries,at 5¢ per qt . ?

2 . What will18 oranges cost, at 35¢ per doz . ?

3 . A mancollected a billOf $300 for me , at 470 commission .

Howmuch was his commission ?

4 . A manput $ 15, which was 163% of his month’

s salary,

inthe bank . What was h is month’

s salary ?

5 . If I sellfor a bookwhich cost me $ 3 , what per centdo I gain?

6 . Out of 500 pupils, 50 are absent . What is the per centOf attendance ?

7 . A garden is 200 ft . long and 100 ft . wide . What will itcost to bu ild a fence around it , at 50If a ft . ?

8 . What is the interest of $ 200,for 90 days, at 3%

9 . What cost 4 lb . 4 oz . Of butter,at 16¢ a lh . ?


10. How many men could you hire with $50, fora week’

s j ob,

if each manreceived 12} per week ?

11 . How many doz . yeas t cakes, at 2¢ each , could be boughtfor 96 f

12 . 75 is 2570 more than what number ?

13 . Sold a watch for $44 , and so lost 12% on the cost . Whatwas the cost ?

14 . One acre Of corn yields 80 bu ., and another acre 20%

more . What does the second acre yield

15 . In14 yd . how many inches ?

16 . Whatwillit cost to fence a garden 10 rd. long and 6 rd .

wide,at $ 1 a rod ?

17 If each boy eats 4} Of a loaf Of bread , how many boys willeat 6 loaves ?

18 . May bought 6} yards Of cloth , at 122 a yard . How manycentsdid the cloth cost her ?

19 . How many half-pint mugs cana person fillwith 4 gallonOf milk

20. If your school-room is 40 ft . long and 30 ft . wide, whatis the distance around it ?

21 . If a cup holds of a pint, how many cups ina gallon ?

22 . A rectangular field containing 1 acre is 40 rods long.

How many rods around the entire lot ?

23 . Find the sum of 4} and i f .

24 . What per cent of a number is 11’s of it ?

25 . Two men , 96 miles apart, travel toward each other, oneat the rate of 5} mi . an hour, the other 45. Inhow many hourswillthey meet ?

26 . Change i and 335 to decimals, and add them.

570 ARITHMETIC.

27 .From a schoolOf fifty pupils, six are absent . What per

cent are present ?

28 .What willit cost to fence a piece Of land that is 6 rods

square , at $3 per rod

29 .What willone busheland one peck Of chestnuts cost , at

8 ¢ a quart ?

30. A can do a piece Of work infour days ; B cando it infivedays . Inwhat time can A and B do it if they work together ?

988 . BostonPublic Schools. Examinati onQuestions. Slate.

1 . A merchant bought 48 bales of cotton, and then sold the

lot for losing What was the cost per bale ?

2 . Mrs. Stetson Owned three pieces Of land , containing

acres, acres, and acres. She gave to her son acres.

How many acres had she left ?

3 . If bbl. of flour cost how much will bbl.

cost at the same rate ?

4 . What is the cost Of sawing a pile Of wood 20 ft . long , 4 ft.

wide , and 6 ft . high , at a cord ?

5 . After increasing thewages Of his workmen a manufacturer paid them a day . What did he pay them before ?What should a bookseller charge for a book for wh ich he paid

at the rate Of $54 a dozen,that he may make 20% onthe cost ?

What is the per cent profit orloss when a hundred logs whichcost are sold at 78 if each ?

6 . What is the du ty on100 watches, worth $ 100 apiece , onwhich there is a specific duty of $ 10 each ,

and anad valoremduty Of 50%

7:Paid $ 25 for an insurance policy on my house . If the

rate 18 for how much is my house insured

572 ARITHMETIC.

17 A farmer bought 6 cows through an agent . He sentto pay for the cows and a commission Of How

much did each cow cost ?

18 . What are the avails Of a 90—days note for $600, datedMay 16, 1890, and discounted June 20, 1890, at the RocklandNationalBank , rate Of discount beingOnthe loth day Of November , 1893 , you lent William Rogers

How much does he owe you tod ay , the rate Of interest being 4

19 . I sold a carriage for $ 300, which was six-‘fifths of thecost . What per cent did Imake ? Write out the analysis .

20. A manbought wheat for and sold it at a gain ofWhat did he receive for it ?

2 1 . What number diminished by 5% of itself equals 665?

22 . A lawyer collected 65% of a note Of $950, and chargedcommission . What was his commission ?

23 . Divide three million by six thousand,and multiply the

quotient by .024 .

24 . What willbe a broker’

s commission,at for selling a

farm Of 673 acres 52 per acre ?

25 . How many square feet ina mat 84 ft . long and 7} ft .wideA field contains 199} sq . rd . If it is 18} rd . long

,how wide

is it

26 . How much must Ihave invested 570 that my incomemay be per year ?

27 . A merchant sold 87 cases Of shoes,12 pr. to the case , for

What was that a pair ?

28 . Find the cost Of the following : 36 rolls of paper at 331464 yd . matting at 33 15; 973 yd . carpeting at 7 windowshades at 7 curtains at 10 chairs at and51» days

’ work at per day ?


29 . is 15% more than what ?A cord Of wood , costing sold for What was the

gain per cent?

30. A man paid for a house and for repairs $ 150,

and then sold it for 18% above the entire cost. What did hereceive for it ?

989 . Massachusetts CivilService. Clerks. Limited Examination.

1 . Write infigures the following number :

Fifty-seven million six thousand thirty-one .

Write in words the number expressed by the followingfi res :gu

2 . Add these across, placing the totals inthe Space indicated ;then add the totals :

Totals .

3 . What is the sum of I} , i}, and if , using the lowest commondenominator ? G ive the answer inmixed numbers.

Divide by 4. G ive the answer inmixed numbers .

4 . SubtractH fromH,and change the result to a decimal.

TO eighty-one and three hundredths add seven and fifty-eightthousandths.

5 . Multiply byDivide twenty-five by twenty-five thousandths.

574 ARITHMETIC.

6 . How many yards Of paper will it requ ire to cover thewalls and ceiling Of a room 20 feet long , 15 feet wide, 10 feethigh , the width of the paper being ten inches ?

7 . If i Of a pound Of butter costs i of a dollar , what will5}lb . cost ?

8 . If the rent of a house worth is $ 334 a month ,

what is the value , at the same rate,of a hou se that rents for

a month ?

9 . If or 27 per cent Of the inhabitants Of a town , arevoters, how many inhabitants has the town?

10. If the tax rate is on what is the assessedvalue Of property that pays a tax Of

990. Philadelphia Public Schools. OralWork.

1 . 12)x 9] 12

2 . Of 48 is 4 times what number ?

3 . A horse was sold for $80, which was i Of the cost . Howmuch was lost on the horse ?

4 . What part Of 5 gal. is 3 qt . ?

5 . 24 is 60 per cent Of what number ?

6 . If a bushelOf English walnuts costs what will6

quarts cost ?

7 . A man had $600 Inthe bank . He drew out 163 per centOf it . How many dollars remained 111 the bank ?

8 . A man put 5 gal. 2 qt . Of syrup into bottles holding two

quarts each . How many bottles did it requ ire ?

9 . (a) What is the interest of $ 300, for 30 days , at 6 percent ?

(5) A lost 40 per cent Of his money , and had $ 750 left .How much had he at first ?

576 ARITHMETIC.

Bills Of exchange are either foreignor domestic . A domesticbillOf exchange is called a draft, the term billof exchange beinggenerally applied only to foreign bills.

DOMESTIC EXCHANGE.


William F . Smith , OfMemphis, Tenn .,owes JohnM. Thomson ,

Of NewYork ,He purchases from 8Memphis banker

,

Joseph E. Washington, a sight draft for the above amount on the

ChemicalBank Of NewYork . The following is the form of thedraft :

$ 3 ,475fi 96 . MEMPHIS , TENN. , Aug. 9, 1893 .

At sight , pay to the order of John M. Thomson ThreeThousand Four Hundred Seventy-five and 1

3056 Dollars, value

received,and charge to the account Of

To CHEMICAL BANK . JOSEPH E. WA SHINGTON .

NewYork.

1 . What must William F . Smith pay for the above draft,the

rate being premium per

(A draft for costs

2 . Find the cost Of a Boston draft on NewYork forat 12 ¢ discount per

(A draft for costs

3 . What will a St . Louis merchant have to pay for a draftonNewYork for at 50¢ premium per

4 . If the rate of exchange is 50¢ discount per whatis the face of the sight draft onBoston

,that can be bought in

NewYork for

5 . When the premium is per Mr. Brown paysfor a draft on Lou isville . What is the face Of the

draft ?

REVIEW . 577

6 . At 4% premium ,find the cost of a

'

eight draft for

Add .

0

7 . A t 75¢ discount per how much will cost a sightdraft onMilwaukee for

50 perM. s .473

25 perM. .237

8 . Paid for a sight draft onMilwaukee. What wasthe face Of the draft , the discount being

9 . I sent a commission merchant to buy grain . Howmuch will he Spend for grain

,if his commission at is

included inthe amount sent ?

(Let 3: amount spent for grain. commission.)

10. A farmer ships produce to a commission merchant,which

the latter sells for charging 2 per cent commission . Forthe remainder Of the money he buys groceries and dry-goods

,

charging 2 per cent commission on the amount spent . What isthe cost of the goods purchased ?


1 . A joiner worked onMonday 9 hr . 45 min.,onTuesday

and Wednesday 10 hr . 45 min. each day , on Thursday and Fri

day 10 hr . 15 min. each day ,and onSaturday 6 hr . 45 min.

What was the average length of his day’

s work ?

2 . A watch that loses 35 seconds inanhour was set right

at noon onMonday . What time did it Show at 6 RM. the fol!

lowing Thu rsday ?

3 . There are 5 boys whose heights are 4 ft . 9 in. ,5 ft . 1 in. ,

4 ft . 5 in. ,3 ft . 11 in. , and 4 ft. 4 in.

, respectively . What is

their average height ?

578 ARITHMETIC.

4 . A manhad a plot Of ground 20 yards long and 12 yardswide , which he planted in cabbage . How many plants d id herequ ire

,if the rows

,which ranlengthwise , were 2 feet apart and

2 feet from the fence surrounding the plot, and the plants intherows 16 inches from each other and from the fence ?

G et the correct number Of rows, and the correct number Of plants ina

row. Howmany plantswould have beenneeded if the rows rancrosswise ?

5 . How long would it take a person to count 8 m illion silverdollars

,at the rate Of 100 a minute , and working 8 hours a day ?

6 . The front wheelOf a wagonis 13 ft . 4 in. inc ircumference .

How many revolutions will it make ina journey Of 14 miles ?How many more revolutions willit make than the hind wheel

,

the circumference Of the latter being 17 ft . 6 in. ?

7 . The wheels Of anengine being 16 ft . 8 in. incircumference

,and the number Of revolutions 150 per minute

,how far

does it go inanhou r ? G ive answer inmiles and rods.

995 . CircularMeasure.60 seconds 1 minute60minutes 1 degree

360 degrees 1 circle.

8 . If the equatorial circumference of the earth ismiles, howmany miles apart are two places on the equator

,the

distance between them being

9 . What is the length of a degree on a circle whose diameteris 18 feet ?

The circumference diameterx

10. The 60th parallelof latitude is a circle abou t one-half aslong as the equator . How far due east of Christiania is St .

Petersbu rg, both si tuated on this parallel, the former being 10°

east Of G reenwich , and the latter 30°

east ?

1 1 . How many miles north Of the equator is a place inlatitude 46° 22' 30" Take 694miles to a degree .

580 ARITHMETIC.

4 . What willbe the cost Of a sight draft for atdiscount per

5 . Find the cost of a 60-days draft for 2: dollars, premium

25 per

6 . Find the cost of anx—day draft for discount

7 . Find the cost of a 30-days draft for premium2: dollars per

8 . Paid for a 60-days draft, at {70 premium .

What was the face of the draft ?

9 . A time draft for at $ 1 premium per costAt how many days

’

sight was it drawn?

10. At what rate did Ipurchase a 90-days draft for $900,its

cost being

LONGITUDE AND TIME.

NOTE—This topic should be taught inconnectionwith the study of

MathematicalGeography . The globe should be used to show the pupils

that allplaces onthe same meridianhave the same time, that a d ifference

inlongi tude of 15 degrees produces a difference intime of 1 hour, and thatthe more easterly of two places has the later time.

1 . The difference intime being 1 hour for each 15 degrees,

find the difference inlongitude between two cities differing intime 3§ hours.

2 . Two places differ inlongitude 61 degrees. What is theirdifference intime ?

3 . London is 75° east of Philadelphia . When it is 1 O’clockat Philadelphia, what is the time at London

4 . When it is 2 PM. at London, what is the time at Philadelphia ?

LONGITUDE AND TIME. 581

5 . When it is noon at a city 25 degrees west of Vienna ,what

is the time at the latter place ?

6 . How many degrees of longitude correspond to a time difference Of 3 hours 40minutes ?

7 . What is the difference inlongitude between Philadelphia ,

75° west longitude

,and St . Petersburg

,30

° east longitude ?

8 . When it is 3 RM. at St . Petersburg,what is the time at

Philadelphia

9 . Washingtonis in77° west longitude,and uses standard

time,that is

,the time of 75

° west longitude . What is thedifference between the correct time at Washington and its clocktime ?

10. A town in84° west longitude uses standard time (OfWhat is the correct time whenthe clocks are striking 12 , noon


1 . Find the difference inlongitude between two places differing intime 3 hr . 44 min.

2 . Two places differ inlongitude 37° What is theirdifference intime ?

3 . Chicago is 87° 35' west Of G reenwich . What is the difference intime betweenthe two places ?Is it earlier or later thannoonat Chicago when it i s noon at

G reenwich ? Why ?What is the standard time at Chicago when it is 1 RM. at

G reenwich ?

4 . When a captain’

s observationOf the sunshows that it Is

exac tly noon , the ship ’

s chronometer,keeping G reenwich time ,

reads 30minutes past 2 RM. HOW many degrees west Of G reen

wich is the vessel?

5 . Find the difference intime between two places inlongi

tude 74° 31' and 93° 14' west Of G reenwich , respectively .

582 ARITHMETIC.

6 . When it is noon at a place 11° east Of G reenwich , it is

PM. at another place . Find the longitude of the latter place .

7 . A train ranfrom New York to San Francisco ,miles

,in3 da . 12 hr. 17 min. HOW many miles per hour did it

average

8 . If for $6 I canhave pounds carried 36 miles, howmany pounds canIhave carried 24 miles for the same money ?

9 . At 80 per ounce , what is the valu e of 36 ingots Of

silver,each weighing 2 lb . 10 oz . 15 pwt . ?

10. Find 30 per cent Of 27 yards 8 inches .

11 . The solid contents Of a block 12 feet 6 inches W ide and 3feet 9 inches thick are 27 cubic yards 1 cubic foot 810 cubicinches. Requ ired its length .

12 . A farmer sold 237 bushels 3 pecks Of wheat,whichwas 48

per cent Of his crop . How many bushels,pecks, etc . ,

did he

have left ?

(48% is given; you have to find what What part Of 48% added to

itselfwillgive the requ ired per cent ? DO not find thewhole crop .)

13 . How many spoons, each weighing 2 ounces 12 pennyweights , canbe made from 4 pounds 4 ounces Of silver ?

14 . A mantravels due W est , on the 45th parallel Of latitude ,34 miles per hour for 24 hours. How many degrees has hetraveled, the length of a degree being miles ?

1003 . OralProblems.

1 . A puts $600 into business ; B, $400 ; the profits are $500.

What is the share Of each ?

2 . TWO boys hire a camera for 26 weeks, payingHow much should be paid by the boy that uses it 12 weeks ?

3 . NewOrleans is 90° west Of G reenwich . When it is 2 PM.

at the latter place, What is the time at NewOrleans

584 ARITHMETIC.

8 . Requ ired the amount of $ 550,for 2 yr . 5 mo . 11 da . ,

at

6% per annum .

9 . A bin6 ft . long and 4 feet wide has a capacity Of 75

bushels . How deep is it ? (Cancel.)

10. Subtract sixteenten-millionths from 27 millionths, and

multiply the remainder by

1 1 . A piece Of land is 120 feet wide and 150feet long . Whatpart Of anacre does it contain ?

12 . Find the difference intime corresponding to a differenceinlongitude Of 7° 42' 30"

13 . Find the cost Of feet Of lumber , at perM.

14 . A note is made and dated Sept . 11 ,1888 , and is made

payable in90 days. Find the date Of maturi ty .

15 . Add 137 days to Aug. 14 ,1892 , and give the resulting

date .

16 . Divide 320 acres Of land among A,B

,and C,

so that Ashallhave 15 acres more thanB

,and C shallhave 27 acres more

than B.

17 . From 10 bu . subtract the sum Of bu . , 1125 bu . ,

and 1

bu . 3 pk . qt .

18. At 90¢ a yard , what will be the cost Of a carpet for a

flight Of stairs of 18 steps,each 74} inches high and 10 inches

wide

19 . By What number must 303 be multiplied to produce 6044?

20. Reduce to common fractions .0125, .067§ .006325.

2 1 . Reduce to decimals and find their sum {5 331 , H,

22 . Why does the value Of a fractionremainunchanged whenboth terms are multiplied by the same number ?

23 . Required the height Of a tower that casts a shadow 21 ft . ,

whenunder the same conditions Of time and place a staff 10 ft .

high casts a shadow 23 in.

EXCHANGE. 585

24 . At anelection A and B were the candidates for sherifi’

,

and B received majority . If the whole number Of votescastwas how many votes did each receive ?

25 . A note for $ 125, made J an. 4,1887 , is paidMay 3 , 1888 ,

with interest at 5 per cent per annum . Find the amount paid .

BILLS OF EXCHANGE

Exchange for 180 17s. 6d . NEW YORK ,Dec . 14

,1895.

Sixty days after sight of this First of Exchange (Secondunpaid), pay to the order of John W .Moran Bro . , One Hun

dred Eighty pounds sterling , seventeen shillings, six pence ,

Value received , and charge the same to account Of

To JAMEs LENNONLondon. PETER COMERFORD SON.

NO. 39.


1 . Find the cost of the above billat per£3.

200

20

180 a

103 . £453 .

2s. 6d.

2 . What would be the cost Of a cable transfer of £251 118 .

9d ,at per £ 7

13250 1 of1

Is.

6d .

33 .

Thenewspapers give quotations of foreignexchange forsight and 60-daybills, also for cable transfers.

586 ARITHMETIC.

The NewYork quotations for French exchange give the numberof francs for 1.

Paris cable transfersParis bankers’ 60 daysParis bankers’ sight

1007 . The quotations for German exchange give the value inU. 8 .

money of 4 Reichmarks (ormarks).

Reichmarks (4)60 days 95}Reichmarks (4)eight 95}

3 . Find the cost of a sight bill on Paris for francs, atfrancs for $ 1.

4 . Find the cost of a 60-days bill of exchange on Berlin formarks, at for 4 marks .

5 . What willbe the face inmarks of a sight billOf exchangeon Berlin that canbe bought for at for 4 marks ?

6 . A NewYork merchant pays for a 60-days billon Paris . What is the face Of the bill, the rate Of exchangebeing francs for $ 1 ?

7 . At per 36, what will be the face of the sight billonLondon that canbe bought for

18750 £ 307 741 , etc .

18750 6121875061 450

61£23 remainder20

46041 , newdiv idend.

8 . Bought goods inLondon amounting to £437 58 . 10d . lessHow much willI have to pay inBoston for a sight billof

exchange at to settle the account ?

9 . What willbe the cost inChicago fora 60-day billonParis ,“

that willpay for the following articles ? Rate , 1 franc 19}18 pieces silk , 44 meters each ,

at 25 francs per meter, less 74

3 pieces Of cloth , 50meters each , at 20francs per meter , less 5Packing charges

,francs.

CHAPTER XIII.

PARTIALPAYMENTS. RATIOANDPROPORTION. SQUARE

ROOT. SURFACES ANDVOLUMES.PARTIAL PAYMENTS.

1008 . U. 8. Rule.

DULUTH,MINN. , J an. 5, 1889.

Ondemand , I promise to pay to the order Of OwenMcGeeThree Hundred Dollars, value received , with interest at 7 percent .

$30019095 J . RANDOLPH PAGE.

Endorsements :May 20, 1889, $ 100; Oct . 30, 1889, $ 100;March 6, 1890, $50.

How mu ch was due Jan . 5, 1891

Find amount of $300 J an. 5,1889

,to first paymentMay 20, 1889, 4 mo.

15 da. (by compound subtraction).Deduct first payment,BalanceMay 20, 1889,Interest on to Oct. 30,

5 mo . 10 da. ,

Amount,Less second payment,Balance Oct . 30

,1889,

Interest on Oct. 30 toMarch 6, 4 mo. 6 da .,

Amount,

Less third payment,BalanceMarch 6

,1890,

Interest on March 6 to J an. 5, 9 mo . 29 da.,

Due J an. 5,1891,

PARTIAL PAYMENTS. 589

NOTE.- Find time by compound subtraction.

1 . How much is due June 3, 1896, on a demand note for

wi th interest at dated June 3, 1893 ,

bearing eu

dorsements of payment Of $ 500, Sept. 18 , 1894 ; $ 600, J an. 3 ,

1895?

2 . A demand note for $600, bearing interest at wasgiven Feb. 18 , 1892 . A payment Of $250 was madeMay 28 ,1893 ; one Of $ 150 was made Oct . 8

, 1893 .How much is due

J an. 23,1895?

3 . Note for interest , dated April 15,1891.

Endorsements : $50, Sept . 20, 1891 ; $ 100,May 26, 1892 ;June 20, 1893 . How much is due Dec . 27 , 1894 ?

1010. Face ofnote,

Interest from April15 to Sept 20, 1891, 5 mo. 5

Amount due Sept. 20,1891

,

If the $50 payment were deducted , and interest computed onthe balance, the makerwould be charged interest on

more thanthe face of the note,and this the lawdoesnot

allow. Interest is takenon untilnext payment ,May26

, 1892, 8 mo . 6 da.,

Amount dueMay 26, 1892,As the two payments are not large enough to meet the interest

nowdue, the interest is againcalculated onthe originalfromMay 26, 1892, to June 20, 1893 , 1 yr. 24 da .

,

Amount,Less $50 100 (three payments),Balance due June 20,

1893,

Interest on to Dec . 27, 1894 ,1 yr. 6 mo . 7 da.

,

Due Dec . 27 , 1894 ,

1011 . By the United States rule for"

partial payments, theamount of the principalis found to the time when the payment

,or

the sum Of two or more payments, equals or exceeds the interest .From this amount deduct the payment or sum Of payments.

Use the balance then due as a newprincipal,and proceed as

before.

590 ARITHMETIC.

4. ALBANY,N.Y.

,March 5, 1893.

One year after date , I promise to pay JohnHarrigan, ororder , Nine Hundred Dollars, value received , with interest at sixper cent .

ANDREW T. SULLIVAN .

Endorsed as follows : June 5, 1893 , $ 10; Sept. 5, 1893 , $50;J an. 5, 1894 , $ 120.

What was dueMarch 8 , 1894 ?

5 . Find the amount due April16, 1893 , on a note fordrawnJune 1, 1891.

Endorsements : $ 150 Sept . 16,1891 ;

$50 Sept. 16,1892 .

Interest

1012 . Inthe United States courts , and inthose of some Of the states,

interest for a portionof a year is takenby days, uponthe basis Of 365 daysto the year.

To make thework easier for the pupils, however, the year Of 360 days

should be used inthe examples given, and the time betweendates should

be found by compound subtraction.

1013 . U. 8. Civil Service. State Department Examination.

1 . Express one hundred billion infigures,both according to

the American and to the English method . (Art .

2 . Express infigures, both in the form Of decimal fractionsand of common fractions, the following amounts : Four m illionths , twenty-one thousandths, one hundredth , and one tenth .

3 . Divide by .16.

4 . Add 3 Of g to and from the sum subtract

5 . A note datedMay 20,1886, for is payableMarch

with interest at 6 per cent . October 12,1886

, $625was paid on the note . How much will be due

,principal and

interest, at i ts maturity ?

592 ARITHMETIC.

5 . Two pictures were sold for $99 each ; on one there wasa gain Of onthe other a loss Of Was there a gain ora loss on the sale Of both , and how much ?

6 . Write the analysis Of the fifth .

7 . NEW YORK ,J an. 1

,1874.

One year after date I promise to pay Noyes Bros. CutlerEight Hundred Dollars for value received , with interest .

$80010095 . JAMESM. SCOTT.

Endorsed as follows : Apr. 1,1874 , $ 10 ; July 1, 1874 , $ 35 ;

NOV . 1,1874 , $ 100.

What was due J an. 1, 1875 ?

8 . What is the difference on a billOf $ 780, between a discount Of and a discount Of 35 and 5% Off ?

9 . A house that .cost rents for What percent does it pay on the investment ? Analyze .

10. HOW many cords ina pile Of wood 42 feet long,12 ft.

high and 8 ft . wide ? Find its cost at per cord .

PRESENT WORTH AND TRUE DISCOUNT.

1016 . Problems are frequently met with inbooks, inwh ich the pres

ent worth is asked of a sum Of money payable at a fu ture date .

1017 . 1 . What is the present worth Of $ 150 payable in1year 6 months , interest 6%By th is is meantwhat sum at 6% interest willamount to $ 150 in1 year

6 months ? Or,

G iventhe amount rate time 1 yr. 6 mo . to find principal.

1018 . By“true discount is meant the d ifference betweenthe sum

payable at a fu ture time and its present worth .

”


2 . What is the “ tru e discount Of $ 150, payable in1 year6 months, interest 6%The amount $ 150, rate time 1 yr. 6 mo . are given. Find the

interest.Let .C principal

150

Interest amount a:

1019 . WorcesterPublic Schools. ExaminationQuestions.

1 . What principal, on interest for 2 yr . 6 mo . at willgain $850?

2 . What is the cost of insuring a house,worth for

Of its value at including for the policy ?

3 . If Boston is 71° 3 ' 30"W . L. and Washington is 77° 2' 48"

W . L. , what is the difference intime between the two cities?

4 . At 9¢ a cubic foot , what willbe the cost of a block of

stone 9 ft . long , 4 ft . wide, and 5 ft . 6 in. thick ?

5 . If a steeple 150 ft . high casts a shadow of 275 ft . ,how

long a shadow willbe cast by a man 6 ft . tall, at the same timeof day ?

6 . If 4 men dig a ditch 24 rods long In20 days,how long a

ditch can5 men dig in8 days ?

7 . What Is the true discount Of $ 756, due in1 yr . 4 mo .,at

6% 7

(See Art.

8 . The tax to be raised ina certain town is The

taxable property is valued at What is the tax on onedollar ?

9 . For what sum must a 60-days note be written to yi eldat a bank , discounting at

10. If anagent receives to invest insilk , after deducting his commission at how many yards can he purchase at

per yard ?

594 ARITHMETIC.

11 . What will be the proceeds of a 60-days note for $500.

dated June 4 , 1879, and discounted at a bank July 1 , 1879, at6%

12 . At what rate will$ 142 gain interest in3 years ?

13 . Mr. FOX buys one-fifth Of anacre Of land forFor how much a square foot must he sellit to gain

14 . What is the duty,at 50f a pound and 30% ad valorem ,

on 700 yards of French broadcloth,invoiced at per yard

,

1 yd . weighing 1} pounds ?

15 . What willbe the amount, at compound interest , Of $340,

at for 1 yr . 3 mo . ,the interest compounded semi-annually ?

1 .

43

2 . 3 .

48

bu . 3 pk . 7 qt . 6 . 7 rd . 1 yd .

4 8

l 3 3 l2 4 2

1 6 3

596 ARITHMETIC.

7 . If I sell10 shares of railroad stock for and gain9% on the cost , Whatwas the cost ?

8 . What is the interest Of $660, for 3 mo . , at

9 . What is the premium for insuring on my houseat

10. How many quarts Of peanu ts inOne bu . and three pk . ?

11 . What principalat 6% simple interest willgain$ 36 in 1year and 6 months ?

12 . What per cent is gained on goods sold at double the cost ?

13 . What is 8% Of fifty bushels ?

14 . How many cubic inches ina ten-inch cube ?

15 . What is the interest Of for 60 days, at

16 . HOW many years will it take $ 20 to gain $20 at 5 percent simple interest ?

17 . Bought 2 chairs at 1 wash-tub for 1 tablefor and 5 doz . glasses at 48 ¢ a dozen . G ave a ten-dollarbillinpayment . How much change did I receive ?

18 . is Of my property . How mu ch am Iworth ?

19 . What is the interest on $ 700, for 15 days, at

20. Bank discount on a note for 65 days,

discountedat date ?

2 1 . At what rate will$2 gain$20 in5 years ?

22 . Cost Of 700lb . of coalat $ 7 a ton ?

23 . My desk is 1} ft . long and 1 ft . W ide . How many inchesaround it ?

24 . If a man spends 50¢ a day during April,May ,and June ,

what does he spend inthe three months ?

25 . A grocer bought 15 barrels of flour at $5 a barrel. At

what price must he sellthem to gain $ 30?


26 . HOW many weeks will4} tons of coallastMrs. Bright,if

she uses 150 Of a ton each week ?

27 . I canbuy 2 pairs Of shoes for 12 shillings. How manypairs at the same rate canIbuy for£ 3 ?

28 . Seven-eighths Of James’

s vacation willbe equal to sevenn inths Of yours ; yours willbe 63 days. HOW many willhis he ?

29 . If two-thirds Of your age is eight years and four months,how Old are you

30. 5 quarts equalwhat decimalof a peck ?


1 . What is the cost Of carpeting a room 16} ft. long, 12 ft.wide

,with Oil-cloth 1}

'

yd . wide , at a yard ?

2 . What is the present worth Of due in4 yr. , at

3 . What must be the face Of a note which,discounted at a

bank,for 27 days and grace , would yield $95?

4 . If Ilose 10% by selling goods at 18¢ a yard , for Whatmust they be sold to gain 20%

5 . I sold of my estate , Or worth . Iam worth,

inaddition to my real estate , How much am Iworthinall?

6 . The edges Of a large cubical box are 5 ft . long . HOW

many sq . ft . Of paper willcover the ou tside of the box

7 . Mr. Jones insured his house , worth for one yearforg of its value , at What would the insurance companylose if the house should burn

8 . If Ilose 9% by selling land at an acre,what shall

I gain by selling i t at anacre ?

9 . A house lot containing sq . ft . is 179 ft . long . HOW

wide is it ?

98 ARITHMETIC.

10. What is the value of a pile of wood 25 ft . x 8 ft . X 8 ft . ,

at per cord ?

1 1 . I bought a horse for $ 250, paid for shoeing , thensold him for What per cent did Ilose ?

12 . What is the value of a pile of wood 40 ft . long , 4 ft . wide,

and 5 ft . high , at a cord ?

13 . How many square yards inthe walls of a room 42 ft . long,

15 ft . wide , and 9 ft . high ?

14 . By buying a cargo of coalat $6 per ton , and selling it at

$ 8 a ton, I gained $ 198 . How much d id Ipay for it ?

15 . Make out a receipted bill for the following : 325 yd . of

silk at per yd . ; 296 yd . of lace at per yd . ; 480

yd . of ribbon at per yd . 45 doz . gloves at 15 per doz .

16 . My dividend is quotient What is the divisor ?

17 . I gave away and of 4} bushels of chestnuts . Whatwas left ?

18 . The distance around a square field is 16 rods . What isthe field worth , at a sq . ft . ?

19 . Sent a broker to invest incotton worth a

pound,first deducting h is commission of How many

pounds did he buy , and what was his commission ?

20. Find the face of a three-months note,discounted at

to yield $856.

21 . Isold 80 yards of broadcloth for$240, thereby losing 2070onthe cost . For what should I have sold it per yard to havegained 15% on the cost ?

22 . A man bought 80 casks,of 85 gallons each

,for

80 gallons leaked out . For what must he sellthe remainder pergallon to gain on the cost ?

23 . Each of two men sold his horse for $ 180. One madethe other lost 20% onthe cost . Cost Of each horse ?

600 ARITHMETIC.

2 . Express the following inwords : (See Art.

(a) XOIXLXXIV . (b)(c) (d )m,

x 49, g.

3 . The quotient arising from the division of by a

certain number is 51, and the remainder is What is thedivisor ?

4 . What is the value of the following ?

zi + li — 3i i x i x i

5 . Ingoing 1 mile 94 rods- 2 yards 1 foot, a carriage wheelmakes 526 revolutions. What is the circumference of thewheel?

6 . What is the weight (intons, cwt . , qr.,etc .) of the water

which fills a cistern 9 feet 8 inches long , 9 feet 4 inches wide ,and 6 feet 9 inches deep , a cubic foot of water weighingounces ? (1 ton pounds.)

7 . Ona note dated Oct . 16,1886

,for with interest

at 6 per cent , the following payments were made : Jan . 28 , 1887 ,

$575 ;May 22 ,1887

, $25 ; and Aug. 4,1887 , $948 . What was

due NOV . 25, 1887 ?

8 . Anowner of 6 per cent bonds sells them at the market

quotationof 118,and invests the proceeds in4} per cent bonds.

The latter investment yields him the same income as the former .What did he pay per hundred for the 4} per cent bonds ?

9 . A grocer pays 18 ¢ per lb . for coffee and roasts it,it losing

10 per cent of its weight inthe process . What must he chargeper lb . for the roasted coffee inorder to make a profit Of 20 percent

,allowing 4 per cent for bad debts ?

10. Money being worth 8 per cent per annum,what is the

true discount on$450, which is due in5 months and 21 days,without interest ? (See Art .

MENSURATION. 601

SURFACES AND VOLUMES.

1 . If a piece Of cloth is 20 yards long and 4} yd . broad,how

broad is another piece of cloth 12 yards long that contains as

many square yards as the former ?

2 . An iron beam 16 ft . long , 2} ft . broad,and 8 in. thick ,

weighs lb . What is the length of a similar beam whosebreadth is 3} ft . , thickness 7} in. , and weight lb . ?

3 . What willit cost to carpet a room 22} ft . long by 15} ft .

wide with carpet 2} ft . wide , costing per yd . ?

4 . What is the length of a box 6} ft . wide and 7} ft . high ,that will exactly contain 12 boxes 4} ft . long

, 3} ft . wide,and

2} ft . deep ?

5 . What is the value , at $ 120 per acre , of a square fieldwhose side is chains ?

10 sq . chains 1 acre.

6 . What is the area insquare feet Of a triangle whose baseis 18 ft . 4 in. ,

and whose altitude is 11 ft . 10 in. ?

7 . What is the area Of a circle whose diameter is feet,

the area Of the circle being .7854 times the area of the squarethat willjust enclose it ?

8 . Find the capacity , inb ushels, Of a bin22 ft . long, 14 ft.

wide,12 ft . high ?

9 . How many gallons willa tank hold , its dimensions being4 ft . 1 in. by 3 ft . 8 in. by 2 ft . 3 in. ?

10. HOW many square yards are there inthe walls and theceiling of a room 21 ft . long, 18 ft . wide

,12 ft. high ? Make a

diagram .

11 . A tank 5} ft . by 6 ft . by 7 ft . can be emptied by two

pipes, one of which discharges 9 gallons per minute and the

602 ARITHMETIC.

other 7 gallons per mi nute . How long will it take each toempty the tank ? How long willi t take both together ?

12 . A parlor is 18 feet long, 15 feet wide . Make a diagramshowing how carpet 27 inches wide can be laid without cuttingthe carpet lengthwise . Which would be the better way to laycarpet 30 inches wide inthe above room ?

13 . Calculate the number of running yards of carpet 30 in.

wide needed for the floor of the above room,including 4} yards

wasted inmatching the pattern .

Find the cost of carpeting the room at 95 cents per runningyard for carpet , 5 cents per square yard for lining , and 10 centsper running yard for sewing and laying .

14 . A rug 18 feet long, 15 feet wide , is placed inthe centreof the floor of a room 21 feet long, 18 feet wide . What is thewidth of the strip left uncovered ? Find the area of the uncovered space ?

15 . A room is 18 feet wide , 24 feet long, 9 feet high . Thereare two doors 4 feet wide,7} feet high ; two windows 4 feet wide ,6 feet high ; and a fire-place 5 feet square . Howmany squarefeet of plastering willthere be on the walls and ceiling

,deduct

ing for a baseboard 12 inches wide ? How many running feet ofbaseboard willbe neededDraw “ development of the above room

,showing the fou r

walls and the ceiling , and locating the doors, the windows, andthe baseboard .

DOnot use baseboard where it is not requ ired .

16 . At the rate Of for a pile Of lumber 25 ft . long,20

ft . wide , 10 ft. high , what is the value of a pile 50 ft . long , 40ft . wide, 20 ft. high ?

17 . If it costs 14 to paint the walls and the ceiling of a room25 ft . long, 20 ft . wide , and 10 ft . h igh

,what willit cost to paint

the walls and the ceiling of a room 50 ft . long, 40 ft . wide

, and20 ft . high ?

604 ARITHMETIC.

2k 415:

4 . Fi nd the value of12 x 3%

4} of 69.

5 . The fence arou nd the four sides of a square field measuresmiles. What is the length infur.

, rd ., yd .

, and ft . of thefence on one side of the field ?

40 rods 1 furlong (fur.) 8 fur. 1 mi .

6 . A merchant buys 42 gallons of alcoholat per gal.,

and keeps it for 3 years. He then finds that he has lost 7 gallons by leakage and evaporation . Estimating the value of moneyat 6 per cent per annum ,

how much per gallon must he charge ,inorder that he may realize a profit of 16§ per cent on his out

lay (including interest)?

7 . At per cu . yd .,what will it cost to remove an

embankment 240 ft . long , 38 ft . wide,and 9 ft. high ?

8 . What is the weight (intons, cwt ., etc .) of the water that

fills a tank 4 ft . 10 in. long, 4 ft . 8 in. wide,and 5 ft . 4} in.

deep,a cubic foot of water weighing ounces?

1 ton lb .

9 . What sum placed at interest at 8 per cent per annumwillamount to in2 years 11 months and 27 days ?

10. What is the difference between the interest and the truediscount of $ 380, due in2 years and 5 months, at (SeeArt .

1027 . Kansas City Public Schools. ExaminationQuestions. Oral

Problems.

1 . By selling anarticle for a man gained Howmany dollars would he have gained if he had sold the article atanadvance of 50% over cost ?

2 . A mansold two cows for $30 each . Onone he gained25% on the other he lost Did he gain or lose

,and how

much 7


3 . $12 is 4 per cent of A’

s and B’

s money . If A has twiceas much as B, how much has each ?

4 . What is the interest Of $60, for 5§ years, at 6%5 . What is the amount Of $300 for 8 years 10 months, at

6%6 . What principal, in3 years and 4 months, at willgive

$40 interest ?

7 . What is the present worth of $52 , due 5 years hence , atsix per cent ? (Art .

8 . Inwhat time will$50, at give $ 18 interest

9 . At what per cent will 12, in3 yr. 4 mo .

,amount to $ 14 ?

10. If I buy an article for $ 75 and sell it for $50, what ismy loss per cent ?

1028 . Massachusetts CivilService. Glerks. GeneralExamination.

1 . Write infigures the following number

Five million thirteen thousand six .

Write inwords the number expressed by the following figures

2 . Add these across, placing the totals inthe space indicated ;then add the totals.

Totals.

3 . What number must be added to the sum Of i » , i , and {4to make

606 ARITHMETIC.

4 . A traveler walked 23} m iles the first day, 3§ miles morethe second day than the first, and 3 4 miles more the third daythan the second . How far did he walk inthe three days

5 . Multiply by divide the product by andsubtract the quotient from 11 .

6 . If be divided between two persons so that oneshallreceive one-third more than the other, how much will eachreceive ?

7 . How many bricks, 8 inches long and 4 inches wide , willbe needed to make a sidewalk 26 feet long and 4 feet wide ?

8 . If it costs to carry lb . 356 miles, what willit cost to carry lb . 890miles ?

9 . If a post 11 feet inheight casts a shadow 9 feet 2 incheslong

,what is the height of a house which casts a shadow 45 feet

10 inches long ?

10. A house rents for $ 30 a month, and the Owner pays $ 75

a year for taxes and repairs . What is the value of the house,if

h is net profit is 5 per cent per annum ?

11 . A loaned B a sum of money at 4 ; per cent interest perannum . At the end of 18 months, B paid the debt, principaland interest , inall What was the sum borrowed ?

12 . Find the interest on for 5 months and 21 days,

at the rate Of 7 per cent a year .

13 . Find the interest on $640, from Sept . 3 , 1888 , to Oct . 30,

1889,at 6 per cent per annum .

14 . At compound interest,what will$200 amount to , in1

year and 3 months,at 6 per cent

,interest compounded semi

annually ?

15 . If a 5-months note for $ 760, datedMarch 13 , is discounted at a bankMay 23 , the rate being 7 per cent a year ,what willbe the proceeds?

608 ARITHMETIC.

The signof square root is

Find the square root of 169.

102 100. 20’ 400. The square root is between10 and 20;therefore, 10 a secondnumber.

169 10’ x second) second

169 100 20X second second

20x second second 69.

From this it appears that the second number is 3 , since20x 3 3’ 69.

It may be shown inthis way10 (first number)169

10’ 100

Trialdiv isor— twice 10 20) second number)60

9

3’ 9

Am . 10 3 13 .

Find the square root of40 (first number)

40'

40x 2 80, trialdiv isor second number)480

36 6’

Am . 46.

1040. Instead of multiplying the trialdivisorby the second number,and thenascertaining whether the remainder is the square of the second

number, the secondnumber is added to the trialdiv isor and this sum is multiplied by the second number.

40 (first number)

(2 x 40) 6 86) second number)516

Am . 46.

REVIEW .

1041 . Inpractice ,thework is shortened by omitting the ciphers.

First, po int 03 inperiods of two figures , commenc

ing at the right . Find the greatest square inthe first4 6 Ans '

period , and place the robt inthe quotient . Subtractthe square from the first period . Bring downthenext 16

period . Multiply the first quotient figure by 2, and 86) 516use it as a trial divisor. Place the second figure in 516the quotient. Afiix it also to the trialdivisor. Mult iply the two figures inthe trialdivisor by the second quotient figure.

Divide (Arts . 385, 616)

?‘

b

wlfi

fl

Write answers (Art .

1 . 4 .

—z

610 ARITHMETIC.

1044 . AnInvoice (English).

Invoice of 3 bales.

Linen G oods forwarded by railtofor shipment thence per Anchoria to NewYork ,and for account and risk ofMessrs . Robinson Co .

®CO. # 2 30 pcs . Bord . Crash

30 u u t

# 3 60 Checked G c

# 4 60

Less d isct .

1 . Find the duty inU. S. money at 50% ad valorem .

46

2 . What is the cost inEnglish money of crockery amountingto £ 166 138 . 4d . less a discount of 5 and 5%

1045. Ratio is the relationwhich one number has to anotherof the same kind .

1046. The first term of the ratio is called the antecedent ; thesecond

,the consequent.

The ratio of 3 to 6, $9 to $ 18 , 15 cows to 30 cows may beexpressed i f , 125 ,H. They are each equalto 5.

1047 . OralExercises.

Express the ratio inlowest terms

1 . 175 to 700 4 . 3 quarts to 4 gallons”8 4 NOTE.

-The denominations must2 . $ 19 to $95 be the same.

3 . to 3 quarts to 16 quarts 18,

612 ARITHMETIC.

5 . E earns in6 days as much as D earns in8 days. Findthe ratio of E’

s daily earnings to D’

s.

6 . One wheel makes 300 revolutions in 2 minutes, thesecond requires only 14 minutes to make the same number .

Find the ratio of the number of revolutions made by the firstwheelin1 minute to the number made by the second wheelinthe same time .

7 . A circle whose diameter is 1 ft . has a circumference offt . What is the ratio of the diameter to the circumference ?

8 . One train goes 40 m iles anhour , a second goes 45 milesanhour . What is the ratio of the speed of the first to that ofthe second ?

9 . A window is 6 ft . 4 in. high by 4 ft . 2 in. wide . What isthe ratio of the height to the width ?

10. A father is 36 years old ,his son is 9. What was the

ratio 6 years ago of the father ’s age to that Of the son?


(Be sure your answer is correct.)

1 . One line is 3 rods 4 yards long, the length of another is5 rods 1 ft . Find the ratio of the first to the second .

2 . One candle lasts 4 hours 20 minutes, another lasts 3 hours15 minutes. Find the ratio of the first to the second .

3 . A pound of coffee costs 254k 1 lb . of sugar costs 51354 .

What is the ratio of price of sugar to that of coffee

4 . Mwalks in1 hour 47 min. as far as N walks in2 hours3 minutes. What is the ratio ofM’

s speed to N’

s ?

5 . P earns in days as much as Q, in182 days. What isthe ratio of Q

’

s daily earnings to P ’

s ? Of P’

s to Q’

s ?

6 . One wheelmakes 600 revolutions in84 seconds ; a secondmakes 300 revolutions in31} seconds. What is the

'

ratio Of thespeed of the first wheelto that of the second ?


7 . The circumference of a circle is ft .,and its radius

is 2 ft . What is the ratio Of the diameter to the circumference ?

8 . One train goes 40 miles in50 minutes, another goes 24miles in a half hour . What is the ratio of the speed of thesecond to that Of the first ?

9 . One window is 6 ft . 8 in. x 4 ft . 2 in. ; a second is 4 ft . 8

in. x 2 ft . 1 in. What is the ratio of the area of the second tothat Of the first ?

10. A mother is now 35 years old,and her sonis 3 years

and 6 months old . Fourteenmonths ago, what was the ratio ofthe mother

’

s age to that of her son?

1051 . Newark, N. J Public Schools. ExaminationQuestions.

1 . Sold 60 bushels of oats at 40 bbl. of flour at56 bushels of corn at Make out the bill

,add 1 per cent

for cartage , and deduct 3 per cent for cash , and receipt thebill.

2 . A purchased a farm 156} rods long and rods wideat per acre . Paid one-half cash , gave a mortgage for thebalance . At the close of one year , he paid $500 on the mortgage . How much remains due, money being worth 5 per cent ?

3 . Analyze : A farmer sold 2 barrels apples at $ 341L perbarrel, spent i of the money for tea at 2 of a dollar a pound , and

the balance for coffee at 4 of a dollar a pound . How manypounds of each did he buy ?

4 . What will it cost to dig and wall the cellar of a house41} feet long , 33 feet wide , and 8 feet deep , the wall to be 14feet thick ? The digging willcost per cubic yard , and the

laying of the wall$15 per 100 cubic feet .

5 . A merchant got a shipment of 500 bbl. of flou r insured

for 80 per cent of its cost at 3} per cent , paying premium.

What did the flour cost him per bbl. ?

14 ARITHMETIC.

6 . Find the maturity , term of discount , and proceeds Of the

following note :

NEWARK ,N. J June 12, 1889.

Six months after date , Ipromise to pay James Jones, or order ,Twelve Hundred and Fifty Dollars, with interest at 5 per cent .Discounted at the bank Nov . 15.

7 . A demand note for dated April17 , 1884 ,has the

following endorsements : June 5, $253 ; Aug. 20

, Nov .

17, $ 420. What is due J an. 1

,1885

,interest at 6 per cent ?

8 . A and B, contractors, received for grading aroadway . A furnished 5 men 20 days, and 6 others for 15 days ;B furnished 10men for 12 days, and 9 others for 20 days. Whatwas each contractor ’s share ?

9 . How many menwillbe requ ired,working 12 hours a day

for 250 days, to dig a ditch 750 feet long , 4 feet wide , and 3 feetdeep , if it requ ires 27 men, working 13 hours a day for 62 days,to dig a ditch 403 feet long , 3 feet wide , and 3 feet deep ?

10. I sent to my agent inSt . Lou is to invest inflour . After deducting his commission at 24 per cent , how manybarrels of flour can he buy at $54 per barrel?

INTEREST AND DISCOUNT.


(Solve the first tenby aliquot parts.)

Find the amount :

1 . 3 yr . 5 mo . 15 da .,

2 . 1 yr . 10mo . 25 da .,

3 . 2 yr . 1 mo . 16 da .,

4 . 7 yr . 7 mo . 7 da . ,

7 mo .

11, of 7 yr. 7 da . what part of 7 mo . ?

616 ARITHMETIC.

REVIEW.

1055. Find products

1 . 648 x 42 . 976 xH3 . x 87}4 . x

5 . x 875

6 . 874 x 99

7 . 848 x 125

8 . 375 x 999

9 . 792 x 25

10. 457 x 16

LONGITUDE AND

1056. NOTE. Making diagrams, as shownbelow,may assist the pupil

to solve the problems .

1 . G iventhe longitude of A as 95° east , and that of B as 74

°

east , and the time at A as 1 30 P .M. , to find the time at B .

Since the latitude of B has no bearing uponits time, both places may befocated uponthe same line running east and west.

Time difference hours.

Time = 7 Time l z 3OP .M.

74° 95°

Longitude difi'

erence

Locate the prime meridian(that of thenthe meridians of 74° and 95°

east. Mark above the last two the names of the places, B and A . Write

above A its giventime, 1 30 P .M.

To find the time at B, we must find the difference of time betweenB andA . The difference inlongitude is 95

° 74° The difference intimeis (21 15)hours.

NOTE. Remember that the more easterly of the two places has the latertime.

x 250

983 x 51

x 62}176 x 234

x 375

895 x 444864 x 486

975 x 318

374x 6}484x 4§

LONGITUDE AND TIME. 617

2 . A is situated in71° west longitude , B in107° west longi

tude . What time is it at B,whenit is noonat A ?

Time difi'

.

Time ? 12M.

107° 71°

Long. difi’

.

3 . Find the longitude of B, whose time is 8 : A .H. ,

whenit is 7 15 A .M. at A , whose longitude is 156°

48 ' west.Time difference

7 15 Am. 8 10 30 AM.

156° 48’ Longitude 7

Longitude difference

Since B has the later time, its locationis east of A . The difference intime, beingnearly anhour, shows the difi

'

erence inlongitude to be nearlyFind the exact difi

'

erence. Is it to be added to 156° 48' or subtractedfrom it , to give the longitude of B?

4 . Whenit is A .M. at A , in57°24' west longitude, it

is 10 A .M. at B . Find the longitude Of B .

Time difference 7} hours.

2 40 AM. 10 AM.

57° 24'

Longi tude difi'

erence 15° x 7}Ifwe go 110

°eastward from A , we shallreach the prime meridianaflser

going howmany degrees and minu tes ? How many more degrees andminutes mustwe travelto reach B? Is B ineast or inwest longitude ?

5 . When it is noonat B, what is the time at A , the formerbeing inlongitude 44° east , and the latter inlongitude 57

° west ?Time difference

Time 12M.

Longitude difierence Why ?

618 ARITHMETIC.

1057 . Find the longitude or the time

Longitude of .4 . Time at A . Long itude of B.

6 . 63° east 9 AM. 54

° east

7 . 57°25' east 83

°20' east

8 . 156°48' west 3 15 P .M. 7

9 . 11 42 A .M. 56°25' west

10. 2°15' west 6 53 A .M. 67

°

48' east

11 . 27°10' east 27

°10' west

12 . 18°4 ' east

13 . 74°56' west

14 . 4 ' 30" east

15 . PM.

SQUARE ROOT.

1053 . Find the square root

5

2 45. 6 . guys

3 444 7 ..ggyg

4 . {gt 8 . 4594 12 .

NOTE—Before extracting the square root of the following, reducemixed numbers to improper fractions.

13 . 12} 16 . 4444

14 . 114 17 . 6444

15 . 244 18 . 10444

620 ARITHMETIC.

DRILLS.

1060. Find sums

1061 . Find answers :

150 23 48 162 (26 61) 174 41 35 165 (28 + 47)172 19 66 154 (36 44) 163 38 43 171 (32 33)

183 37 42 184 (39 35) 155 49 24 180 (18 28)

161 12 71 173 (57 17) 181 47 33 153 31)

1062 . Find products

36 x 31 54 x 51

36 x 29 54 x 49

45 x 41 63 x 61

45 x 39 63 x 59

1063 . Find quotients

576 18 448 14

693 -z 21 533 13

608 19 924 22

848 16 943 23

1064 . Find answers

315 x lg 32 x 39;

378 x IH 45 x 194497} 3 1054 12

854 4 . 11 1094 4

REVIEW . 621

1 . Paid 92¢ for coffee , for butter,and for lard .

How much was my bill?

2 . Ihad $ 150. Spent $ 23 for a suit of clothes and $48 fortools. Howmuch was left ?

3 . What is the area of a field 36 yd . by 31 yd .

4 . 600 hours how many days ?

5 . What is the cost of a cow if Ipay $630 for 15 ?

6 . How many ounces in29} lb. ?

7 . 1094 lb . sugar are divided among 4 people. What is theshare of each ?

8 . At llflaf per lh .

,how many lb . iron canI get for

9 . What willbe the cost of 51 tons iron at $ 17p er ton ?

10. What willbe the average age of 9 boys, each 12 years old ,

and 6 boys,each 10 years old ?

11 . At 42 miles per hour , how long willit take a train to go882 miles

12 . At 25¢ per hour , what willa man earn in18 days of 10

hours ?

13 . What will be the net price of an article whose catalogueprice is the discount being 90 and 10%

14 . A manhad $ 181 inbank . What will be his balanceafter taking out 47 and 33 ?

15 . How many feet in14 rods ?

16 . 77 yards how many rods ?

17: How many sq . yd . are there ina floor 103 yd . long and6} yd . wide ?

18 . Cost of 372 eggs at 15? per doz .

622 ARITHMETIC.

19 . A man owns 3 farms containing 65 acres,86 acres, and

98 acres, respectively . Howmany acres does he own ?

20. What is the area of a piece of glass measuring 81} by 61inches ?

2 1 . What is the value inU. S . money of 50 marks at 23156

cents ?

22 . How many francs will a calf cost , if 18 are worth 630francs ?

23 . A man spends per year . What is the averageamount spent per month ?

24 . What willbe the cost of 16 lots at each ?

25 . At $ 87} each , how many horses can be bought for

TABLE.

1066. The following shows the value of goods exported toEurope from the United States during 1891 .

Find the missing value , and the per cent received by eachcountry (2 places of decimals)

COUNTRIES . VALUE.

1 . G reat Britain and Ireland ,

2 . Germany ,

3 . France ,

4 . Belgium ,

5 . Netherlands,

6 . Spain,

7 . Russia inEurope,

8 . Portugal,

9 . Allothers,

624 ARITHMETIC.

8 . Compu te simple interest on at fromFeb . 3

,1889, toMay 15, 1891.

9 . Redu ce 4 of 15, of 1} to a simple fraction .

10. Make out a receipted bill for the following articles , soldby Richard Roe to JohnDoe onMay 1, 1891 :

25§ lb . of butter per lb .

12} bbl. of flour per bbl.

35} lb . of coffee per lb.

10} lb. of rice per lb .

PROPORTION.

1068 . A ratio is generally expressed by the sign(z). This isanother form of the division sign

1069. Two equalratios form a prop ortion.

n 13 z 39

3 :9 : z 13 z 39.

1070. Supply missing term

1 21 14 x

45—90

2 . 5 . 3 + x = 12 + 20

3 . 6 . ? 1 19 z z28 z 76

7 . llb . 1 oz . :2 1b.

8 . 3 qt . 1 pt .

- 1 gal. x ii -z

9 . 4 bottles 3: bottles 6 pints 15 pints.

10. 3:men 9 men 16 acres 36 acres.

PROPORTION. 625

1071 . The first and the last term of a proportion constitutethe extremes ; the second and the third, the means.

5 and 27 are the extremes. 15 and 9 are the means. The proportionisread z 5 is t0 15 as 9 is t0 27.

1072 . The proportion 3 4 a: y may be written 2: 3/Clearing of fracti ons, we have 3y 4x ; i .a. ,

the product of theextremes is equalto the product of the means.

Solve the following :

1073 . Make the product of the extremes equalto the productof the means

,after canceling any factor common to an extreme

and a mean.

3

1 .

6

2 .

4

-t z z5 za: { :x zzi z-fi»

z :

1 . If 9 eggs cost 25 what will3 dozencost ?

2 . If 7 lb. flour cost 23 f , what willbe paid for 49 lb . ?

3 . For $5 I canget 12 straw hats. How many can I getfor $20?

4 . A wheelmakes 75 revolutions in5 minutes. Howmanydoes it make inanhour ?

626 ARITHMETIC.

5 . $ 100 principalgives $6 interest . How much willbe theinterest of $450 principal?

6 . A merchant pays 75¢ freight for 125 lb . of merchandise .How much willbe the freight on lb. at the same rate ?

7 . A locomotive goes 3 miles in4 minutes. How far does itgo inan hour ?

8 . 4 horses caneat a certain quantity of hay in10 months .

How long willit last 20 horses ?

9 . 12 men can do a piece of work in15 days. How longwill36 men require ?

10. 15 kilos cost 270francs. What willhe the cost of 5 kilos


NOTE.— Solve by proportionor inany otherway .

1 . If 9 cows cost $267 , what will be the cost of 36 at thesame rate ?

Insolving such examples by proportion, we say

9 cows $267,

36 cows $ 3 .

The ratio of the cost, 267 93, must be the same as the ratio of thenumberof cows, 9 : 36. Making the proportion, we have

Canceling,

2 . 7 bbl. sugar cost Find the cost of 42 bbl.

3 A wheelmakes 248 revolutions in5 minu tes. How manydoes It make in1 hour 20minutes?

4 . A locomotive goes kilometers in4 minutes . Howfardoes it go inanhour ?

628 ARITHMETIC.

1077 . What single discount equalto each of the followingdouble discounts ?

1 1 . 30 and 30%

12 . 20 and 25%

13 . 25 and 20%

14 . 15 and 3070

15 . 40 and

Which the better discount for the buyer ?

2 1 . 30 and or 40 and

22 . 50 and or 40 and

23 . 20 and or 30 and

24 . 20 and or 30 and

25 . 30 and l5% ,or.25 and

26 . 30 and or 50 and

27 . 40 and or 20 and

28 . 40 and or 30 and

29 . 20 and or 60 and

30. 40 and or 30 and

Find the value of x

List price , $ 250 ; selling price , $x ; rate , 40 and

List price , 800 ; selling pri ce , $ 684 ; rate , 3: and

List price , $ 33 ; selling price , $90 ; rate , 33} and

List pri ce , $600 ; selling pri ce , $ 378 ; rate , 30 and x% .

List price , $ 16 ; selling price , $x ; rate , 30 and 2070 .

SQUARE ROOT. 629

List price , 2:f selling price,27 rate , 50 and

List pri ce , $5 ; selling price , rate,a: and

List price , $ x ; selling price , rate , 30 and 3070 .

List price, $ 200 ; selling price , $ x ; rate , 25 and

10. List price, selling price , 60¢ rate,50 and

a

ah

-la

SQUARE ROOT.

1080 Find the square root of

6 5 2

425 104

1215 6 51

26 04 Am . 652.

1081 . Infinding any figure of the root after the first, we multiply theother figure or figures by 2 for a trialdivisor.

1m Find the square root of

4 5 6 7

4 85

60 74

6 38 89 Am .

1m . Find the square root

1 .

2 .

3 .

4 .

5 .

630 ARITHMETIC.


1 . What is the profit on 9 boxes of oranges, each containing20 dozen, bought at per hundred and sold at the rate of

18 for

2 . How long willit take a train to go 176 miles at the rate

of feet per minute ?

3 . If .0375 of anacre of land is worth $9, what is 33, acre

worth ?

4 . A t £ 1 13 . 7d . per barrel, how many barrels of flour canbe bought for £ 161 173 . 6d . ?

5 . A ,B

, and C buy a house for A fu rnishesB,

0,the remainder . The yearly rent

,less expenses

,is

$ 576. To what amount is each entitled ?

6 . If 580 tiles, each 6 inches square , will cover a certainarea

,how many tiles

,each 4 inches long and 3 inches wide

,

willbe needed to cover the same area ?

7 . A manreceives commission onh is yearly sales .

What is the amount of his sales, if he is allowed 4 per cent commission

8 . At what rate per cent will$360 produce interestin2 months 12 days ?

9 . Find the square root of

10. What will be the capacity,ingallons

,of a tank 9 feet

long, 6 feet 8 inches wide , and 6 feet 5 inches deep ?

11 . What decimalmultiplied by willgive the sum of g,175 » 4. 09375, and

12 . A dealer bought a lot of coal at per ton. Whatwas the totalcost if he gained by selling i t at per

ton?

632 ARITHMETIC.

3 . What is the product of the sum and the difference of 41and

4 . Subtract 1} of 4 from H ; and find the value of 131 of

163 . 6d .

5 . Add 75, Qof g of 7g, andH.

6 . Reduce g, of 8 sq . rod to the fractionof anacre , and findthe value of 37; of a toninpounds and ounces .

7 . Reduce to its lowest terms, and 312:to itssimplest form .

8 . Add 1, i , Q, and multiply the sum by and sub

tract the product from 1.

9 . Find the value of 9fi meters at 43 francs per meter .

10. Divide 2} by and add the quotient to 151 .

11 . Multiply 2125 by 163, and divide the result by I} of12 . Reduce 7s . 6d . to the fractionof a and 7 hr . 12 min. to

the fraction of a day .

2 f of 5}13 . Reduce to Its S implest form —

1? of 1

14 . Add together £3 and 1} of shillings.

15 . What fractionalpart of 7 A . 127 sq . rd . is 5 A . 81 sq . rd . ?

16 . What must be added to 3 ofgto make it equalto 136 of 34?

17 . 4 of a number is 148 . What is the number ?

18 . Ift of a field is worth $ 325, what is the field worth

19 . If f} of a house is worth what is the value of 4?

20. If13, of a ship is worth 23 . 6d . , what is 35, worth ?

11; 3

5, 23 . 6d.

Deduct 31,

if:


was the invoice value per yard , and the cost per yard afterduties and charges were paid ?

2 . A man sold his house and lot for payablecash , in9 months

,in18 months, and the balance

in28 months, with 6 per cent simple interest. What was thewhole amount paid ?

3 . A garrison of men is provisioned for 100 days. At

the end of 30 days, 600 men are withdrawn , and at the end of

60 days, 900men are added . How long willthe provisions last?

4 . The duty on certa in colored cottons is 51¢ per squareyard and 20 per cent ad valorem

,or uponthe value . What

would be the duty on 267 pieces of su ch cottons, 30 inches wide ,each piece containing 37 linealyards, and costing 7¢ per linealyard ?

5 . What willbe the result , if i of g of 3} be multiplied by 4of itself

,and the product be divided by f ?

6 . A collector of internal revenue deposited inthe treasuryretaining 2} per cent of the amount collected .

What amount did he collect ?

7 . What willit cost to transport pounds of mailmattermiles

,at per 100 pounds per 100miles ?

8 . What is the duty Ch 25 tons 2 cwt . 3 qr. of iron at $40

per ton (1 ton lb .)

make on the remainder of the cargo,which he sold at 40 cents apound ?

634 ARITHMETIC.

DOMESTIC EXCHANGE.

Find the value of as.

(Do not allowdays of grace inthe

FACE orRATE orEXCHANGE.

DRAFT.

1 . $ 100 20f perM. premium

2 . a: par

3 . a:

4 . x premium

5 . $200 $1 perM. discount

6 . $600 premium

7 . 75¢ perM. discount

8 . par

9 . $800 a:

10. $400 $2 perM. premium

APPLICATIONS OF SQUARE ROOT.


1 . How many inches inthe side of a square table t containing 529 square inches ?

2 . The surface of a square piece of board contains 3 sq . ft.

97 sq . in. What is the length of one side infeet and inches ?

(Reduce area to square inches.)

3 . How many rods long is a square field containing 90 acres?How many yards of fence would be needed to enclose it ?

case of sight drafts .)

RATE or Cos'r orTERM.

INTEREST. DRAFT.

sight 9:

6 days 6%

60 days 6%

sight

30 days 6% x

z

a: 6%

93 days a:

24 days 6

9 days a:

636 ARITHMETIC.

largest square,C,out a right-angled triangle of the same dimon

sions as those of the original triangle m . Cut another triangleof the same dimensions from the left-hand portion (Fig.

Place one of these triangles below the remainder of the square C,and the other at the right , as inFig. 4

,and the resulting polygon

will be exactly equal insurface to the two squares A and .B

(Fig.

1 . What is the bank discount on a 60-days note for$800?

2 . What is the amount of $520, for 3 years and 6 months ?

3 . A note for $850 was given January 1, 1880. The payments on it were : July 1, 1880, December 1, 1880,

August 13 , 1881 , What was due January 1,1882

, by the U. S. rule ?

4 . A note of $900, dated February 28 , 1887 was endorsed asfollows : July 12

, 1887 , $500 ; August 30, Feb

ruary 28 , 1889, $25. What was due September 1 , 1889 ?

5 . A 60-days note was made and discounted at a bank on thel0th of June ; the proceeds of this note were $340. What wasthe face of it ?

6 . What is the cost of a 60-days draft for$500, at a discountof 2 per cent ?

7 . What is the compound interest of $325, at 5 per cent, for3 years and 6 months ?

8 . What is the exact interest of $200, for 73 days, at 5 percent ? (Take 365 days to the year .)Express five-eighths of one per cent by a commonfraction in

its lowest terms, then by a pure decimal, and again with the signof per cent .

9 . Write a proportion . Thenwrite some of the principlesof proportion , and show that they are true of the proportion youhave written .

COMPOUND INTEREST. 637

Solve by proportion , giving reasons

10. If 1 of an acre of land cost $ 100, how much land will750 buy ?

If 5 mencando a piece of work in13 days, inhow many dayscan31 mendo it ?

COMPOUND INTEREST.

1 . Find compound interest of for 3 years, atinterest compounded semi-annually .

Do not use unnecessary figures . 11, each 6 mo .

first 1year

second 1year Carry to only 4 places of decimals.

etc . etc .

2 . Compound interest of at for 2 years, interestcompounded quarterly .

3 . Compound interest of at for 3 years,interest

ompounded semi-annually .

4 . Amount of at compound interest for 5 years, atinterest compounded annually .

5 . Amount of $800 at compound interest , for3 years, atinterest compounded semi-annually .

1%first 1year196

1%second 1year

195

1%th ird 1year

etc.

638 ARITHMETIC.

STOCKS AND BONDS.


Brokerage is calculated onthe par value. The d ividends are based onthep ar value.

1 . Find the cost of 240 shares mining stock,par value $ 10,

at 871, brokerage

2 . Paid for 120 shares R. R. stock, par valu e $ 100,

brokerage 170 . What was the value of the stock per share ?

3 . Bought 150 shares canalstock at 871, brokerage 195 , pay~

ing for it What is the par value per Share ?

4 . How much brokerage is paid by the buyer of 275 sharesbank stock ,

par value $ 100, brokerage 170 ?

5 . A broker sells for a customer 200 shares stock,par value

$25, at 1021. If he retains 170 brokerage , how much does he

pay over to the former owner of the stock ?

6 . A manbuys 60 Shares bank stock , par value $ 100,at 450,

no brokerage . If the annuald ividend is what is his incometherefrom ? What per cent does he receive on his investment ?

7 . A manufacturing corporation makes a year overallexpenses. The stock consists of shares

, par value $50.

What rate of dividend canbe declared ?What per cent on his investment does a man receive who has

bought his stock at 175,no brokerage ?

8 . A railroad company ’

s stock consists of shares, parvalue $ 100. Its profits for the year are out of whichmust be paid the interest for the year on worth of

bonds, at What rate of dividend can be paid the stockholders ?

9 . A capitalist bought 360 shares stock ,par valu e $25, at

1681. He paid therefor,including brokerage

,

What was the rate of brokerage ?

640 ARITHMETIC.

SQUARE ROOT.

1094. SlateExercises.

Find square roots to three decimalplaces

1 . 7 s . 38 5 . 850 7 . 807

2 . 14 4 . 74 c . 758 8 .

What is the square of .1 ? Of .3 ? Of 11 ? Of 12?

1096. How many decimal places inthe first two answers ?Inthe last two ?


Find square roots to three decimalplaces

1 . .10 5 . .4 9 . 13 .

2 . .40 6 . .9 10. 14 .

3 . .90 7 . 1 1 . 15 .

4 . .1 s . 12 . 16 .

1098 . U. S. Civil Service.

Department.

1 . Express the following in figures, using such monetarysigns as may be necessaryNinety-six million seven hundred six dollars six cents and two

mills.

2 . Express the following in words : (a)MCCCXCII. (b)

3 . Multiply by and divide the produ ct by.365.


4 . Add the following numbers—3 ; and 67711.

Subtract from their sum

5 . The revenues of a post-omoe in1884 were In1885 they were What was the per cent of decrease ?

6 . The following stock of postage stamps, stamped envelopes,etc . , was found on hand at Clinton(Neb.) post-office at close of aquarter :

29 sheets 1f stamps25 2 100 stamps to each sheet.33 4 46

62

37 15 ; 15°38 boxes lf postalcards, 500 cards ineach box.

12 newspaper wrappers,250 ineach box

, per

thousand .

12 boxes stamped envelopes, 500 ineach box , and 368 envelopes

(part of a box), per thousand .

State the totalvalue Of this stock .

7 . An inspector found that the postmaster at had , onthe 5th day of July

,1881, unlawfully paid to one AB,

out of the moneys of the government . This sum was recoverablewith interest at 6 per cent from the time it was received untiltime of payment . Onthe 19th ofDecember, 1885, the inspectordemanded restitution of this sum unlawfully paid , and AB, toavoid su it , paid the amount due. What sum must the inspectorcollect and deposit as postalrevenue to satisfy the demand ?

8 . Five registered letters, containing , respectively , 150,

$202, $ 78 , $ 104, and $ 116, were rifled of their contents by apostal clerk . When arrested , he made restitution of

which an inspector was requ ired to apportion pro rata amongthe senders of the letters. What amount should he pay to eachout of the sum recovered ?

642 ARITHMETIC.

9 . An inspector, being entitled to a salary of per

annum, was on duty twelve days . He was entitled to $4 per

diem for his traveling expenses, and to reimbursement forpaid as stage fare , steamboat fare , and for telegraphing on government business during that time . State hisaccount for services per diem

,and expenses for the twelve days.

It is assumed that the twelve days’

service was rendered inthemonth of November

,occu rring ina quarter which has ninety-two

days,the quarterly pay being $350.

10. Under the lawthe salary or compensation Of postmastersat fourth-class Ofli ces consists of

(1) Allthe box rents, i f there are any .

(2) Commissions on the amount of postage-due stamps, ordinarypostage stamps, stamped envelopes, postalcards, and newspaperand periodical stamps canceled on matter actually mailed at

their Offices,and on amounts received from sales of waste paper

,

dead newspapers, printed matter, and twine , at the followingrates : Onthe first $50 or less, 100 per cent ; on the next $ 100or less

, 60 per cent ; on the next $ 200 or less, 50 per cent ; onthe excess above $350,

40 per cent .Inno case is the compensationof a postmaster Of this class to

exceed $250 inany one quarter .During the year

,a postmaster

’

s account was as follows

Cancella Cancella Cancella Cancella Sales oftionof tionof tionof tionof waste Sales of

QM " B“ rent.2-cent l-cent postal stamped paper, twine.

stamps. stamm. cards. envelopes . etc .

.74

.95

What was his compensation each quarter ? What was hissalary for the year ?

644 ARITHMETIC.

15 . A court yard 84 ft . by 36 ft . is to be paved with flagstones measuring 6 ft . by 3 ft . How many stones willbeneededWhat willbe the cost of the work at per sq . yd . ?

16 . How many rods of fence will beneeded to enclose the field shown inthediagram

1 chain 66 feet.NOTE. A right angle contains 90 degrees.

17 . Find the length of the fourth side ofthe following piece of ground .

How many yards of fence are needed toenclose it How many acres does it contain

18 . What is the length of the diagonalof a rectangular field90 yd . wide, 120 yd . long ?

19 . The dotted line inthe accompanyingdiagram indicates a path through the field .

How many yards are saved by taking thepath instead of following the road ?

20. Find the length (inrods and a decimal) of the diagonalof a square 40-acre field .


To the following tenproblems the wrong answers are very frequentlygiven.

1 . Sold a horse for $250, losing $50. What is the loss percent

2 . If 3 boys solve 3 problems in3 minutes, how long willittake 6 boys to solve 6 problems ?

3 . Two boys go fishing ; one brings 7 cakes for lunch , theother brings 5 cakes. A third boy joins them at noon

,and pays

12¢ for his share of the dinner. Howshould the first twodivide the money received ?

REVIEW . 645

4 . If 100 per cent is gained by selling an article for $ 1, howmuch would be gained by selling it for $2 ?

5 . A boy had a slate 5 inches by 7 inches . He buys onetwice as large . G ive the dimensions of the newslate.

6 . A man W ishes to put up on the front of his lot a fence 30feet long . If the posts are 6 feet apart, how much will theycost at 252 each ?

7 . One-half the money received by a newsboy is profit.

What per cent does he make ?

8 . 50 per cent of a number multiplied by 30 per cent of thesame number equals 60. What is the number ?

9 . Three-fourths per cent Of a number is 90. What is thenumber ?

10. Animporter receives some cases Of goods numbered consecutively . Howmany cases are there , if the number Of the firstis 28

,and of the last 75 ?


1 . If a bar of Silver weighing 2 lb . 3 oz . 6 pwt . is worth£ 6 138 . 7d . , What is the value inEnglish money of a Similar barweighing 15 lb . 7 oz . 4 pwt. ?

2 . A quantity of provisions would last a ship’

s crew 20 days,

allowing each man 2 lb . 4 oz . daily . What should each man beallowed so as to make the provisions last 4 days longer ?

3 . If 40men are able to do a piece of work in10 hours, howmany extra men must be employed to finish it in8 hours ?

4 . If it requ ires 40yd . carpet 2 ft . 9 in. W ide to cover a floor,how many yards of carpet 2 ft . 6 in. wide would be needed ?

5 . How long W illit take a train to go 112 miles, at the rateof 46 miles in1 hour 20min. 30 sec . ?

6 . Change yards to rods, yards, etc .

646 ARITHMETIC.

7 . If a beam 5 ft . 6 in. long , 10 in. Wide , and 8 in. thi ck ,weighs 924 lh .

,find the length of another beam Of the same

materialwhich weighs lh .,and whose end is a square foot .

8 . A field 110 yd . long and 44 yd . wide contains anacre .

What is the area Of a field 220 yd . long and 88 yd . W ide ? Of

one 440yd. long and 176 yd . wide ?

9 . A ship with a crewof 32 menhas provisions that will

serve for 45 days, at .a daily allowance of 3 lb . for each man. If

it then picks up another vessel’

s crew consisting of 16 men , whatmiist be the daily allowance , to make the provisions last for 40

days ?

10. If a steel bar 12 ft . long , 4 in. broad , and 21 in. thick ,weighs 480 lh . , What is the weight of another steel bar 18 ft .

long , 3 in. broad , and 2 in. thick ?

11 . If 8 horses eat bushels of cats in9 days,how many

days will bushels last 14 horses ?

12 . A person deposits intwo banks 750 and respec

tively,at the same rate . The latter sum draws 18 more interest

per year thanthe former . What is the rate per cent ?

13 . TWO men have saved each . One has loanedat and the remainder at What rate must the

other man receive for h is money in order to get the sameinterest ?

14 . I owe payable in3 years 4 months. I have atpresent What rate of interest must I receive to pay mydebt at maturity ?

15 . A person loans 1 Of his capitalat and the other halfat 470 . He receives annually $ 40more interest from the formerthanfrom the latter . What is h is capital?

16 . A

'

certainsum loaned at 4% produces $ 30 less interestthana sum $400 greater , loaned at HOW much is loanedat each rate ?

648 ARITHMETIC.

4 . You sellto John Smith, tod ay,the following

25 lb . Coffee 181 cents,100 lb . Sugar 41 cents ,15 lb. Tea 45 cents

,

200 lb . Flour 21 cents,200 lb . Potatoes 11 cents.

Make out your bill, and receipt it .

5 . Find one-half of 3 miles , 7 furlongs, 15 rods, 7 feet, and5 inches . (1 mi . 8 furlongs .)

6 . Find the G reatest Common Divisor and the Least CommonMultiple of 18 , 24 ,

30,36.

7 . What is the d iagonalOf a square lot 1371 feet square ?

8 . Sold a house and lot at a loss of thereby losing21per cent of the cost . Find the cost price .

9 . If a horse dealer buys a span of horses at 10 per cent lessthantheir value , and sells them at 10 per cent more thantheirvalue , what per cent does he make ?

10. If a boy buys peaches at the rate of 5 for 2 cents, and sellsthem at the rate Of 4 for 3 cents, how many must he buy and sellto gain

EXACT INTEREST.

Exact interest is used by the Government inits calculations. 365 days

are takento the year.


1 . Find the exact interest of $280 from April14 to Sept . 6at

Time 145 days. Ans. $280x i x145

100 365

2 . Find the exact interest on fromMarch 4 to Dec .

15 at 6 per cent .


3 . On$ 384 at 7155 per cent for 75 days.

On$438 at 5% from Jan . 1 toMarch 15.

5 .

‘

05 at 417. for 87 days.

6 . On at 5% from April29 to Sept. 20.

7 . On$584 at 31% fromMay 16 to Dec . 1.

8 . On at from J an. 4 , 1893 , to June 2 , 1894 .

9 . On at 6% from April1, 1892, to Feb . 15, 1893 .

10. On at 5% from Aug. 19,1892 , toMarch 4 , 1894.

1105. Unless exact ” or accurate ”

interest is specified , use 360 daysto the year.

1106. N. Y. StateExaminations forSecond Grade Certificates.

1 . William Wilson sold goods to the amount of One

half of his sales showed a profit of 25 per cent on the cost , andthe remaining half a loss of 161L per cent on the cost. Requ iredthe totalcost of the goods.

2 . Find the amountMarch 4 , 1890, of at interestfrom Nov . 3

,1889

,at 4 per cent per annum .

3 . If .01125 Of the value of a piece of property iswhat is the whole value ?

4 . If I sell1 Of my farm for 1 of what the farm cost me,what is my gain per cent ?

5 . Reduce 240 rd . 4 yd . ft . to the decimalOf a mile .

6 . Inwhat time will$ 726 amount to at 6 per centper annum ?

7 . Which is the higher rate of freight on wheat,

perhundred or per bushel(60lb .) and What per cent ?

8 . Write inwords :

(a) .267 ; (5) (c) 7206690 3 (d) 2007-111 .

650 ARITHMETIC.

9 . Find the interest on $865 for 13 days at 51 per cent perannum .

10. If 40 per cent of the selling price of an article is profit,what is the per cent of gainon the cost ?

11 . What number added to 41 times itself willequal601?

12 . Divide 1 by .000031.

13 . Reduce to lowest terms (a)111-1 ; (5)111.

14 . A hotel is insured for 1 of its value , at 11 per cent premium , and the premium amounts to $ 150. Find the valuation

of the hotel.

15 . If 4 men eat 64 pounds of bread in2,weeks, how many

pounds will16 meneat in7 weeks at the same rate ?

16 . Divide .75 of 171 by 1 of .035.

17 . Make and solve a problem inwhich avoirdupois weightis changed to troy weight .

18 . Find the cost of pounds of hay at $ 15 per ton.

19 . Multiply 171 131 by 1 Of 27 .

20. Find the cost of plastering the walls and the ceiling of ahall72 ft . long , 50 ft . wide

, and 22 ft . high ,at 181 cents a square

yard , allowing 972 sq . ft . for Openings and baseboards.

21 . A certain quantity of paper w illmake copies of an

octavo book (8 pages to the'

Sheet). Howmany 00pies of a 12mobook (12 pages to the sheet)willthe same paper make ?

22 . Inwhat time willa note for $ 200, drawing 51per centinterest, double itself ?

23 . A reservoir 8 ft . X 4 ft . x 3 ft . 9 in.,holds how many bar

rels of water ? (1 bbl. 311 gal.)

24 . Find the diagonalof a square park containing 20 acres.

652 ARITHMETIC.

8 . 5 houses are sold for and is gained .

Find (a) cost of 3 houses, and (b) the gainper cent.

9 . Which willcost the more, and how much more , 15 sharesof N. Y. ,

N. H. , H.,at 85,

or 13 N. Y . N. E.,at 102 , if the

brokerage ineach case is 170 ? (Par value per share

10. Anagent receives to invest inland atper acre , after taking ou t a commission of 2170 . Howmanyacres canhe buy ?

MEASUREMENIS .

1 . What is the area of a triangle whose sides measure 15, 16,and 17 inches

,respectively ?

15From the half sum of the three sides subtract each

16 side separately . The square root of the continued prod17 not of the half sum and the three remainderswillbe the

24 15 9

24 16 8

24 17 7

2 . Find the area insquare feet of a triangle whose sidesmeasure 35 ft . , 84 ft . , 91 ft .

3 . Find the area of a triangle whose sides measure 21, 28 ,and 35 rods, respectively .

4 . Inthe following field,AB measures

39 rods ; BC,52 rods ; CD ,

25 rods ; AD,

60 rods ; and the diagonal, AC, 65 rods.A

Find the area of the field insquare rods.

5 . Find the area of an isosceles triangle whose base is 30 yards, its equal sides measuring 25yams.

6 . What is the altitude of an isosceles triangle, base 96 ft. ,equalsides 64 ft . ? Find its area.

PARTIAL PAYMENTS. 653

7 Find the area Of anequ ilateral triangle,each Side being

8 . Find the area of a right-angled triangle,base 42 ft . ,

hypotenuse 70 ft .

9 . Find the area of an isosceles triangle , altitude 48 ft . ,equal

sides 50 ft .

10. Place two equ ilateraltriangles, sides 2 inches, base to base ,making a rhombus. Find its area , also the length of each diagonaL

11 . Find the radius of a circle whose circumference is 132 ft.x diam . circum .)

12 . Find the area of a circle whose radius is 4 inches . (Area

1 circumference x 1 diameter .)

13 . Find the area of a circle Whose radius is 3: inches.

14 . Find the radius of a circle whose area is sq . in.

15 . Find the area of a circle whose circumference is ft .

PARTIAL PA YMENTS.

BROOKLYN, N.Y June 19, 1894.

Ondemand,Ipromise to pay William R. Budd , or order , Two

Thousand Four Hundred Fifty-four Dollars, value received ,with interest at 6 per cent .

ARTHUR TOWNSEND.

The following payments are endorsed on the note

July 5,1894

, $ 200.

July 29, 1894 , $450.

Sept . 18 ,1894 , $ 700.

Oct . 25,1894

, $ 300.

Find the amount due J an. 2,1895.

654 ARITHMETIC.

Ifno payments had beenmade, therewould be

And interest from June 19 to J an. 2, 197 days,

Totaldue,

The credits are : Payment July 5, 1894,Interest on$200, July 5 to J an. 2

,181 days,

Payment July 29, 1894,

Interest on$450, July 29 to J an. 2, 157 days,

Payment Sept. 18, 1894,Interest on$ 700, Sept. 18 to J an. 2, 106 days,

Payment Oct. 25, 1894,

Interest on$300, Oct. 25 to J an. 2, 69 days,

Balance due,

1109 . By thei

merchants’ rule , interest is calculated on the

face of an interest-bearing note from its date until settlement,

and interest is allowed on all credits from their payment untilsettlement.

1 . A note for $500, with interest at is dated July 25,

1893 . Payments are made : $ 100, Sept . 18 ; $200, Feb . 5, 1894.

How much is due April1, 1894

2 . Find amount due Sept. 15, 1894 , on a demand note forwith interest at dated J an. 18 , 1894 . Payments of

and $500were madeMarch 30 and June 17,respectively .

3 . June 12,1892, Robert Colgate bought goods amounting to

$600. Dec. 31, 1892, he paid $ 300; April5, June1,1893 , he settled the account . HOW much did he pay on that

date , if he is charged 6% on the purchase from its date,and is

allowed 6% interest on his payments ?

4 . T. J .Minturn loaned Chas. A . Dorsey $ 500, Sept. 1, atPayments of $ 200 each were made Oct. 1 and Nov . 1.

How much is due Dec. 1 ?

656 ARITHMETIC.

8 . 39 rods, 44 rods, 17 rods.

9 . 65 rods, 34 rods, 93 rods.

10. 13 chains, 37 chains, 40 chains.

ExaminationforState Scholarships, CornellUniversity.

1 . Reduce the fraction8? x I

; 067111:111 of

t

2 . Divide 1 by to the quotient add the divisor, and fromthat sum subtract the dividend . G ive the fractionalpart of theanswer ina decimal.

3 . A rectangular field , containing 71 acres, is 30 rods wide .

HOW much willit Cost to bu ild a fence from one corner to thediagonally Opposite corner at $ 2 a rod ?

4 . If the interest on $300, for 1 yr . 8 mo . ,is $ 36, find what

would be the interest on for 3 yr . 4 mo . 24 da . ,at the

same rate .

5 . Find the proceeds of a bankable note for $500, at fourmonths, when discounted at Six per cent .

6 . Reduce .0468 T. to a compound number (lb . and

7 . Divide 871 by 191 of 2511

.

8 . Find the amount of $635, from Nov . 13, 1888, to June 6,

1889,interest at 5% per annum .

9 . Define notation ; numeration .

Write w ith figures two million,forty-three thousand , eighty .

Write inwords

10. Find the prime of

CHAPTER XIV.

EQUATION OP PAYMENTS. MENSURATION OP SURFACES

ANDVOLUMES. BOARDMEASURE. ANNUAL INTER

EST. GOVERNMENT LANDS. METRIC SYSTEM.EQUATION OF PA YMENTS.

1 . A friend loans me $ 800 for 6 months without interest.How long ought I to loan him $400 to cancelthe obligation ?

2 . InWhat time would the interest on $450 be the same asthe interest for 3 months on $600?

3 . W borrows from X $ 200 for 5 months and $400 for 2months . How much money should W '

loanX for one month inreturn for the accommodation

4 . A manOffers a lot for $ 600, payable $ 300 in2 months,and $300 in4 months. Howmu ch credit should be given to abuyer who wishes to pay the $600 at one time ?

5 . Mr. Jones has bought $600 worth of goods on 6 months’

credit , and $300 worth on 3 months’ credit . .For what timeshould he give a note (without days of grace) for the Wholeamount , 900

I

1 . Inwhat time would the interest on be the same asthe totalinterest on the following amounts :

$ 100 for 1 month , $200 for2 months, $ 300 for 3 months, $400for 4 months ?

658 ARITHMETIC.

Interest on$ 100 for 1 month Interest on $ 100 for 1 month200 2 months 400 1

300 3 900 1

400 4 1600 1

1000 3000 10

2 . A person owes $400 payable in4 months, and $500 payable in13 months. What would be the average time for thepayment of the whole indebtedness of $900?

The debtor is entitled to the use of $400 for 4 months, wh ich is equaltothe use of for 1 month . He is also entitled to the use of $500 for 13months

,wh ich is the same as for 1 month . He is entitled ,

inall, tothe use of for 1 month , wh ich is equal to the use of $900 for howmany months ?

By equation of payments is meant a method of ascer

taining at what time severaldebts due at different times may besettled by a single payment . The time thus found is called theaverage time , or the equated time .

3 . Find the average , or equated, time for the payment of thefollowing

$ 600 due in2 years

$ 500 due in11 years

$ 300 due in1 year

$ 400due in9 months

$ 250 due in8 months

$450 due in6 months

$500 due in3 months

$600 payable incash

660 ARITHMETIC.

14 . How many bushels of bran worth 40 cents per bushelshould be mixed with bran worth 30 cents per bushelto make100 bushels Worth 36 cents a bushel?

a: number of bushels at 40!100 a: number of bushels at 30f

value (incents)of one kind

z ) value of otherk ind

Totalvalue howmany cents ?

15 . How many bushels of corn Worth 60¢ per bushelShouldbe mixed with 80 bushels of corn worth 50¢ per bushelto makea mixture Worth 529

‘ per bushel?

16 . A cando a piece of work in20 days, B cando it in30days . They work together and receive $5 per day as the wagesof both . What Should be A’

s Share of the totalamount received ?

Howlong does it take both together to do thework ? Whatwould Areceive per day if he d id the work alone ?

17 . A partnership is formed between A with a capital of

and B with a capital of Six months thereafter ,they take inC with a capitalOf How Should a profit of

be divided at the end of the year ?

18 . Three merchants shipped a cargo of iron by sea . A sent180 tons , B

‘

sent 105 tons, C sent 315 tons . During a storm thesailors were obliged to throw overboard 180 tons to save thevessel. What portion of the loss should each merchant sustain ?

19 . If pure milk is reduced invalue from 24 ¢ per gallon to20¢ per gallon by the addition of water , how many quarts of

water have been placed ina canthat contains 40 quarts of theadulterated . article ?

1118 . Brooklyn Public Schools. ExaminationQuestions. Dic

tated Examples.

1 . Add 798 ; 584 ;

888 ; 68.

EXAMINATION QUESTIONS . 661

2 . Multiply by

3 . Divide by 789.

4 . Howold December 1,1892 , was a boy born November

25,1876 ?

5 . A man has inbank . What will remainafter hehas taken out $478 ?

6 . How many hours inthe month of January ?

7 . Inhow many years,months

,and days, will 100 amount

to 111, at simple interest ?

8 . A and B contract to haula pile of lumber for 105. A

furnishes 3 teams, and B 4 teams. HOW mu ch does B receive ?

9 . A farmer makes 675 gallons of cider . He has but 12barrels, each of 45 gallons

’ capacity , to store it in. How manymore such barrels does he need ?

10.

'What willbe the cost of 36 yards Of cloth at per

yard ?


1 . A manpaid $ 999 for the rent of a house from June 29,1889

,toMay 5, 1891. What was the rent per year ?

2 . What per cent of 3 pounds 7 ounces is 7 pounds 9 ounces ?

3 . At 50 cents per running yard , what willbe the cost offencing a square field containing 10 acres ?

4 . A t the rate of 20problems anhour for A , and 15 in55minutes forB,

inwhat time canboth together solve 100problems ?

5 . What sum willamount to in2 years 5 months11 days, at 41 per cent, simple interest ?

6 . Find the entire surface of a cube, Whose edge measures15 inches.

662 ARITHMETIC.

7 . A dealer buys books at each , less 331 and 10 percent

. At What price per copy must he sell them to gain431

per cent ?

8 . Abraham Lincoln died at the age of 56 years 2 months

3 days, after serv ing as president 4 years 1 month 11 days.

G ive the date of his birth , the date Of h is inauguration being

March 4, 1861.

9 . A dealer buys 150 barrels of flour. He sells one-third of

it at per barrel, losing 10 per cent . The remainder he

sells at a profit of 6 per cent. What is h is net gain or loss ?

10. Sixty per cent of 661 per cent of a number equals 810.

What is the number

112 1 . St. PaulPublic Schools. ExaminationQuestions.

1 . Bought 1 of a box Of candles, and having used 1 of them,

sold the remainder for 11 of a dollar . How much would a box

cost at the same rate ?

2 . Reduce 131 , .621, .37116 , 1, to decimals, and find their sum .

3 . A ladder 40 feet long is so placed ina street,that with

out being moved at the foot, it will reach a window on one side33 feet , and on the other Side 21 feet from the ground . What isthe breadth of the street ? Illustrate .

4 . A grocer has a false balance by which he gives but 12ounces to the pound . What is the realvalue of a barrelOf sugarthat sells for $ 28 ? (Solve by proportion .)

5 . Explain partnership , telling what it is, its purpose andits advantage .

6 . Four men hired a coach for 13 , to convey them to theirrespective homes

,which were at distances from the place of

starting as follows : A’

s 16 miles, B’

s 24 miles, C’

s 28 miles,and

D’

s 36 miles. What ought each to pay ?

7 . Analyze the fourth .

664 ARITHMETIC.

10. A log of wood is 15 inches broad and 11 inches thick .

What length of it willmake 10 cubic feet ?

11 . What rate per cent is required to make gaininterest in11 months and 19 days ?

12 . How long will it take to gain at 7per cent ?

13 . A merchant imported 120 tons of English iron , costing 14pence per pound , on which he paid a duty of 20 per cent . The

freight was 5 shillings sterling per ton. Whatwas the totalcostinU . S . currency ? (1 ton lb . £ 1

14 . Reduce the following complex fraction to its simplest form

I x se. a ”3

15 . SamuelAdams, a contractor , had the following dealingswith the Treasury Department : He furnished , January 3 ,

1883 ,

lb . of twine at 12 a lb . April4,1883

, 25 doz . gold pensat $ 25 a doz . ;May 7 , 1883 , 645 reams letter paper at $2 aream ; July 9, 1883 , 45 doz . qt. Arnold ’

s ink at $ 3 a doz . ;

October 30, 1883 , envelopes,at $2 a thousand ; and

December 5, 1883 , 8 doz . inkstands at a doz . Hewas paidcash as follows : February 4 , 1883 , $ 175 ; April30, $ 350 ; July15, $ 700 ; November 5, and December 31, 1883 , he wasallowed on settlement $45 for cartage , and charged $ 75 forbreakage , and $60 for shortage on envelopes. State his accountinthe blank below , with proper heading, and show the balance

,

if any, due him . (No interest .)


1123 . Questions fromExaminations forN.Y. State Certificates.

1 . The large wheelOf a bicycle is feet incircumference .How many times must it revolve inrunning a mile

2 . How many rods of fence are requ ired to enclose a squarelot whose area is sq . ft . ?

3 . Property worth is insured for of its value, at {L

Of one per cent .‘

What will be the loss, includi ng premium , in

case of totaldestruction by fire

4 . Requ ired the cost of laying a pavement 5} rods long and8 ft . 6 in. wide

,at per square yard .

5 . The sum of two numbers is 12 bu . 1 pk . 3 qt . , and theirdifference is 1 bu . 2 pk . 5 qt . What are the numbers ?

8 . Make and solve a problem illustrating how the principalis found when the interest, rate per annum ,

and time are known ,using 1 yr . 1 mo . 1 da . as the time .

7 . Make and solve a problem illustrating the application of

square root infinding the side of a right-angled triangle.

8 . How many acres of land,inthe form of a square , may be

enclosed by 160 rods of fence ?

9 . Find the square root of .441 correct to two decimalplaces.

10. Reduce 17 lb . 11 oz . avoirdupois weight to pounds, ounces,pennyweights, and grains

,troy weight. (1

'lb. avoirdupoistroy grains .)

11 . A note for and interest , amounted to2 yr . 7 mo . 13 da. after date , when it was paid . Required therate per cent per annum .

12 . The premium on a fire insurance policy , at of 1 percent , was 140. Requ ired the face of the policy .

13 . What is the length (inrods) of the diagonal of a squarefield containing 40 acres?

666 ARITHMETIC.

14 . ReduceH‘H to its lowest terms.

15 . Find the solid contents of a cube , the area of one face of

which is 256 square feet .

16 . What elements are requ ired to find the per cent of gainor loss in a business transaction ? Illustrate by an originalproblem .

17 . A car contains pounds of wheat . Find the value

of the load at 92£5per bushel. (1 bu . 60lb .)

18 . Find the area of a triangle whose bas e is 22 ft . 8 in. , andaltitude 19 ft . 9 in.

19 . The list price of a certain stove is $38, and the retaildealer is allowed commercial discounts Of 20 per cent

, 5 percent

,and 3 per cen t. What price does he pay for the stoves ?

2 0. If 45 horses eat 11 tons Of hay in30 days, how many tonsshould last 84 horses 56 days ?

21 . If a ton of coallasts a family 21 days, what will be thecost of coal used by it from Oct . 17 , 1888 , to April 25, 1889 ,

exclusive of either day named , at per ton ?

22 . A farmer sold acres of his farm of 112 A . 96 sq . rd .

What per cent of his farm had he left ?

23 . Find the cost Of a pile of 4-foot wood, 27 ft . long and 6

ft . high,at per cord .

24 . Find the interest on 1 , for 3 yr . 1 mo . 29 da .

,at 5 per

cent per annum .

25 . Four menhire a pasture field together . The first pastures 4 cows 18 weeks ; the second , 5 cows 12} weeks ; the third,11 cows 65weeks ; the fou rth , 9 cows 16 weeks. What part of

the rent should each pay ?

26 . How many rods of fence will be requ ired to enclose afield inthe form of a right-angled triangle

,whose area is 13}

acres and whose base measures 48 rods ?

668 ARITHMETIC.

15 . Of the rectangle , Fig. 2 . 16 . Of the rhombus, Fig. 3 .

17 . Of the trapezoid , Fig. 4 .

FIG . 2. FIG . 3 . FIG . 4.

18 . Of the trapez ium ,Fig. 5. 19 . Of the rhombus, Fig. 6.

20. Find the altitude , AB ,Of the following triangle , Fig. 7

(First find the area.)

FIG . 5. FIG 6. FIG 7 .

21 . Find the diagonal(inrods) of the square whose area is 5acres.

22 . Find the area of a hexagon , composed'

of six equilateraltriangles, each side being 6 inches. Fig. 8 .

FIG . 8 . FIG . 9. FIG . 10.

23 . What is the area of the circle circumscribed about theabove hexagon , Fig. 9 ?

24 . What is the area of the square inscribed ina circle whosediameter is 10 feet

,Fig. 10?


25 . What is the area of the sector,Fig. 11 , the

radius of the circle being 10 inches, and the arc ,z

I

of the circle 72° ll

(A sector is a portionof the surface of a circle inclosedbetweentwo radii .)

112 5. Newport, B. I., Public Schools. Examina FIG . 11.

fionQuestions. OralProblems.

1 . What would be the cost of 120 books at each ?

2 . Change {if to a decimalof three places.

3 . How much wood in three piles containing,respectively

,

of a cord,

of a cord , and i Of a cord ?

4 . What is the percentage of gain incase of railroad stockbought for $80 per share and sold for 90 per share

5 . Change mills to dollars.

Slate Problems.

1 . Find the sum of the answers of the following examples

.0179

30872 .179

2 . Define underwriter, policy , and premium .

3 . Ipaid for insuring on my house,for5 years.

What was the per cent of the yearly premiums ?

4 . What is the balance of a billof after two discountshave been made ; the first Of 2070 on the the other of

5% on what then remained ?

5 . What is the accu rate interest of at fromDec . 28

,1890,

toMarch 8, 1891 ?(365 days to 1 year.)

670 ARITHMETIC.

6 . Find the date when the following note becomes due,the

time for which it was discounted , the bank discount, and theproceeds : A 4-months note for $225, dated December 10, 1890,and discounted January 1, 1891, at 7

7 . What is the amount of $261, for 3 years, at compounded annually ?

8 . There was shipped to Liverpool from NewYork inoneweek inspecie . What amount of English currencycould be bought with it ?

9 . How much money must be invested inthe U. S . 44’

s toyield a quarterly income of $225, bonds selling at (Thebonds pay interest on the face value .)

10. Isend to my agent at Havana What part of thissum willremain to invest insugar , after deducting his commission Of 2% on what he lays out ?

1126 . Kingston, N. Y., Public Schools. ExaminationQuestions.

1 . If 4 Of a lot cost what is the value of Of it ?

2 . What is the freight on lb . iron at per ton ?

3 . What is the value of 10 lb . 7 oz . 16 pwt . of gold at $ 75a pwt. ?

4 . How many days fromMarch 16 tillDec . 25 of this year ?

5 . The dividend is the quotient 17,the remainder 102 .

What is the divisor ?

6 . Divide the L. C.M. of 132 and 156 by their G . C. D.

7 . The product of three numbers is .0728 ; one of them is

another .07 . Find the th ird .

8 . What is the greatest length Of boards that I canuse infencing my garden , the sides of which are 168 ft.

,182 ft .

, 280 ft . ,

and 252 feet ?(Find G . C. D. of the givennumbers.)

672 ARITHMETIC.

3 . A note of dated July 5, 1888 , was paid June 6,1890, with interest at 8 per cent. What was the amount paid

4 . Two men hire a pasture for $420. A puts in300 sheepfor 5 weeks, and B puts in450 sheep for 6 weeks. What shouldeach pay ?

5 . A managrees to dig a cellar 30 feet long, 24 feet wide ,and 6 feet deep . What per cent of the work has he done whenhe has removed 16 cubic yards ?

6 . How many boards 16 ft . long , and 4 in. wide, are re

quired to floor a room 48 ft . long , and 32 ft . wide ?

7 . What is the length of one side of a square field containing15 A . 1 sq . rd . ?

8 . What is the present value or worth of due in

10months 24 days, when money is worth 5 per cent ? (See Art .

9 . How much walking does a mansave by crossing diagonallya field 28 rods long, and 21 rods wide , instead of going along theend and side ?

10. Inorder to have an annual income of what summust be invested at

1128 . Elmira,N. Y.,Public Schools. Examination Questions. Oral

1 . At 10f a quart , what are 3 bu . 1 pk . 5 qt . of chestnutsworth ?

2 . How many yards in288 inches ?

3 . What decimalof a number is per cent Of it ?

4 . $ 120 is per cent of what number ?

Slate Problems.

1 . Anagent sold 5 tube of butter , weighing respectively 46545, 48 , and 50lb . ,

at anaverage of 32¢ a pound . He paid


freight and cartage His commissionwas 5 per cent .How much was his commission

,and what were the net proceeds ?

2 . Anagent’s commission,at for selling woolwas $250.

What did the sale amount to ?

3 . Anagent sold goods to the amount of and receivedcommission . What was the rate ?

4 . What is ratio

5 . What are the terms of a ratio ? Name them .

6 . What is a proportion

7 . The extremes of a proportion are 16 and 25,and one of

the means is 20. What is the other mean ?

8 . Change the following equation to the form of a proportion : 310 Tfi-h .

9 . Prove that the expression you have given is a tru e proportion.

10. If a quantity of flour willlast 21 soldiers 16 weeks, forwhat time willit last 24 soldiers ?

11 . If 9 men can do a piece of'

work in28 days, how manymencando the work in12 days ?

12 . If a traveler walks 140mi . in8 da . , walking 7 hr . 9. day ,

how many miles canhe walk in12 da . of 8 hr . each ?

13 . Find the cost of insuring property worth forQof

its value , at f% .

14 . At 4 per cent , how much insurance can be effected onastore for $ 120

15 . A schoolhouse is to be built at a cost of The

property Of the town is assessed at What is therate of tax , and what tax willa man pay whose property is as

sessed at

674 ARITHMETIC.

SPECIAL

1129. G ive sums

182+

G ive answers

150 172+ 19—66 183 —37) 161+ 79 12

154 —35 173 (57 18) — 36

155— 49+ 24 — 33 165 —28) 34

153 151+46—24 182 — 33) —46

1131 . G ive products

44 x 20 63 x 32

44 x 22 54 x 38

44 x 18 54 x 42

63 x 28 33 x 48

1132 . G ive quotients

676 -z 13 527 17

602 14 738 18

690 15 950 19

672 16 924 21

1133 . G ive results

84 x 11 2111+ 13

48 x 2; 214; 14

36 x 9; 1301 21

48 x 19; 1554, 22

676 ARITHMETIC.

15 . A degree of longi tude inlatitude 45° is about 70% of thelength of a degree on the equator . Calling the latter length 69miles, how long is a degree of longitude inlatitude

16 . At $44 per acre , how much land can be bought for 968 ?

17 . A number of marbles divided among 29 boys gives each16 marbles, and leaves a remainder of 26. How many marblesare there ?

18 . What is the monthly salary of a clerk who receivesper year ?

19 . Howmany revolutions ina mile , ft .,are made by a

locomotive wheel16 ft . incircumference ?

20. How many feet of fence are there around a lot 49 ft .

wide, 87 ft . long ?

2 1 . Howmany bricks 8 in. by 4 in. by 2 in. would make acubic foot ?

22 . 13 is one factor of Find the other two primefactors.

23 . What are the three equalfactors of 343 ?

24 . What is the square root of

25 . At 44 miles per hour , how long will it take a mantowalk 37§ miles ?

26 . What willbe the cost of 9 dozen hats at each ?

27 . A 90-days note is drawn April 6. Onwhat date is itdue ?

28 . How many square yards are there inthe entire surfaceofa cube whose soli d contents are 8 cubic yards ?

29 . What are the proceeds of a 60-days note for $ 100 discounted at a bank the day it is drawn at

MEASUREMENTS. 677

LEEDS , Aug. 24, 1892.

Statement of certain goods, viz . : woolenmantlings, consigned '

by Johnson Birmingham toMessrs . Hamburger Bros. of NewYork , and dispatched from Leeds via Liverpool, per S . S.

Umbria.

295 lb. net .

Find the duty inU. S. money at 44¢ per lb . and 50% ad

valorem . £ 1

1136. NOTE. 2/3 2 shillings 3 pence. 1 9 1 shilling 9 pence.

MEASUREMENTS.

113 7 . Find the area of each of the following triangles and its

altitude .

Whenthe area of a triangle is knownand the length of the base, howcanits altitude be calculated ?

678 ARITHMETIC.

1 . Base , 51 ft . ; other sides, 20 ft . and 37 ft .

2 . Base , 21 yd . ; other sides, 13 yd . and 20 yd .

3 . Base , 148 rods ; other sides, 39 rods and 113 rods.

4 .

5 .

Base , 28 chains ; other sides, 17 chains and 25 chains.

Base , 75 inches ; other sides, 20 inches and 65 inches.

1138 . Find the areas of the following quadrilaterals

6 . G iven AB ,17 ; B0, 10; OD,

2O; DA , 13 . AO 21.

7 . G iven AB ,25 ; B0, 39 ; CD,

34 ; DA ,50. AC’= 56.

8 . G iven AB , 37 ; B0, 15 ; CD, 39 ; DA ,17 . AO: 44 .

9 . G iven AB ,111 ; B0, 45 ; CD,

25 ; DA ,113 . A0= 132 .

10. Given AB , 113 ; BC, 17 ; CD, 39 ; DA ,111. A0= 120.


1 . A and B rented a field for a year for $ 175. A put in6

horses for the whole time,B put in5 horses for 11 months and 3

horses for 5 months. Howmuch of the rent had each to pay ?

2 . A bankrupt surrenders property worth for thebenefit of three creditors to whom he owes $ 750, and

respectively . How much should each creditor receive ?

3 . Four persons rented a pasture for 26 weeks. K put in50 sheep and L 60 Sheep for the whole time

,Mpu t in70 sheep

for 20weeks , and N 90 sheep for 22 weeks . How much of the

rent, $ 130, had each to pay ?

680 ARITHMETIC.

1 1 . If 76 boards, each 14 feet long and 10 inches wide,are

worth how much would 50 such boards be worth ?

12 . If 7 men receive $ 126 for 5 weeks’ work

,how much

should they receive for 9 weeks’ work ?

13 . If for 7s . 6d . I canbuy 9 lb . of raisins,how many pounds

canIbuy for £ 56 16s. ?

14 . A field of grain was to be cu t down by 40men in10 days .

Eight of the men , however , failed to come . How long did ittake the others to do the work ?

TABLE.

BrooklynAssessments and Taxes for 10years.

A ssessments . Tax Levies.

For State ForCountyPurposes . P urposes .

TOtalLevy .

Average

Tax

Rate

per

0

Valuation

Find for each year the totaltax levy,and the tax rate indol

lars, cents , and mills per Of assessed valu e .Find the average assessment per year ; the average tax levy

for state , county , and city purposes ; and the av erage tax rate .

SURFACES OF SOLIDS. 681

SURFACES OF PRISMS AND CYLINDERS.


NOTE. The pupils should be encouraged to make cardboardmodels of the forms studied .

1 . Find the convex surface of a square prism,

one Side Of its base being 4 inches and its height6 inches. Draw the development .NOTE. The convex surface is the surface exclusive of the

bases.

2 . Find the convex surface of a triangularprism , each side Of whose base measures 4 inchesand whose altitude is 6 inches. Draw the development .

3 . Find the convex surface of an hexagonalprism,

each side of its base being 4 inches and its altitude 6inches. Draw the development .

4 . Canyou show that the convex surface of a

prism is found by multiplying the perimeter of thebase by its altitude (height)?

5 . Find the convex surface of a cylinder,the

diameter of its base being 4 inches and i ts height 6inches.

6 . How do you find the entire surface of a prismor cylinder ?

7 . What is the entire surface of a cube whose sideis 7 inches ? Of a cube whose side is 2: inches ?

8 . The entire surface Of a cube is 216 sq . in. What islength of one side ?

9 . The convex surface of a cube is 144 sq . in. Find

entire surface .

682 ARITHMETIC.

10. Find the entire surface of a square prism ,one side of

whose base measures 4 inches, and whose altitude is 6 inches.

11 . The convex surface Of a square prism is 600 sq . ft . ,the

altitude is 15 ft. What is the length of one side of the base ?

12 . The entire surface of a square prism is sq . in. One

Side of the base measures 15 inches. What is its convex surface ?What is its altitude ?

13 . Find the entire surface of a square prism whose convexsurface is 540 sq . in.

, and whose altitude is 15 inches.

14 . What is the entire su rface of a cylinder whose base has adiameter of 1 foot, and whose altitude is 1 foot ?

SURFACES OF PYRAMIDS AND CONES.

15 . The convex surface Of a square pyramidconsists of how many equaltriangles ? Find theconvex surface when one side Of its base measures 4 inches and its slant height (AX ) 6 inches.

Draw the development .

16 . The convex surface of a pyramid is equalto the perimeter of the base multiplied by what ?

17 . Find the entire surface of the above pyramid .

18 . Calculate the entire surface of a square pyramid whoseslant height is 18 inches , the area of its base being 144 sq . in.

19 . Find the entire surface of a triangular pyramid whosethree convex faces and the base are equ ilateral triangles, eachside measuring 2 inches.

20. Draw the developed convex surface of acone , the diameter of whose base is 4 inches

,and

whose slant height is 6 inches.

Calculate the convex surface .

21 How many square inches of paper wouldbe required to cover the side and the base of a cone 6 inches indiameter at the base , and having a slant height of 10 inches ?

684 ARITHMETIC.

5 . At $2 a rod,what is the difference inthe cost of fencing

a lot of land 20 rods square,and another lot containing the same

area which is 40 rods long ?

6 . A manhas a straight fence made Of three lengths of wire .

The three wires are supported by 11 posts, the distance betweenthe centers of . the posts being 10 feet. These posts are each 6inches in diameter , with holes through which the wires pass.

The wires are fastened on the outs ide of the end posts by twisting them around spikes driven into the posts, and it took 3 inchesOf wire to fasten each end of each wire . Howmany feet of wireare requ ired for the fence ?

1143 . BostonExamination Questions. Mental.1 . A room is Q as wide as it is long . Its length is 20 ft .

How many sq . ft . are there inthe floor ?

2 . What will5 tons of granulated sugar cost at per lb .

3 . If 5 yd . of cloth cost 90 what will8 of a yd . cost ?

4 . What willbe the cost of 1 pk . 1 qt. 1 pt . of nuts at 10¢per qt .

5 . What is the value Of an acre of land at 10¢ ‘ per sq . ft .

6 . Anagent insured a house for me at a commission of

His commission was 15. How muchwas the house insured for?

7 . A gold-digger who had 3 lb . of gold dust, lost 9 oz . What

per cent was left ?

8 . What is the interest of $50, for3 yr . 7 mo . 12 da . , at 670

9 . How many days fromMay 16 to July 5 ?10. Add 379 and 297 .

1 1 . What is the bank discount on a sixty-days note for $400,at 6%12 . A dealer sold flour at a profit Of a barrel

,and

gained What was the cost ?


13 . Inwhat time will$50 double itself at

14 . What would 51 lb . of butter cost at a lh. ?

15 . Mrs. Allen bought 7 chairs at $4 apiece , 2 tables at $9apiece , and a carpet for $ 33 . She paid two $50 bills. Howmuch change was due her ?

16 . If $ 1 is paid for insuring a piano worth $500, what is therate of insurance ?

17 . Inwhat time w illany sum of money double itself, at 6%18 . Into how many lots

,containing g of anacre each , can

eight acres be divided ?

19 . A manlends at and at What isthe difference inthe amount of yearly interest due on each ?

20. A manowning 3 of a ship , sold i of his share . Whatpart of the ship did he stillown ?

2 1 . Find the sum of the prime numbers as far as 12.

22 . 2} 31 4;

23 . Interest of for 30 days, at 6%

24 . Interest of for 6 months, at 4%

25 . How many rings, each 2 pwt . 12 gr. ,canbe made from 4

1b . of gold ?

26 . How far apart are two places whcse difference intime istwo hours and three minutes ?

27 . Find the number of sq . in. on the surface of a block 10in. x 4 in. x 3 in.

28 . How much would you pay for 3§ yd . of cloth at a

yard

29 . Oil is worth a pint . How many pints canI buyfor$ 6 ?

30. Sold oranges for i f apiece , gaining How muchdid they cost apiece ?

686 ARITHMETIC.


1 . Find the interest of from June 24, 1871, to April

1, 1874, at

Find the amount of fromMay 27 , 1881, to February12 , 1885, at

2 . If a manowning 4570 of a steamboat sells Of his sharefor what is the value of the whole boat ?

3 . A farmer having 6 bu . 8 qt . of cranberri es, lost by decay7 pk . 7 qt . What had he left ?

4 . Sold tea for 114% of its cost, and made a profit of 7¢ a

lb. Find selling price .

5 . Ing of an acre of land how many bu ilding lots, each 60ft . by 121 ft. ?

6 . What is the balance of a billof after two deductionshave been made : the first

,10% on the the other , 5%

on what remained ?

7 . I bought a store for a certainsum ,and after paying a tax

of on the cost and 170 more for insurance , I sold it forwhich exactly covered the cost , tax , and insurance .

What was the cost ?

8 . I send to my agent of which he is to lay out

what he caninland at anacre , after deducting his commission

‘

of 5% on the cost of the land . How many acres canhebuy ,

and what is h is commission?

9 . Find the interest of for 2 yr . 3 mo . 7 da . ,at -70 .

Find the bank discount and the proceeds of a 4-months notefor $450,

dated June 10,1889

,and discounted July 25, at

For what sum must a 90-days note be drawn so that when itis discounted at a bank at the proceeds shallbe

10. John Johnston, of Vermont , sent to Smith Bros.

, OfBoston,

to be sold on commission,the following goods : 25 tons of hay ,

2

688 ARITHMETIc.

2 1 . What is the difference between four thousand nine andsevenhundred eighty-six ten-thousandths, and four hundredthousand nine and seven hundred eighty-six millionths.

If 9 of a lb. of coffee costs $3, what will1 lb . cost ?Discover a fraction which , multiplied by 3, equals 3.

22 . A box is 10 ft . long,5 ft . wide , and 4 ft . hi gh . How

many yards of carpeting one yard wide willit take to cover thebox on allsides ?

23 . A note for with interest at is givenMay 10,1887 How much money would pay it June 20, 1890?What should I get June 10 for a 90-days note for $ 375,datedMay 5, and discounted at

24 . What oh } Of -i of% is 4b?

25 . What must be the face of a note,given for 125 days, to

Obtain from a bank,discount being at

26 . Franklin Park contains 525 acres of land . What is itworth at $0371 a sq . ft .

27 . A Owns 12 A . 13 sq . rd . 2 sq . ft .,and B Q of an acre of

land . Howmuch is it allworth at 10¢ per sq . ft . ?

28 . How many inches in116 Of a mile ?

29 . Bought a horse for $90, and sold h im for $95. Whatper cent of gain ? Bought another horse for $95, and sold himfor $ 90. What per cent was lost ?

30. Bought land at per acre , and sold it again at $ 75per acre

,thereby making How many acres were

bought ?

VOLUMES OF PRISMS AND PYRAMIDS. OF CYLINDERS

AND CONES.

1145. Slate Exercises.SUGGESTION — Have the pupils construct of cardboard a hollow square

prism of convenient size, and a pyramid hav ingbase and altitude respectivelyequalto those of the prism. Let them use sand orwater to ascertainhow

VOLUMES. 689

many times the contents of the pyramid must be takento exactly fill the

prism.

Volume q n’

sm or cylinder area of base x altitude.

or cone area of base x 1 altitude.

1 . Find the volume Of a square pyramid,the area of the base

being 9 square feet and the altitude 6 feet.

2 . What is the volume Of a square pyramid whose altitudeis 12 inches, one side of the base being 10 inches ?

3 . The base Of a prism is a triangle whose sides measure 3 ,4,and 5 inches

, respectively . Find the solidity, its altitude

being 10 inches.

4 . The base of a prism 19 feet high is a rectangle whosesides are 9 feet and 13 feet. How many cubic yards does itcontain

5 . Find the volume of a prism whose bases are equ ilateraltriangles

,each side being 4 ft . ,

and the height of the prism being12 ft .

6 . How many cubic feet are there ina stone roller 6 ft . long , 8 ft. in circumference ?

7 . Find the volume of a cone whosealtitude is 18 meters, diameter of base 6meters.

8 . How many gallons of oil(231 cu . in.)willfilla cylindrical

tank 51 ft. high , radius of base 3 ft .

9 . How many pounds of water would the above tank hold ?

(1 cu . ft . oz .)

10. Measure accu rately the interior dimensions of a quart ora pint cup , and calculate its volume .

NOTE. Howmany cubic inches ina quart , liquid measure ?

690 ARITHMETIC.

1 1 . Measure the interior dimensions of a peck or a bushel,and calculate its volume .

12 . Pour a quart or a pint of water into a paper box having arectangular base , and calculate the number of cubic inches of

water inthe box .

What would be the depth of a quart of water ina box whosebase measures 51 by 3 inches?

13 . Find the volume of a cone whose altitude is 18 yd .,cir

cumference of base yd .

14 . Find the slant height of a cone whose altitude is 12

inches,the radius Of the base being 5 inches.

15 . Calculate the convex surface , the entire surface, and the

volume of a square pyramid whose altitude is 24 inches , one sideof the base being 14 inches.

1146 . U. S. CivilService ExaminationforClerkships.

1 . According to the census of 1880, the value ofmanufacturedproducts of Bostonwas Brooklyn

,

Chicago , NewYork , Philadelphia,

St . Lou is, What was the totalvalue of manufactured products of the cities named ?

Express infigures the following numbers

2 . One millionone thousand one hundred one .

3 . Two hundred two millionone hundred thousand one, and

six hundred-thousand ths .

Express inwords the following numbers

4 .

5 .

6 . The cost price of beef is 10143 per pound , and of flour 3113

per pound . A rationconsists of 1 lb . 4 oz . of beef,and 1 lb . 6

oz . of flou r . What will be the cost of rations at theabove rates ?

692 ARITHMETIC.

13 . How much gold at 1111 canbe bought for in

currency ?

14 . Richard Wells, a contractor, furnished to the InteriorDepartment, January 1, 1882, 645 barrels of flour at perbarrel; January 16, bushels of cats at 57 75 per bushel;April4, pounds of bacon at 9¢ per pound ;May 3 ,bushels of corn at 74¢ per bushel; and June 20, 325 barrels of

pork at per barrel. January 31, 1882 , he was paid cash ,$885 ; February 5, $450; April 11, and May 30,

Oninspection , June 25, 345 pounds of bacon and35 barrels of pork were condemned and rej ected ; and on settlement, June 30, 1882, he was charged $ 75 as penalty for failureto deliver goods intime , according to the terms of the contract ,allowed a credit of $65 for cartage , and paid the balance duehim incash . State inthe form below , wi th proper headings ,Wells’s account with the Interior Department.

LUMBERMEASURE.

1147 . Lumber is measured inboard feet. A board foot is 1foot long , 1 foot wide, 1 inch thi ck .

A board 16 feet long,1 foot wide

,1

“

inch thick , contains 16board feet .A board 16 feet long , 9 inches wide , 1 inch thick , contains16 X 1)board feet, or 12 board feet . A board of the same lengthand width , 2 inches thick , contains (12 x 2) board feet, or 24board feet.Inpractice , the term boardfoot is seldom used, the word foot

alone being generally employed .

LUMBER MEASU “ BBS”

: 93

0 F031“

1148 . Fi nd the number of feet (board ach of the

following boards and planks

1 . 16 feet long, 12 inches wide , 1 inch thick.

2 . 14 6 1

3 . 12 8 1

4 . 14 12 1

5 . 16 3 1 :

e. 12 9 2

7 . 14 3 2

s . 16 9 2

9 . 14 6 2

10. 12 8 2

1 1 . 18 9 3

12 . 9 12 3

13 . 20 6 3

14 . 15 4 3

15 . 12 3 3

16 . What is the cost,at $30 per thousand feet, of 15 planks,

each 16 feet long , 9 inches wide , 3 inches thick ?

17 . Find the number of (board) feet of lumber requ ired tofloor a dock 36 feet long , 17 feet 6 inches wide, the planks being21 inches thi ck .

18 . Find the duty , at $1 per thousand feet , on the followinglumberimported from Canada

150 boards, 13 feet long , 8 inches wide , 1 inch thick ;

60 planks, 14 feet long , 9 inches wide , 2 inches thick ;

40 scantlings, 15 feet long, 5 inches wide , 4 inches thick .

94 ARITHMETIC.

19 . At $ 18 per thousand, what willbe the cost of the boardsnecessary to enclose a field 160 yards long , 120 yards wide , witha fence 4 boards high, each board 6 inches wide , 1 inch thick ?

20. The owner of a lot, 25 x 100 feet , wishes to enclose i t bya tight board fence 6 feet high . He pays 25f each for theposts, which are placed 61 feet apart. The boards, costing $25perM. , are 1 inch thick , and are nailed to two continuous horizontalstrips of scantling, 3 inches wide , 2 inches thick , costing

$ 18 per thousand . How much is paid for the material,no

allowance being made for waste ?


1 . (a) Write a negotiable note requ iring endorsement. (b)Whena manis unable to write .h is name , inwhat way does hesigna note ?

2 . Write equivalent fractions for each of the following , andfind their sum : 1, 1, 11, 2131 , 111.

3 . How many bu ilding lots , each 75 ft . by 125 ft . ,can be

laid out on 1 A . 46 P . (sq . rd .) 181 sq . yd . ? Analyze .

4 . A young maninherited a fortune , 1 of which he spent in3 mo .

,and 1 of the remainder in 10 mo .

, when he had onlyleft . What was the inheritance ?

5 . If Ilend my neighbor $200 for 6 mo . ,how long ought he

to lend to balance the favor ?

6 . What number is that which being increased by its half, itsthird

,and 18 more

,willbe doubled ?

7 . Two ships sail from the same port ; one goes due north128 miles, and the other due east 72 miles. How far are theships from each other ? Illustrate .

8 . B and C, trading together, find their stock to be worthof which C owns They have gained 40% on

their first capital. What did each put in?

696 ARITHMETIC.

9 . How many square yards are there ina circular walk,the

radius, AB ,of the inner edge of walk being 10

feet, and that Of the outer edge , AC, being 15feet ?

(Find the difi'

erence betweenthe area of a circle of 15ft . radius , and that Of a circle of 10 ft. radius .)

10. A circular flower-bed 20feet indiameteris surrounded by a walk 5 feet wide . How many square feet ofsurface does the walk contain ?

(If you have to subtract 100 times from 225 times howcan

you shortenthework ?)

11 . How many square inches are there inthe surface Of a

frame 3 inches wide,around a looking-glass 6

inches indiameter ?

(Area x

12 . What is the ratio between the surfaceof the above frame and that of the looking

glass ?(Indicate Operations and cancel.)

13 . What is the area of a walk 5 feet wide around the out

side of a square plot containing 400 sq . ft . ?

(What is the area of the large square, includingthewalk ?)

14 . The ou ter edge Of a walk 5 feet wide ,surrounding a plot of ground , measures 120feet

,the inner edge measures 80 feet . How 0

many square feet does the walk contain ?

(The“average length of the walk is 100 ft. ; that is, i ts

length measured ona line along the center of the walk .)

15 . Find the ratio between the area of a triangle whose sidesmeasure 16

, 30, and 34 feet , respectively , and the area of

another whose sides are 32, 60, and 68 feet.

MENSURATION. 697

SURFACE OF SPHERE.

1151 . Take a wooden hemisphere and drive a tack into thecenter of its curved surface . Commencing at the tack

,carefully

wind a waxed cord about the cu rved surface, inthe way a boy

winds a top . When this surface is exactly covered,cut the cord .

Wind the same cord around a tack driven into the plane surface Of the base of the hemisphere , pressing it closely to the surface . When the latter is entirely covered, just one-half of thecord willbe used .

If a sphere is cut through inany direction , the section madewill be a circle . The section formed when the sphere is cut

through the center is called a great circle.

The above experiment shows that the surface of the hemisphere is equalto that of two great circles of the same sphere .

1152 . The surface of a sphere is equal to that of four greatcircles.

Since the surface Of a great circle Of the sphere is 1 diameterx 1 circumference, the surface of the sphere is 1 diameter x 1circumference x 4 diameter Of sphere x the circumference .

Calling the radius of a circle R, and using the G reek letter1r instead of we have

Diameter of circle 2R.

Circumference of circle 2 11-12.

Area of circle «R’. (1 of 2R x 1 of 2 11

Surface Of sphere 4 1rR’.

698 ARITHMETIC.

1153 . SlateExercises.

16 . Find the surface of a sphere whose radius is 1 inch .

Of a sphere whose diameter is 2 inches.

Of a sphere whose circumference is inches.

17 . At 10 cents a square foot , what willbe the cost of gilding

a sphere 12 inches indiameter ?

18 . Find the ratio between the surface of a sphere 1 foot ind iameter , and the convex surface of a cylinder 1 foot high , thediameter of the bas e 1 foot.

19 . What is the ratio between the surface of the above sphereand the entire surface of the cylinder ?

20. Find the surface of a sphere whose circumference is 20

inches.

1154 . U. S. CivilService. Bookkeeper’s Examination.

1 . The tax assessed by a certaintown is its propertyis valued at What rate per cent is the tax , and howmuch willbe paid by a manwhose property is assessed at

2 . Add these across,placing the totals inthe spaces provided ,

and find grand total

Total.

700 ARITHMETIC.

1156. Extract the cube root Of

We see by inspectionthat 20 5th

; ggbe

tio

zt is

zgietweer

é 20

an a 18,

a . I] 3

tract from the cube of (20

20 8 000. The remainder 3 x 20 remainder

is equal to the second 3 X 29 X 5 300

numbermultiplied by the sum 5’ 25

of three times the square of

the first etc . Usingas a trialdivisor, the second number is seento be 6 orless.

Taking 5 as the second number, we add to the three times theproduct of the first and second and the square of the second

making a totalof Multiplying th is sum by the second number, we

get wh ich is equalto the difi'

erence between and The

second number is, therefore, 5, and the cube root of is 25.

1“

40 8 8 7

512

19 200

960

Am . 48. Am . 87.

Inthe last examplewe point ofi’ three places, beginning at the right, and

find the greatest cube inthe first period , placing its cube root as the firstfigure of the answer.

root Of the following

6 .

7 .

8 .

9 . m10. 1941

MENSURATION. 701

VOLUME OF SPHERE

1158 . Cut up a sphere (a round potato ,for instance) into a number of

smallpieces, passing the knife ineach case through the centerof the sphere.

Each piece is a solid,having for its base a portionof the surface of the

sphere, and for its altitude the radius of the sphere.

Whenthe pieces become very numerous, the base of each may be con

sidered a plane, and the solid a pyramid . The volume of each pyramid is

equal to the base x 1 altitude ; and the totalvolume of all, which is the

volume of the sphere, is equalto the totalsurface of allthe bases, which isthe surface of the sphere, multiplied by 1 altitude, that is, 1radius .

Surface of sphere 4 1rR'

,

therefore , volume of sphere 4 1rR’X 1B 111

-R“.


1 . Find the volume of a sphere whose radius is 3 inches.

2 . If the diameter of a sphere is 3 inches, what is its volume ?

3 . What is the ratio between the volumes of two sphereswhose diameters are 1 foot and 2 feet, respectively ?

4 . Find the ratio between the volume of a sphere 1 foot in

diameter,and that of a cube whose side is 1 foot .

702 ARITHMETIC.

5 . The radius of a sphere is 18 inches. What is the circumference Of a great circle ? The surface ? The volume ?

6 . What is the weight of an iron cannon-ball 12 inches indiameter , considering the weight of a cubic foot of water asounces, and considering iron times as heavy as water ?

7 . Find the ratio betweenthe volume of a sphere 4 inches ind iameter , and that of a cylinder 4 inches inaltitude , radius of

base 4 inches.

NOTE. Indi cate the volume of each , and cancel.

8 . A manhas a cubical block of hard wood,its side measur

ing one foot , which he wishes made into a sphere one foot indiameter . What decimalpart of the block is cut away ?The volume of the sphere is abou t what fraction of the volume

of the cube ?

9 . How much would be cut away inmaking a cylinder one

foot indiameter and one foot high , from the above cubicalblock ?Abou t what fraction Of the volume of the cube is the volume

of the cylinder ?

1160. Chicago Public Schools.

1 . A generalwished to remove pounds of provisionsfrom a fortress in9 days. It was found that in6 days 18 menhad carried away but 18 tons. How many men would be re

qu ired to carry away the remainder in3 days ?

2 . A , B,and C enter into partnership . A puts in$500 for

4 months,B $400 for 6 months, and C $ 800 for 3 months ; they

gain$ 340. Find each man ’

s Share Of the gain.

3 . A school-room is 40 feet long, 30 feet wide , and 14 feethigh . Find the difference between the length of a diagonaldrawnon the floor and one drawn from the floor to the ceiling .

4 . A merchant bought worth of goods for cash,and

sold them on 4 months’ credit at 15% advance . He got the noted iscounted at 6% to pay the bill. Howmuch did he make ?

704 ARITHMETIC.

14 . Find the cost of a draft for payable in60 days,when exchange is 1% discount, and interest 5 per cent.

15 . West Point is inlongitude 74° and St . Louis inlongitude 90° When it is 4 o ’clock RM. at West Point, whattime is it at St . Lou is ?

16 . If 108 men can bu ild a fort in121 days of 121 hours each ,inhow many days can 84 men bu ild it by working 101 hoursdaily ?

17 . A can do 1 of a piece of work infour days, and B can do

1 Of it in5 days. Inwhat time canthey do the whole worktogether ?

18 . A man desires to secure anannualincome of $650 for hisdaughter. How much 5 per cent stock must he buy at par todo it ?

19 . A square is inscribed ina circle whose diameter is 84inches. Find the area of the four segments of the circle outsideof the square .

20. Find the difference between the volume of a cylinderwhose diameter and height are 12 inches , and the volume of asphere whose diameter is the same .

1161 . NewYork Public Schools. Examinationfor Admissiontothe City College.

1 . (a) Define a common fraction ; (b) a compound fraction ;(0) a proper fraction.

(d) G ive the rule for the multiplication of one decimalfractionby another .

(e) What is discount ?

2 . Add the following fractions

as 9 of t of a ,53 1°

(b) Find the value Of the following

a le


3 . (a) Write ina decimalform

(b) How do you most easily multiply a decimalby 10?

(c) Divide 16 by 4 ; by 400 ; by .004 .

(d)Multiply 41 by .0006.

4 . Inmultiplying by 1 explain why canceling the 3 inthenumerator and the denominato r willgive a correct result.

5 . Divide by .5, and explain the reason of the rule for

pointing off the decimalplaces inthe quotient .

6 . What sum Of money put at interest for 2 years 9 monthsand 9 days

,at

"

6 per cent per annum,willproduce

7 . A certain cistern canbe filled by one pipe in10 hours,

by another in6 hours,and canbe emptied by a thi rd in5 hours.

Inhow many hours canit be filled if allthree pipes are openedat once ?

8 . Animporter bought inFrance pieces of a certaingoods at $40 per piece . The duty paid on importing the goodswas 50% of the cost

,and the freight and other charges were in

all For how much per piece must he sellthe goods soas to make

9 . Two men start from two towns 105 miles apart and walktoward each other . They meet at the end of 15 hours. The

first has traveled 3 miles per hour . At what rate has thesecond traveled ?

10. If 10men,working 8 hours per day , can build a certain

wall in6 days, how many hours a day must 12 menwork tobuild the same wallin4 days ?

11 . (a)What ismeant by the ratio of one quantity to another ?

(b) What is meant by the G reatest Common Divisor Of severalnumbers? G ive anexample .

(0) What is meant by a minuend ? By a quotient ?

(d)When is a common fraction said to be inits lowest terms ?

706 ARITHMETIC.

12 . (a) Reduce to its simplest form the following expression

21X 1 X of 11.

(6) Find the valu e of the following :

i i t 3

(0) Reduce 11 to a decimalfraction .

13 . (a) Find the value of 7 11 of a mile inthe lower denominations.

(b) What is the effect upon the . value of a fraction if we multiply the denominator by 3 ?

(0) Explain the reason of the last answer .

14 . (a) Write indecimalform the followingTenten-thousandths.

One thousand twenty-four ten-millionths.

Thirty-two and four thousandths.

(6)Multiply .0036 by

(0) Divide by .0016.

15 . (a) If a cipher is added at the right of the decimal,what

effect has this on the value of the decimal?

(6) Explainthe reason of this.

16 . (a)What part of 1 is 1?

(b) What is the easiest method of dividing a decimalby 10?

(0) Reduce 20 square rods to the dec imalof anacre .

17 If by selling a house for a builder gains 121 percent

,what per cent would he have lost by selling it for

18 . A,B,

and C begana partnership onJanuary 1. A put

in B,and C, A t the end of six months

B put in more , but A withdrew At the end of

the year they had on hand incash,and goods valued at

At this time the property was divided and the firm

708 ARITHMETIC.

25 . (a) Write indecimalform

Ninety millionths ; thirty ten-thousandths ; ten and twenty

five thousandths .

(6) Divide .064 by .000016.

(a) Add .003 ,

(d ) Subtract from

26 . (a) What is the effect on the value of a decimalof movingthe decimalpoint two places to the right ?

(6) Explain the reason of this.

(0) Inthe multiplication of two decimals, how many decimalplaces are to be pointed off inthe product ?

(d )Explain the reason of this rule .

27 How much is 51 tons of coalworth , if 17? tons are Worth

$ 100?

28 . A person expended 16 per cent of all he was worth inbuying 20 per cent of the stock of a mining company . If theentire stock of the company sold for what must theperson have been worth

29 . Four men undertook to do a piece of work in 18 daysand worked at it 6 hours a day for 10 days, when they foundthey had finished only i f of it . How many more men did theyhave to engage , inorder to finish the job by the time agreedupon

,provided they allworked thereafter for10hours each day ?

30. A provision merchant bought 100 barrels of apples at a

farm house at $ 1 per barrel, and paid 5 per barrel to havethem taken to the railroad station . Then he paid $50 freighton them to NewYork , and $ 20 cartage inthe city . They weresold at once for $ 3 per barrel, but the commission merchantcharged him 10% commission onthe sale . Also , when some of

the barrels were opened,the apples were found to be damaged

,

and he had to repay the purchasers $ 20 on account of this.How much did he gain inall?


3 1 . (a) What is an improper fraction ?

(6) What is a mixed number ?

(c) What is interest ?

When are four numbers said to be inproportion ?

32 . (a) Divide 1§ byg(6) Reduce the following fractions to equ ivalent fractions hav

ing their least common denominator :

,amg of ; of? it Of § .

(c) Add together the results obtained in(b).

3 3 . (a) What fraction of is i ?

(6) Find the value of 11; of a. mile inrods, yards, feet, and

inches.

(0) Reduce 356 to a decimalfraction .

34 . What is the effect of multiplying the numerator and thedenominator of the same fraction by 4 ? Explain the reason .

35 . (a) Write indecimalform the following

Tenand ten hundredths ; nine millionths ; thirty thousandths ;thirteen hundred and forty-two hundredths .

(6) Add .003 ,

(0) Divide .048 by

3 6 . (a)Multiply .26 by .0035.

(6) Divide .006 by 100, by the shortest method .

(0) Explain the reason of this method .

3 7 . If 4 men,working 8 hours per day , can mow a meadow

in3 days,how many men , working 9 hours per day, can mow a

meadow three times as large in4 days ?

3 8 . A , B,and O entered into partnership for one year. A

put in B,

and 0,At the end of six

710 ARITHMETIC.

months,A withdrew and 0 put in more . The

profits at the end of the year were What was eachman ’

s share ?

3 9 . A bought merchandise from B for and gave hisnote for six months

, without grace , with interest at Justwhen the note was due

,he sold the goods to C for tak

ing C'

s note at 3 months , without interest , which his bank discounted for him the same day . After paying his note to B

,how

much money had he remaining ?

40. A drover bought a drove Of 50 cattle for Hesold k of them at a gain of 10% on the average price , and k Of

them at a gain of Half of the remainder,however

,were

so injured ina railroad accident that he could Obtainonly $ 100for them . For what price apiece must he sell the rest so thathis totalloss shallbe 100?

CUBE ROOT.

Find the cube root of

When the root containsmore than two figures , continue

,as shown inthe accom

panying example, taking for

div isor three times the square3 X 202

of the first two figures con 3 X 20x 1

sidered as tens, plus three times 1

’

the product of the first two 3 X 2109:

figures consi dered as tens by 3 X 210X 5the th ird figure, plus the square

of the th ird figure.

52

677 375

Find the value of the following

712 ARITHMETIC.

10. Find the entire surface of a cylinder 10 inches long, and8 inches indiameter .Find the number of cubic inches inthe same cylinder .

NewYork Public Schools. Admissionto NormalCollege.

1 . Define the following terms employed inarithmetic

Quantity,number

,abstract number, multip lication, p rop ortion.

Explain the reason formultiplying the second and third termstogether and dividing by the first term insolving an example insimple proportion .

.Li fi . f i i éf2 .

fr ois what part o

i 0f 5i?

3 . Columbus i s 83° 3 ' west longitude ; and when it is 37 min.

33 see. past 1 P .M.

,it is 11 o ’clock A .M. inSanFrancisco . What

is the longitude Of the latter city ?

4 . Divide thirty-two hundred-millionths by sixty-four tenthousandths.

5 . A,B

, and Ogained by speculation of which A ’

s

share was twice as much as 0’

s, and B’

s five times as much as

C’

s. How much did each gain?

6 . A manowes $600, Of which one-third is to be paid inoneyear, and the remainder in two years . What is the presentvalue

,money being worth 6 per cent ? (Art .

7 . I bought a watch for $ 120 and set su ch a price onit that,

after falling $ 12, I still made 15 per cent on the purchase .What per cent did I abate from the asking price ?

8 . A pole was broken 52 feet from the bottom,and it fellso

that the end struck 39 feet from the foot . Required the lengthOf the pole.

9 . Extract the cube root of 225} to two places of decimals.

10. Sold a horse so that i of the gain equaled 135 of the cost .What was the gain per cent ?


tions. OralProblems.

1 . If i yd . costs what will yd . cost ?

2 . If 3 men can do a piece of work in4 days,how long will

it take 24 men to do it ?

3 . What principalat interest for 5 years, at 6 per cent, willproduce 12

,simple interest?

4 . A stack of hay willkeep a cow20weeks,and a horse 15

weeks . How long willit keep them both ?

5 . Divide 7} lb. Of tea into two parcels,one of which is If lb .

heavier thanthe other.

6 . Sold a cow for $24, losing thereby 4070 of the cost price .Had I sold her for advance on the cost, what would Ihave received for her ?

7 . What will460 lb. of tea cost at $ 4 8 per lb.

8 . If 12 oz . of bread are destroyed inmaking a gill of

whiskey, how much willbe destroyed inmaking a gallon

9 . If the weight of air is 15 lb. on the square inch, what is

it on the square foot ?

10. Seven is three-fifths of what number ?

11 . What is the value of 960 lb . of wheat at per bu . of

60 lb . ?

12 . At what rate will$400 make simple interest,in1 yr. 3 mo . ?

13 . What is the brokerage on at

14 . What will feet of lumber cost $25 perM.

15 . A and B are partners ; A puts in16, of the stock, and B

the remainder ; B’

s gain is Find A’

s gain .

714 ARITHMETIC.

1167 . Public Schools of British Columbia. ExaminationQuestions. Slate Problems.

1 . Find the value inU. S . money Of of 3} gu ineas .

(1 gu inea 21 sh illings. £ 1

2 . Reduce to its simplest form

{t of fi - g of fi;

3 . What is the difference between the compound interest onfor 3 years, at and on for 14 years, at the

same rate ?

4 . A trader gets 500 barrels of flour insured for 75% of thecost , at 2} per cent, paying premium . At what price perbarreldid he purchase the flour ?

5 . A cando a piece Of work in27 days, and B in 15 days ;

A works at it alone for 12 days,B thenworks alone for 5 days

,

then C finishes the work in 4 days . Inwhat time could Chave done the work by himself?

6 . If 72 mendig a ditch 20 yd . long , 1 ft . 6 in. broad,4 ft .

deep , in 3 days Of 10 hours each , how many men would berequ ired to dig a ditch 30 yd . long

,2 ft. 3 in. broad

, and 5 ft .

deep,in15 days of 9 hours each ?

7 . A manhas beenreceiving 12% on his capital inCanada .

He goes to England to live , and invests it inthe 3 per cents , at944, and h is income inEngland is What was his income inCanada

, £ 1 being equalto $48 63?

8 . A t what price must anarticle that cost $30 be marked so

that after deducting 40% from the marked pri ce,30% profit

may be realized ?

9 . Wri te a ninety-days promissory note for which you should

get $ 240 at the bank,discount being

716 ARITHMETIC.

A vessel is worth For what sum must the ownerinsure her, so that, if she is lost, he may receive both the value

of the vessel and the sum paid for insuring her , the rate of

insurance being 1§ per cent ?

6 . Extract the square root of and find the value

V2 1

vs 1

7 . A contractor undertook to complete a piece of work in84days, and engaged 30men to do it. After 40 days

,he put on 20

more men,and thenfinished the work 2 days too soon . How

many days behindhand would he have been if he had not engaged the additionalmen?

to four decimalplaces.

8 . The walls of a room are to be papered . The room is 23feet 8 inches long

, 15 feet 10 inches wide , and 11 feet 9 incheshigh ; there are two windows, each 9 feet 6 inches high and 5feet wide ; a fireplace , 4 feet 6 inches high and 6 feet wide ; anda door, 7 feet 6 inches high , 3 feet 6 inches wide . Find the costof the paper required

,at per rollof 12 yards , the paper

being 26 inches wide .

9 . Divide $336 between two persons so that one may haveas much as the other.

10. A manbought a horse for $365, giving his note due in30 days. He sold the horse , taking inpayment a note for $435,due in4 months . What was his gain per cent ?

(Interest at 6 per cent per annum.)

1 1 . Inwhat time will the interest on £57 18 . 8d . amount to£2 118 . fi d . at 74} per cent per annum ?

12 . Which is the greater, the square root of 281 or the cuberoot of

13 . A contractor engages what he considers a sufficient number of men to do a piece of work in84 days ; but he ascertainsthat 3 of his men cando respectively 4, 1, and 4 less than an


average day ’

s work , and two others 4 and 115 more . Inorder tocomplete the work inthe given time , he procures the help of 17

additionalmen for the 84th day . How much less or more thanan average day ’

s work on the part of these 17 men is requ ired ?

14 . The floor of a hall is 260 feet long and 93 feet wide .

Find the cost of covering it with carpet and oil-cloth ; the Oilcloth to be laid along the sides and ends a yard wide

,and the

carpet to extend 6 inches over the oil-cloth . Carpet, a

yard , 38 inches wide ; Oil-cloth , 90 cents a square yard.

15 . A mancanwalk from A to B in1 hour less than a secondman

,and

,when they start from opposite ends of the distance and

walk incontrary directions, they meet at a point which is twiceas far from A as it is from B. If the first manwalks 5 milesper hour

,what is the distance between A and B?

16 . A trainleaves B at 9 A .M.,and runs to C at the rate of 15

miles an hour ; and another trainleaves A at noon, and , runningthrough B to C at 25 miles anhou r, reaches C half anhour laterthan the train from B. Find the distance from A to C,

the distance from A to B being 15 miles.

1169 . U. S. NavalAcademy. Competitive Examinationfor Appointment as Cadet Engineer.

1 . Add 3101; to the difference between .014 and , gm and

reduce the result to a decimal(six places). What per cent of£ 31 178 . 4d. is £5 19s. 6d . ? Reduce .2083 (the 83 is repeatedindefinitely) to a common fraction .

so“)2 . What will it cost to make a road 104 rods long and 60

feet wide , the land costing $ 154 an acre , grading at the rate of

$200 a mile , and fencing one side at 50 cents a yard ?

3 . Find the square root of and the value‘ ofx/B V5 vs

VE—v

’

é meach to six places of decimals.

718 ARITHMETIC.

4 . The French gram is the weight of a cubic centimeter ofwater ; a kilogram is grams. Find the weight inkilograms Of ananchor weighing pounds

,given that a cubic

foot of water weighs ounces,and the centimeter is .3937

Of aninch .

5 . In loaning money , banks deduct the interest , includingthree days Of grace , inadvance . For what sum shalla personwho wishes to u se give his note for 60 days, interest 8per cent per annum ?— A personhaving a note for due

at a bank ,pays in$200, and requests that a new note may be

made payable at 90 days, the sum paid serving to pay the interest on the new note as well as to reduce his debt . For whatsum shallthe note be issued

,the rate of interest being 8 per cent

per annum ?

1170. U. S. Military Academy, West Point. Examination forAdmission.

1 . Which is the greater,and by how much

,orH?

2 . What number subtracted 88 times from will leave013 as a remainder ?

3 . What is the value Of a pile of wood 127 feet long, 4 feet

wide,and 3 feet 8 inches high , at $ 7 a cord ?

4 . Reduce 3 acres 107 square rods 27 square yards 7 squarefeet 23 square inches to square inches .

5 . Divide among Thomas, Richard , and Henry insuch a way that Henry shall receive 3 cents for every 5 centsthat Thomas gets , and Richard shallreceive 2 cents for every 5cents that Henry gets.

6 . Reduce 272 liqu id quarts to dry quarts.

7 . A pipe discharging 3 gallons 1 pint a minute fills a tub

in4 minutes 20 seconds. Another pipe discharges 83 quarts a

minute . If both pipes discharge together into the tub,howlong

willthey take to fillit ?

720 ARITHMETIC.

GOVERNMENT LANDS.

1173 . Insurveying government lands, a line is runeast and west,called the base line, and one perpendicular to it, called the principal

Parallellines are runnorthand south , and east and west, 6miles apart, forming squares,

called township s. The row of

townsh ips adjoining the prin

cipalmeridianis called Range 1

East or West,according to its

location. The rowof townshipsnorth of the base line is called

Township 1,North ; the row

above, Township 2, North, etc .

The townsh ip inthe diagrammarked by a star is designated3 T. S. ,

R. 2 E. (th ird townsh ipsouth of base line, inthe second range east of the principalmeridian).

1174 . A township ,which contains 36 square miles, is div ided into sec

tions one m ile square, numbered as inthe diagram ,No . 1 being found at

thenortheast corner. Each sectioncontains 640 acres.

TOWNSHIP

1175. Sections are div ided into half-sections (320 A .)and quarter-sections (160 and the latter are subdiv ided into half quarter

-sections(80 A .) and quarter quarter-sections (40

METRIC SYSTEM. 721


1 . Find the cost of the SW . 1 of the N. 4 of sec. 13 , T. 7

N.,R. 4 E.

,at $ 18 71} per acre .

2 . What will be the cost of fencing,at 75f per rod, the W .

i» Of the NW . 1 Of sec . 36?

3 . Mr. Thompson owns sec . 1,and his brother owns sec . 30 of

the same township . What is the length of the shortest linebetween the boundaries of the two farms?

4 . A road runs east and west between townships 4 and 5,south . Another road runs north and south between R. 7 and 8

east . How far is it by road from the north-east corner of T. 5 S .,

R. 10W ., to the north-west corner of T. 7 N.

,R. 8 E. ?

5 . How many feet of boards, 6 inches wide , would be neededto bu ild an Open fence

,4 boards high

,around the N. 1} Of the

SW . 1 of sec . 16 ?

6 . The owner of secs. 19 and 20 has sold the W. h of NW . 1Of sec . 19 ; also the N. 1, the N. i of S E. 1, and the S E. 1 of

the SE . 1 of sec . 20. Draw a map Of the land he stillowns, andcalculate its area .

METRIC SYSTEM1177 . The metric system

,which is used innearly all the

countries of continentalEurope,is based upon the meter. The

length of the meter is one ten-millionth part of the length Of themeridian from the equator to the poles —about inches.

1178 . The subdivisions of the meter are denoted by the Latinprefixes milli centi (Th ), deci For the multiples,the G reek prefixes deka hecto kilo andmyria are used .

1179 . It will be noticed , in the table below , that small

letters are used for the abbreviations of the Latin prefixes of the

722 ARITHMETIC.

subdivisions,and capital letters for the G reek prefixes Of the

multiples.

The following is the table of

1180. Measures of Length.

10millimeters (mm.) 1 centimeter (cm.)10 centimeters 1 decimeter (dm .)10 decimeters 1 meter (m .)10meters 1 dekameter (Dm .)10 dekameters 1 hectometer (Hm.)10 hectometers 1 kilometer (Km .)10 kilometers 1 myriam

-eter (Mm.)

1181 . The units of this table incommon use are the centimeter

,the meter

,and the kilometer .

1182 . A personwho wishes to buy 124 meters of cloth , would not ask

for 1 hectometer 2 dekameters 4 meters, any more thana NewYork merchant would tella personwho owes h im that h is b ill is 3 eagles 8

dollars 7 dimes 5 cents.

Long d istances are expressed inkilometers. The thickness of

wire is giveninmillimeters.

1184. Problems.

1 . What will be the cost infrancs Of 380 m . 75 Of dressgoods at 2 f. 60 per meter ? meters francs .)

2 . How many square meters ina piece of carpet 26 m . 50

long, 85 cm . wide ?

3 . How many square meters ina circle whose diameter is15 meters

4 . Anare is a surface 10 meters long , 10meters wide . Howmany ares ina field 135 meters long

,69 meters wide ?

5 . Find the area in ares Of a right-angled triangle whosebase is 245 meters

,hypotenuse 875 meters.

724 ARITHMETIC.

15 . How many pounds ina kilo,when a cubic foot Of water

weighs oz . ? (See problem

16 . What would be the circumference of the earth inmiles ifthe meter measured 40 inches ?

(The motor ism ofwhat part of circumference ?)

17 How many square yards ina square meter ?

18 . How many acres ina hectare ? (See problem

19 . How many rods ina hectometer ?

20. How many cubic feet ina stere ? (See problem

21 . How many troy grains (7 ,000 to av . 1h.) in a gram ?

(See problem

22 . How many kilometers ina mile ?

1187 . Measures of Surface.100 sq. mm. 1 sq . cm .

100 sq . cm. 1 sq . dm.

100 sq . dm. 1 sq . m. sq. yd .

1188. The square meter is the principalunit of surfaces, such as walls,ceilings, floors, etc.

100 centiares (ca .) 1 are (a .) sq . yd.

100 ares 1 hectare (Ha.) acres.

1189 . The are is the principal unit Of surface of smallplots of land.

The area of a farm is . expressed inhectares, of a country insquare kilo

meters.

cu . mm . 1 cu . cm.

cu . cm . 1 cu . dm.

on. dm. 1 cu . m . cu . ft.

1191. The principalunit is the cubic meter.

METRIC SYSTEM. 725

The stere (cubic meter) is used formeasuringwood .

10 decisteres (dst.) 1 stere (at ) cu . ft .

10 steres 1 dekastere (Det.)The store is the only unit used .

Dry and LiquidMeasures.10milliliters 1 centiliter10 centiliters 1 deciliter Dry . Liquid.

10 deciliters 1 li ter (l.) 908 qt. qt.

1 dekaliter pk . gal.

10 dekaliters 1 hectoliter bu . gal.

10 hectoliters 1 kiloliter10 kiloliters 1 myrialiter

The liter and the hectoliter are the principalunits.

Table ofWeight.

10m illigrams (mg.) 1 centigram10 centigrams 1 decigram

10 decigrams 1 gram (gr.)10grams 1 dekagram

10 dekagrams 1 hectogram10 hectograms 1 kilogram (kilo) lb.

10kilograms (Kg ) 1 myriagram

10myriagrams 1 *

qu intal

10 qu intals 1 tonneau (ton)

The kilo is the ordinary unit. Heavy articles are sold by the

tonneau .

23 . What willbe the duty on 150 liters of wine at 50fi' per

gallon

24 . Paris is 466 Km . from Lyons. How many miles apartare the two cities ?

25 . Find the duty , at 89' per sq . yd . ,

on360 meters of dressgoods 72 centimeters wide . (Cancel

726 ARITHMETIC.

Kansas City Public Schools. ExaminationQuestions.


1 . A manhas two su its of clothes and an overcoat. The firstsu it is worth $ 12 . The first su it and the overcoat are worthtwice as much as the second su it . The second suit and the overcoat are worth three times as mu ch as the first su it . What is

the value of the overcoat ? Of the second su it ?

2 . A lost $ 22 . He then found .1. as much as he had remaining,

and then he had i as much as he had at first . How muchhad he at first ?

3 . What time is it , if 1» of the time past midnight equals thetime past noon ?

4 . At what time between 3 and 4 is the hour hand 5 minutespaces inadvance Of the minute hand ?

5 .

126 of A

’

s age equals of B’

s age . The sum of their agesis 30years. How old is each ?

6 . A and B are partners. They gain$ 7203 A owns 3. of thestock ,

lacking $ 30,and gains $525. What was the whole stock ?

What was the share Of each ?

7 . A bought sheep for $80. 5 Of them died . He then soldof the remainder at cost

,and received $ 40. How many did

he buy ?

8 . 1 Of A’

s age equals 7} Of B’

s age . B is 10 years Older thanA . G ive the ages Of each .


1 . Add 24 lb . 10 oz . 17 pwt . 22 gr. ; 16 lb . 19 pwt . ; 10 oz .

15 pwt . 21 gr. 45 lb . 9 oz . 18 gr. ; and 13 lb . 11 Oz . 18 pwt.23 gr .

2 . A manwas born Aug. 31, 1855. How Old is be tod ay ?

3 . Divide 19 mi . 180 rd . 2 yd . 9 in. by 7, and multiply thequotient by 12 .

CHAPTER XV.

ALGEBRAIC EQUATIONS.— TWO UNKNOWN QUANTITIES.

THREE UNKNOWN QUANTITIES — PURE QUADRATICS.

AFFECTED QUADRATICS.

ADDITION OF ALGEBRAIC

Exercises.

2 . 6 hundredths 4 . 3 f 5 . 7 a:

8 hundredths 4 x

10 hundredths 2 a:

12 hundredths 9¢ 5a:

hundredths ?f ?x

2 a 7 . 3 x 8 .- 5xy 9 . 9 abc 10. 24 am

4 a 71 4 x 4 my 15 abc 5xyz

6 a 5a: my 6 abc xyz

7 a + 101: 2xy abc 15xyz

— 19 a 3: - ?xy ? abc ? arg/s

1200. In the quantities 2a ,3 x

,5xy,

15abc,the numbers 2

,

3,5,15

,are called coefficients. Whenno coefficient is expressed

,

1 is understood . Thus, abc 1 abc.

Where no signis expressed , is understood .

1201 . What a person has may be represented with a plussign placed before the amount ; debts may be shown by a

minus sign placed before the amount.

728

ALCEBRAIC EQUATIONS. 729

A has $500; B owes $300. If they unite their fortunes, whatwillthey be worth together ?

$300

$200

Both together are worth $ 200.

The sum of 500 and 300 is 200.

1202. If A bad $300 and B owed $500, the would be

$200 indebt.— t soo.

1203 . Add

1 .—2 a 2 . 7x 3 .

— 5xy 4 . 9abc 5 .

4a - 4x 4x31 15abc

6a 2 3: my 6abc

7 a 5x 222 abc

— 5a 6ar — ? xy

1204. Canyou give the rule for addition where the quantitieshave different signs ? Which Sign does the sum take ?

1205. Add

6 . 3 x+ 14 , — 7x+ 9, — 23 , 4x — 5, - 2x, and 3x+ 11.

3 x+ l4— 7x+ 9

23

4x 5

—2x

3x + 11

4a + 3x,— 2a ,

— 7x — 3 a ,— 5x,

— 9a + s .

— 3 b+ c, 4a + 6b, 5b— 9c, — 3 a ,— 2a — 3 b+ 4 c.

1x — 8 , — x + 4, —1x — 3 , 7x+ 16, - 5x - 10.

4x+ 23 , — 8x+ 21, —1x + 11,— x+ 5, 9z —3 .

730 ARITHMETIC.

SUBTRAOTION OF ALGEBRAIC QUANTITIES.


1 . The thermometer inthe morning was 33 degrees, at noonit was 52 degrees. What was the difference intemperature ?

2 . InDecember the thermometer was 10 degrees below zero .

InJuly it was 90 degrees above . What was the difference intemperature

3 . Two cities are inthe same latitude . One is in34° eastlongitude , and the other in17

° west longitude . What is theirdifference inlongitude ?

4 . What is the difference inlongitude between two cities onthe equator, one being in56

° west longitude , and the other in47

° west longitude ?

5 . A boy makes 40 one day and 50¢ the next . How doeshe stand at the end Of the two days ?

6 . How would he stand if he made 4015 one day and lost 50¢the next day ?

7 . A man traveled from the townM,60 miles due north

,

and then traveled 50miles due north . How far is be from his

starting-point

8 . One day a mangoes 50miles due north ; the next day hetravels 70miles due south . How far is he then from his starting

point ?

9 . OnMonday A is worth $250 ; onTuesday he is worth

$ 150. What has he lost ina day ?

10. A manhas $ 150 Jan . 1. Feb . 1 he owes $ 250. Whathas he lost ina month ?

1207 . The degrees above zero on a thermometer may be indicated by a plus Sign those below

,by a minus sign

732 ARITHMETIC.

REM0VING PARENTHESES.

1210. From 84 take the difference between 49 and 25.

84 (49 25) what ?

Would the result he the same if we should write the above

84 49 25?

What sign must be changed ?

1211 . Write the following without parentheses

1 .— 16) 4 . (17 — 8)

2 . 92 5 . 75+ 4 x (15

3 . (43 6 . 75

1212 . Is there any change made inthe signs Of the first ? Inthe signs

of the second ? Of the th ird ? Of the fourth ? Of the fifth ? Of the sixth ?

1213 . Solve the following equations . Prove the correctnessof your answers.

1 . 6(2 :r

NOTE. 6(2 z —5)means 6 times (2 z or 12 33—30.

2 . 4 .- x)

s. s . l5(x — 16 x)

c . ss — 9 A)= Icz

Removing the parenthesis, we have38 —11 + 9 z - 10a:

Transposing, 9 3 loa n —38 11

z 27

Bringing a: to the right side of the equation, and 27 to the left side,we have

27 x

Inpractice ,however, whenthe result is such as the above, z

the signs of both members are changed , and the result becomes2 - 27

ALCERRAIC EQUATIONS. 733

1) 19) 3(:c

6(2 x — 5)— 5x = 12

5x 2: 5)

- 19.

2

18 — 6 24 — 4

2 5

Clear of fractions by multiplying both members Of the equationby 10

,and observe which signmust be changed to preserve the

equality.

1214 . 2

When a: 6,the above may be written

3 23— 6 4 x — 4

2 5

Clearing of fractions,

15x — 30 (8 x— 8)= 2O

Removing the parenthesis,

15x — 30— 8 x + 8 = 20

Transposing, 15x 8 a: 20 30 8

7 a: 42

a: 6

NOTE. The horizontalline betweenthe numeratorand the denominatorof the foregoing fractions has the effect Of a parenthesis , the entire quantityabove the line being divided by the number below.

18—6 24—418—6 2

1 of (3 :e—6)

ARITHMETIC.

x— l x — 2

2 3

x — l x — 2

2 3

x — l x — 2 x — 32

2 30

2 x — 5+

x 7 5x — 3

2 6

4 x + 5 x + 2

6 3

x 1x + 3§ x

2 x= 3 + 21x

— 1@

x x 27 a: x

— 120 9i s6

x + 1 4

a: :L‘

5 x — 62 3x +

2

— e + a

52 + 9 x

736 ARITHMETIC.

9 . A man has a cask of 60 gallons'

capacity . He draws ofi'

one-fourth of its con tents, and then fills it . If it takes 24 gallons to fillit , howmany gallons did the cask originally contain ?

10. A number is divided by 3 , and 40 is subtracted from the

quotient, leaving a remainder of 104 . What is the number ?

11 . The d ifference between two numbers is 430. When thegreater is divided by the less, the quotient is 4 , and the remainderis 76. What are the numbers ?

Let wa less .M y “ 76

less less

12 . A person pays 103 with 29 $2 and $5 bills . How manyare there of each denomination ?

13 . A father is 30 years older than his daughter . In4 years,

his age willbe four times her age. What are their present ages ?

a: and a: 30 present ages . a: 4 and a: 34 ages 4 years later.

14 . The product of two numbers is 180. If the smaller number be increased by 3 , the product of the two numbers will be225. What are the numbers ?

smaller greater.

15 . A man ’

s wages are $ 1 per day more than his son'

s.

For 33 days' work ,the father receives $ 12 more than the son

earns in40 days. Find the wages of each .

16 . The sum of two numbers is 47 ; their difference is 17 .

What are the numbers?

17 . A mother is 41 years old . Her son ’

s age is 5. Inhowmany years willthe son ’

s age he 1 of his mother’s?

TWO UNKNOWN QUANTITIES.

1 . Ipaid a dollar fortwo 2515balls and five bats. How mu chdi d Ipay apiece for the latter ?

ALCEBRAIC EQUATIONS. 737

2 . When three times one number is added to five timesanother , the sum is 84. If the second number is 12

,what is the

first number ?

3 . A girl paid for 1 pound of tea and 21 pounds of

cofi'

ee. The coffee cost 20f per pound . What was the price of

the tea per pound ?

4 . A man sold pigs at $5 each and lambs at $8 each , receiving $42 . He sold 4 lambs. How many pigs did he sell?

5 . Four times a father ’s age added to twice his daughter ’sage amounts to 180 years. The girl is 10 years old . What isthe father ’s age ?

6 . Eight peaches and seven pears cost 44A The peachescost 2¢ each . What is the cost of a pear?

7 . Two pi eces of cloth and eleven pieces of silk contain 152yards. There are 10 yards ineach piece of cloth . How manyyards ineach piece of silk ?

8 . Two-thirds of a yard of linen and three-fourths of a yardof lace cost 40£5. The price of the lace is 32¢ a yard . Find theprice of the linen .

9 . Three and one-half times one number added to four andone-third times a second number equals 60. The second numberis 9. What is the first number ?

Find the value of the unknown quantity

1 . 8 x + 7 y = 44. Whenx= 2, find the value

2 . 3 y + 5 z = 34. Find the value of z ; y= 3 .

3 . 2 x + 11 2 = 152 . z= ?

4 . 14 z + 7 y= 98 . a: 31 ; y= ?

5 . 1x + z = 32.

738 ARITHMETIC.

6 . 9x — 25y = 8 . x= 12.

7 . 31y + 41 2 = 60. z = 9.

8 . l6 x — 19 z = 49. z = 5.

9 . 7 y y==61.

10. 32 a: 50y y 20.

1219. A boy gave 17¢ for 3 lemons and 4 oranges, anotherboy paid 25fi

‘for 3 lemons and 8 oranges. How much did the

lemons cost apiece ?

a: cost of lemons 3 x+ 4 y = 17

y : cost of oranges 3 x + 8 y = 25

Subtracting (1) from (2) 4 y 8

The oranges cost 2¢ each y 2

How much apiece was paid for the lemons ?

11 . If 3 coats and 14 vests cost $ 78, and 2 coats and 14 vests,at the same rate

,cost $66, how much does 1 coat cost ? What is

the price of a vest ?

12 . G iven 4 x + 7y = 53

2 x + 3 y = 25to find the value of y .

First multiply (2) by 2, making it 6y 50. Why

13 . What is the value of x inequation when the valuefound for y is substituted therein ? Substitute the same valu efor y inequation (2) and find the value of x .

1220. Find the values Of a: and y inthe following equations

14 . 33 + y= 15. 2 x+ 3 y= 38.

15 . 2m+ 2y= 30. x + 3 y= 27.

16 . 2x + 3 y = 18 . 4 x + 3 y= 24 .

17 . 2 x + 3 y = 40 3 x + 2y= 35.

740 ARITHMETIC.

2+232 : 17 Clear of fractions.

5 . 22 :

4 820

i r + ta= 42

23 x — 7y= 3 x + 5l11y

= 15x+ 2

x + y = 100,000

5 2

3 x + 7 5.

3 y— 4

7 x — 62

5y + 3


1 . The sum Of two numbers is 37 . Twice the first added tothree times the second is 96. What are the numbers ?

(Let a: first number ; 31 second number.)

2 . The difference between two numbers is 28 . Five timesthe first less twice the second is 197 . What are the numbers?

(:c—y = 28 ; 5x

3 . The product Of the first of two numbers by 5, added to theprodu ct of the second by 3 , gives 37. The product of the firstby 6

,d iminished by 5 times the second , equals 10. Find the

numbers.

4 . Divide 65 into two parts whose difference shallb e 19.

(Let a: and y parts. Solve also by one unknownquantity.)

3 5 . 41x + 31y= 6771x— 51y= l2

3 6 . 3( — 9)4(3 x — 8)= 17 y

—155

37 . 11) 2(y 6

x + 9 32

y— 3 15


5 . A person pays $ 103 with 32 bills, some Of them $2 bills,the others $5 bills. How many of each does he use?

6 . For 25 head of pigs and sheep,a farmer received $ 145.

How many of each did he sell,if he sold the former at $ 7 each ,

the latter at $ 5 each ?

7 . 10 oranges and 4 peaches cost 38 115; 6 oranges and 7

peaches cost 32 Find the cost of anorange . Of a peach .

8 . 5 pounds of tea and 3 pounds of coffee cost 8

pounds of tea and 1 pound of coffee cost What is eachworth per pound ?

9 . A farmer buys a certain number of horses at $ 125 each ,four times as many cows at $45 each , eight times as many sheepat 10 each

,and half as many pigs at $5 each , spending

for all. How many of each does he buy ?

10. A manpaid 75¢ for 2 pounds of raisins and 3 pounds ofcheese . 5 pounds of raisins and 2 pounds of cheese at the sameprice would have cost 94 What did each cost per pound ?

1 1 . The sum of two numbers is 19. The sum of the secondnumber and ten times the first , minus the sum of the first andten times the second

,equals 45. What are the numbers?

12 . Reduce15,to an equ ivalent fraction , the sum of whose

numerator and denominator shallbe 126.

a: numerator ; y denominator.

y 13'

13 . What fraction equ ivalent to 155 has 147 for the di fferencebetween its numerator and denominator ?

(z -y =

— 147. Why ?)

14 . 10 pounds of coffee at 30f per pound are mixed with 2:

pounds of coffee at 25 per pound . What is x equal to , whenthe mixture is worth 26¢ per pound ?

25 3 + (10x 30)

742 ARITHMETIC.

15 . A grocer mixes green tea costing 60¢ per pound withblack tea costing 40¢ per pound . He uses 100 pounds inall,

and the mixed tea costs him 48 ¢ per pound . How many pounds

of each does he use ?

Let z a number of pounds of black tea ; y number of green. Then

a: y number of pounds of m ixed tea.

a + y= 100 ; 40x + 60y

THREE UNXNOWN QUANTITIES.

1223 . 1 . G iven the following :

3 x + 2y = 12 (a)

5 z

2 x + 3 y + 2 z = 35 (c)

to find the values of x, y , and z .

(a) multiplied by 5, 15 a: 10y 5 z 60

(b) 3,

l5x — 12 y + 9 z = 48

Subtract,

22y 14 z 12 (J)

anequation containing only two unknown quantities.

(b) multiplied by 2 , 10x 8 y 6 z

(c) 5, IOx + 15y + IOz

Subtract , 23 y 4 2

an equation containing only two unknown quantities .

Compare the two equations (cl) and (e), which containsame two unknownquantities.

(d) multiplied by 2, 44 y 28 z

(e) 7 , 161y 28 z

Subtract,

205y

3/

744 ARITHMETIC.

f63+

3

i i i”.2 6

3 — 2 3 x 284

5x — 6 3_8 2+13

— 4y- 3 z

12 +5x

g6g= 2 +

3 z

z2z

5 z — 3 3 x — 192_

3 g— z

4 4 6

2 x + y 9 x — 7 3 2 4- 9 4 2.

2 8 4 16

12 24 . Slate Problems.

1 . A man placed 1 of his capital at 5% and the other thirdat At the end of a year , capitaland interest amounted to

What was his capital?

and f x i interest.3 100 3 100

2 . A has 18 chestnuts more than B. If each finds 4 more,

A willhave fou r times as many as B. How many chestnuts haseach

3 . Two mechanics earn together $8 per day. One works 23days and the other 17 days, forwhich they receive together 166 .

What does each earn per day ?

4 . The sum of the first and the second of three numbers is 55,

of the first and the third 62 , of the second and the third 83 .

What are the numbers ?


5 . The sum of two numbers is 53 . Four times the first is 20more than twice the second . Find the numbers.

6 . A certain sum of money is divided among four persons.

The first takes 1 of it, the second takes 1 of the remainder,the

third takes 1 of what then remains,the fourth receives the

balance, $24 . What is the share Of each of the other three ?

7 . A merchant sold a lot of goods for $510, thereby losing

f , of their cost. What did the goods cost ?

8 . A man collected a bill for a physician and deductedof the amount for his services. If he gave the physi cian $ 147,what was the amount collected

9 . Divide 1301 acres of land among three persons, givingthe first 271 acres more thanthe second

,and the second 131

acres more than the third .

10. A merchant has sold 1 of a piece of Cloth , and has remaining 16 yards more than 1 of the piece . How many yards didthe piece contain originally ?

11 . A servant is engaged for a year for $280 and a su it ofclothes ; he leaves at the end of six months, and receives $ 130and the su it. What is the value of the clothes

MULTIPLIC'ATION OF ALGEBRAIC' QUANTITIES.

1225. Multiply z 3 by a: 4.

The product is equalto 2: times (.r. 3) 4 times (a

4 x + l2

12 Ans.

NOTE.- 3

’ is read a:square. The 2 is called aneap onent.

746 ARITHMETIC.

Multiply (a: 7) (x

a: 7

a: 8

Product by z ,x’ 7 a:

Product by 8 , 8 x + 56

39+ 15x + 56

1226. Multiply1 . x + 5) by 4 . by ( x+ 9)

2 . x + 8) by 5 . (3 x + 1) by ( x + 7)

3 . (2 x+ 5) by (x+ 2) 6 . (2 x+ 1) by (2 3 4-1)

x — 5

q

x + 4

x(x—5) x’ — 5 z

406— 5) 4 3: 20

f — x — ZOAm .

12 8 . Find products

NOTE. (a: 8)(a: 9)means a: 3 multiplied by z 9.

7 . (x 10.— 5) 13 . (2 x

s . (x 11 . (2 x 14 .- 3)

9 . (x 12 . (x 1) 15 .— 3)

12 29 . (SB 5)(a:— 4)The product is equal to x(:c 5) 4(x that is

,that

4(x 5) is to be subtracted from x(x

x(a: 5)= x’ — 5x ; 4(x 5)= 4 x — 20.

Placing the subtrahend under the minuend , and changing thesigns of the former (Art. we have

f — 5x

— 4m+ 20

(x — 5)(z 9 x + 20 Ans.

748 ARITHMETIC.


Find value of x , y, z , etc

1 . :c’

3y’+ 25= 100

5 z’

-3(x’—21)=l98

—x’= 49Q

Q

m

fi

w

N

3 4

.

z + 7 z — 5

z — 3 z — 9

20x 30x.

2; — 1 y— 5 y+ 7

12 37 . Slate Problems.

1 . Find the dimensions Of a field , the length of

twice its breadth , its area being square rods.

2 . The surface of the six equal faces of a cube contains 96square inches. Find the length Of one edge .

3 . One number is fourth-fifths of another, and their productis 80. What are the numbers ?

4 . One-thi rd of a number multiplied by two-fifths of the

same number gives a product of 270. Find the number .

5 . Thirty per cent of a number multiplied by forty per centof the same number gives a product of 300. What is the number ?

(x — (x

- 5>

anssssu c EQUATIONS. 749

6 . Thirty per cent of forty per cent of a number is 300.

What is the number ?

7 . The base of a right-angled triangle is Qas long as theperpendicular, and the area of the triangle is 96 square rods.

Find the length of the base . What is the length of the hypotenuse

8 . The base of a right-angled triangle measures :c yd . ,the

perpendicular measures3 3:yd . What is the length of the

hypotenuse ? If the hypotenuse measures 15 yd . , find the lengthof the base .

9 . The base of a right-angled triangle measures 3: ft .,the

hypotenuse measures (:c+ 9) ft. , the perpendicular measures 15ft . What is the length of the base

10. The difference between the squares of two consecutivenumbers is 49. What are the numbers ?

AFFECTED QUADRATI08.

The square of the sum of two quanti ties is equalto the squareof the first twice the product of the first and the second thesquare of the second .

(se— l)(x —2 x + 1

The square of the difference of two quantities is equalto thesquare of the first twice the product of the first and the secondthe square of the second .

(m —n)’ m’ — 2mn+ n

’

(10 lox 3 + s’

750 ARITHMETIC.

1240. Sight Exercises.

Extract the square root Of

1 . 6 .

2 . x’ — 14 x + 49 7 . a.

"

3 . z’ — 18 :c + 81 8 . a

’ — 2 ab+ b’

4 . 9 . 39— 24 :c + l44

5 . 10. x’+ 22 x + 121

1241 . The square of (a: 3) consists of how many terms ? Of

how many terms does (a: consist ? (a:

Supply term necessary to make a complete square

1 . 6 .

2 . z’ - 12 x + ? 7 . :c

’

3 . x’ — 8 x + ? 8 . z

’ - 10x + ?

4 . z’ — 16x+ ? 9 .

5 . 10. x’ — 22 x + ?

G iven

What number must be added to the first member Of the equation to make it a “ complete square ?If a number is added to one member of an equation, what

must be done to the other member to preserve the equality ?

752 ARITHMETIC.

1247 . To make the first member a complete square, youadded the square of what part of the coefficient Of z ?

12 48 . Find values of a:

1 .

2 . z’ — 3 :c = 10

f

3 . f + 5x

4 . :c’ — 7 x = 8

12 49 . When has a coefficient,divide both members by the

coefficient .3 z ’ 9 a: 84 .

Dividing by 3 , :c’ 3 a: 28 .

Completing the square ,

Extracting square root, 3: 4} 1

21

x — 1} — fi* — 7

Ans. 4 or— 7 .

6x’ 6 x = 36 6 . 3 19+ 9 x = 54

9 39 + 9x = 180 7 . 8 x’ — 72 :c —160

7 x’ + 28 x= 147 8 . 7 z ’ + 49x= 56

4 x2 — 40x~= — 64 9 . 3 x’ + 21 x = 54

8 z ’ — 16 :c= 504 10. 5 39 — 25 2:


12 50. Slate Problems.

1 . The sum of two numbers is 12 ; their product is 32.

What are the numbers ?

a:and 12 z numbers. (12 z)a: product.

2 . The base of a rectangle is 50feet longer thanits altitude . Its area is

square feet. How long is thebase

3 . The perpendicular of a rightangled triangle measures 15 yards more than the base .

The hypotenuse is 75 yards. Find the length of theperpendicular.

[ z’ (15 z)’

4 . The hypotenuse Of a right-angled triangle is Iitimes as long as the base . The area of the triangleis 150 square yards. How long is the hypotenuse ?

[Perpendicular VG :c)’

:c’

area 4base x perpendicular. ]

5 . The entire su rface of a square prism is 170

square feet . Its altitude is 6 feet , and oneside of its base is a: feet. Find the valu e so 2 :

of :c.

6 . A garden 50 feet long , 40 feet wide ,has a walk just outside it a: feet wide .Find the area Of the walk .

If the area Of the walk is 784 squarefeet, what is its width T rods

7 . A field,ABC’D

,contains 12

acres . Its length is 1} times itsbreadth . How many rods long isthe diagonalBO?

754 ARITHMETIC.

8 . A flag-stafi

’

, AB ,50 feet high , was broken

Off at the point 0. The broken part, resting on0,reached the ground D,

30 feet from the base of

the staff. Find the length of the part broken off.

9 . A ladder, CE or DE,placed at a point B,

ina street 58 feet widebetween the Oppositehouses, just touchesthe top ofa house ,DB ,

60feet high on one sideof the street , or the top of a house ,0A

,56 feet high on the other side .

Find the length of the ladder.

56’+ x’.

10. ABC is a triangle . The side AB 3

measures 13 feet ; the side BC’,4 feet ;

13

AC,15 feet . Find the altitude BD . D T 0

11 . ABO'D isa trapez ium . AB=34 ft .

BC : 20 ft . OD=4Oft. ; DA 26 ft .

The perpendicular BF measures 16 ft .

Find the length of the diagonalAO andof the perpendicular ED.

756 ARITHMETIC.

1252 . Designationof Angles.

The angle formed by the lines S Tand TUmay called

angle T. It is frequently better to call it

the angle STU or UTS , the letter at thevertex being placed betweenthe two others.

The use of the three letters is necessarywhere there is more than one angle having

its vertex at the same point , as inthe aecompanying figure , where UI , VI , and WX

meet at the point X .

12 53 . Measurement of Angles.Anangle is measured by the

arc Of a circle , the center Of thecircle being at the vertex Of theangle . The angle 1 2 3 is meas

ured by the arc YZ ; the angle4 5 6 by the arc ab.

60 seconds 1 minute,60minutes 1 degree,

360 degrees 1 circle.

The number of degrees inan arc or an angle is determined by a p rotractor.

ELEMENTARY GEOME'I‘RY. 757

To measure an angle,X YZ ,

for instance,produce the lines

YX and YZ . Place the point A Of the protractor on the vertex ( Y) Of the angle, and theedge AC on the line YZ produced . Using thelower line of figures, read Off from the protractorthe number of degrees at the point where the line YX producedcuts the semi-circle .

Inmeasuring the angle DEF:the line ABD

is placed onEF , the point A on the vertex E.

The number of degrees inthis case is read fromthe upper row Of figures.

EXERCISES IN CONSTRUCTION.

12 56. NOTE. Inthe following 100 exercises, the ruler, the compasses,

and the protractormay be used .

The drawing Should be carefully done with a sharp , hard pencil.

1 . Draw an obtuse angle formed by two lines, each one inchlong . Draw anacute angle formed by two lines, each six incheslong. Which is the larger ?

2 . Fold a piece of paper twice , so that the lines made by thecreases willform four right angles.

Fold a piece Of paper so as to make fourangles that are not right angles.

3 . The lines GE and [ J intersect at

If , making four right angles. Which are

is longer , 7 8 , or ed ? Which contains the

greater number of degrees ?

4 . Draw two lines meeting at anangleOf Two lines meeting at anangle of Two meetingat an angle of

5 . Draw two lines making two angles, one of

wh ich measures How many degrees does

the other angle contain

758 ARITHMETIC.

NOTE. A line parallelto the right or the left side of the paperis called a verticalline ; one parallel to

the top orthe bottom of the paper is called K

a horizontalline ; others are called oblique Mlines.

KL is a verticalline,MN is a hori

zontalline, OP and QR are Oblique lines .

6 . To a horizontalline draw a line making two equal adjacent angles. How many degrees does each angle contain ?To a vertical line draw a line making two equal adjacent

angles. How many degrees does each angle contain ?To an Oblique line draw a line making two equal adjacent

angles. How many degrees does each angle contain?

7 . How many degrees are there ina right angle ?

8 . To an Oblique line draw a line making two unequaladjacent angles. How many degrees are there inthe sum of the twoangles ?

9 . How many degrees inthe angle T,i f S

containsV measures How many degrees does U

measure ?If one Of two adjacent angles measures

how many degrees are there inthe other angle ?How many degrees are there inanangle adjacent to one of

47°45'

10. Construct angle 5,

angle 4 ,Measure angle 3 .

How many degrees and minutes w illthere beinangle 5

,when 3 contains 494

° and 4 contains

Whenangle 3 contains 36° 30' and angle 5 contains 79°

how many degrees and minutes willangle 4 contain ?

1 1 . Erect a perpendicular at each extremity of a hori zontalline . At each extremity of a verticalline . At each extremityof anOblique line .

760 ARITHMETIC.

23 . Draw an angle equalto the sum of three angles measur

ing, respectively , and

24 . How many degrees are contained inthe angle made bythe hour and the minute hand of a watch at 1 O

'

clock ? Howmany degrees inthe angle made by the hands of a church-towerclock at the same hour ?

25 . How many degrees are passed over by the minute handina quarter of an hour ? How many are passed over by the

hour hand inan hour ? Inhalf anhour ? In15 minutes ?What angle is made by the hands at 12 :15 ? At At

NOTE. The pupils should , of course, understand that the anglesinquestions 24 and 25 are those formed by imaginary lines passing throughthe

’

centers of the hands. They should know, too, that a geometricallinehas neither breadth nor thickness

Parallels.

Lines lying inthe same plane that willnot meet, no matterhow far they are produced

,are said to be p arallel.

26 . Draw two or more lines that shall be perpendicular to a

horizontalline . Where willthey meet ?Draw two or more that shall be perpendicular to a vertical

line . Where willthey meet ?Draw two or more that shall be perpendicular to an oblique

line . Where willthey meet ?

27 . TO a horizontalline draw two or more lines running inthe same direction, and each making an angle Of 35

° wi th thefirst line . Willthe oblique lines meet ?Draw two or more lines running inthe same direction

, and

each making anangle of 125° with a verticalline . Will the

Oblique lines meet if produced very far?Draw two or more lines running inthe same direction

, andeach making anangle of 74

° with an oblique line ? Will theformer lines meet ?

ELEMENTARY GEOMETRY. 761

28 . Draw two lines making angles of 30° and respectively,

with a third line . Willthe two former lines meet if producedineither direction ?

29 . Draw a line,AB , meeting a hori

zontalline BC at anangle of Drawa third line , DE, parallel to the horizontalline , and cutting the Oblique line . Whatangles does it make with the Oblique line ?Draw a fourth line , FG , parallel to the

oblique line , and cutting both horizontallines.

Mark in each of the twelve angles thenumber of degrees it contains.

Take a stifl'

card , having a square corner, and

cut 05 the corner, making the Side AB equal to

the side BC, and each abou t 4 or 5 inches long.

Measure the angle at A and the angle at C.

30. With a ruler , draw a line DE ; and without moving theruler , draw lines FG ,HI,AB ,

by placing the sideBCOf the triangle firmlyagainst the edge of theruler .

EU, HI, and AB areparallel. Why

3 1 . Holding the triangle against the edge Of ruler J'

If , drawOblique lines AC, EMNO.

If the angle AOB containshow many degrees will

LMJ contain ? How manywillNOJ contain ?

parallel Why ?

32 . Draw a line making an angle Of 70° witha horizontalline . At the pomt P , using onlythe ruler and the triangle , draw a line parallel

762 ARITHMETIC.

3 3 . QR and UV are parallel lines,cut by a line ST. If the angle 6 meas

ures how many degrees does a

measure ?Find the number of degrees ineach of

the other six angles.

34 . Draw WX and YZ intersectingat any angle . With the ruler and thetriangle draw a line parallelto WX .

Mark all the angles that are equalto i . Mark all those that are equal

to j .

1261 . Triangles.

35 . From the extremities Of the line AB , draw lines that shallmake angles of 60° and respectively

,wi th

AB . Prolong the lines untilthey meet at C,forming a triangle .

Measure the angle at C. How manydegrees does i t contain ? How many degreesare there inthe sum of the three angles of the triangle ?

36 . Construct a triangle having one angle of 90° and one of

Measure the third angle .

How many degrees are there inthe sum

of the three angles ?

37 . Draw a triangle having angles at thebase meas uring, respectively , 57

° andWith the ruler and the triangle draw IJ

parallel to DE. Produce DE to H andFE

'

tO G .

How many degrees will the angle a contain ? How many degrees in the angle 0 ?

Calculate the number of degrees inthe angle 6whenyou know

764 ARITHMETIC.

43 . See if you can draw a triangle containing three right

angles .

A triangle containing two right angles.

A triangle containing one right angle .

44 . Try to draw a triangle that shall contain three obtuseangles .

A triangle that shallcontain two obtuse angles.

A triangle containing one obtuse angle .

45 . Draw a triangle containing two angles of 50° and

respectively . How many degrees are there inthe third angle ?Measure each side

,and mark on the side its length .

Opposite which angle is found the longest side ? Oppositewhich

,the shortest Side ?

46 . Draw a triangle having two angles of 75° each . Are anytwo of its sides equal?Draw a triangle having two angles of 50

° each . Are any ofits sides equal?

47 . Draw a triangle having two angles of 60° each . How

many degrees does the third angle contain ?Are any of its sides equal?

1263 . A triangle having allits sides equal, is called anequ i

lateraltriangle .A triangle having two equalsides

,is called eu isosceles triangle .

A triangle having all its sides unequal, is called a scalenetriangle .

48 . Fold evenly on the line AC (Fig. a rectangular sheet ofpaper . Cut off, on the line AB ,

one of thefolded corners. When this piece is Opened out A

(Fig . it makes what kind of a triangle ?Which are the equal sides ? How does thecrease line , AC, divide the base , BB ?If the angle ABC

,Fig. 1

,is and the angle

at C is a right angle , what is the angle BAC?

ELEMENTARY GEOMETRY .

How many degrees inthe angle BAB ', Fig. 2 ?

How does a perpendicular let fall uponthe baseof anisosceles triangle from the Opposite angledivide the angle ? How does it divide the base ?How do the angles at the base of an isoscelestriangle compare with each other as to size ? FIG 2

The unequalside of anisosceles triangle is called the base.

49 . Draw an isosceles triangle having the base a verticalline .

Anisosceles triangle having the vertex below the base .

One having an oblique line for the base .

50. Draw a right-angled isosceles triangle . How manydegrees willthere be ineach of the other angles ?Draw anobtuse-angled isosceles triangle .

5 1 . How many degrees will there be in each angle of anequ ilateraltriangle ?Draw anequ ilateraltriangle having one side vertical.Draw anequ ilateral triangle having its

vertex below the base .

52 . DEF is an isosceles triangle , DEand EF being the equal sides. If theangle 1 measures how many degrees

are there ineach of the other five angles,when the line FG bisects the base ?

53 . If the angle DEE in the abovetriangle measures howmany degrees will there be intheangles 3

,4,5, and 6 ?

54 . ABC is a right-angled tri

angle , the angle at B measuringand the angle at C measuringIf the line AX is so drawn

as to make the angle AXB equal

to find the number of degrees in B

the anglesm ,n, andp , respectively ?

766 ARITHMETIC.

12 65. Quadrilaterals.

A plane figure of four sides is called a quadrila teral.When the opposite sides are parallel, the quadrilateral iscalled a p arallelogram . (Fig. 1 toA rectangle is a parallelogram all of whose angles are right

angles. (Fig. 1 toWhen the four sides of a rectangle are equalto each other, it

is called a square. (Fig. 1 and

The term oblong is frequently applied to rectangles whoseadjacent sides are unequal. (Fig. 3 and

FIG . 1. FIG . 2. FIG . 3. FIG . 4.

A rhombus is a parallelogram allof whose sides are equal, butwhose angles are oblique . (Fig. 5 and

When the adjacent sides of a parallelogram are unequal andthe angles are Oblique , it is called a rhomboid . (Fig. 7 and

FIG . 7. FIG . 8.

A trap ezoid is a quadrilateral having only two of its sidesparallel. (Fig. 9 and

A trap ezium is a quadrilateral having no two sides parallel.

(Fig. 11 and

FIG . 9. FIG . 11. FIG . 12.

768 ARITHMETIO.

60. We have seenthat an equilateral triangle is equ iangue ;

that is, that allits angles are equal. Are the four angles of a

square equal? Are the four angles of a rhombus equal?

61 . A triangle that has three equalangles is equ ila teral; thatis, allits sides are equal. Canyou draw an equiangular parallelogram that willbe equilateral? Canyou draw an equ iangularparallelogram that willnot be equilateral?

62 . Draw a rectangle , base 21} inches, altitude 2 inches.

A rhomboid , base 2} inches, altitude 2 inches.

63 . Make , out of paper, a rectangle and a rhomboid , eachhaving the above dimensions, and endeavor to ascertain, by cut

ting , whether or not they are equalto each other inarea .

64 . Show , by cutting , that any parallelogram can be dividedinto two equaltriangles.

65 . Make three rhomboids of different shapes, the base of

each to measure 3 inches, and the altitude 2 inches. Are theyequalto each other inarea ?

66 . Make three rhomboids of different shapes, the base of

each to measure 3 inches, and the adjacent side 2 inches. Are

they equalto each other inarea ?

67 How do we calculate the area of a rectangle 3 inches by2 inches? Of a rhombus whose base is 3 inches

,altitude 2

inches? Of a rhomboid , base 3 inches, altitude 2 inches ? Of a

right-angled triangle , base 3 inches, altitude 2 inches? Of anobtuse-angled triangle , base 3 inches

,altitude 2 inches ? Of

an acu te-angled triangle , base 3 inches, altitude 2 inches? Draweach of the foregoing.

68 . Draw a rectangle hav ing two adjacent sides measuring 3inches and 2 inches, respectively . A rhomboid having two adjacent sides measuring 3 inches and 2 inches

,respectively . A

trapezium having two adjacent sides measuring 3 inches and 2inches, respectively. A trapezoid having two adjacent sides


measuring 3 inches and 2 inches,respectively . A right-angled

triangle , an obtuse-angled triangle , and an acute-angled triangle ,each having two adjacent sides measuring 3 inches and 2 inches,respectively .

Calculate the area of each, making such measurements as may

be necessary .

69 . Draw three trapezoids of different shapes, whose parallelsides shallmeasure 3 inches and 2} inches, respectively , andwhose altitude shallbe 2 inches.

Draw a rectangle 2 inches high that shallbe equal inarea toeach of the foregoing. How long will be its base ? What willbe the relation of the length of its base to the lengths of theparallelsides of the trapezoids ?

70. Draw three trapeziums of different shapes,each having

One diagonal Of 3 inches,the perpendiculars let fall on this

diagonalfrom the opposite corners measuring 2} and 2 inches,respectively .

What is the altitude Of a rectangle that is equal inarea toeach of the foregoing, when the base of the rectangle measures 3inches ?

1267 . ThOCirols.

7 1 . With the compass points one inch apart , drawa circle.

Drawa radius . How long is it ?

The line made by the compass pencilis called the circumference. A line

drawnfrom the center to the circumference is called a radius.

7 2 . Draw a circle , radius 11} inches. From a point on the circumference, draw a line through the center to an opposite pointon the circumference .This line is called a diameter. How long is the diameter ?

7 3 . Draw a circle , radius Ii inches. Draw a diameter andseveralradii .Write diameter, circumference, radius, each inits proper place .

770 ARITHMETIO.

74 . Drawa circle . Between two points on the circumferencedraw a line that does not pass through the center .

This line is called a chord.

75 . Draw a circle . Init draw two diameters, a radi us , andthree chords. Wri te oneach line its name .

7 6 . Draw a part of the circumference of a circle greater thanone-half of it . Draw the chord .

A part of the circumference is called anarc.

7 7 . Draw an arc less than a semi-circumference . Draw a

chord . Write the name on each .

Canyou make a chord that willbe longer than the diameter ?

7 8 . Draw two equal circles. In the first draw the chord of

anarc of Inthe second,the chord of an arc Of

What is the ratio between the two chords you have drawn

7 9 . Ina circle draw a chord equal inlength to the radius.

How many degrees are there inthe arc whose chord has beendrawn ?

80. Draw anarc of To its extremities draw two radn.

The part of the surface of a circle enclosed by two radii and the intercepted arc is called a sector.

8 1 . Draw a sector of 60°

(a sextant). A sector of 90°

(a

quadrant).

82 . Draw an arc of 120 degrees . Draw the chord.

The part of the surface of a circle bounded by anarc and itschord is called a segment.

83 . Draw several circles having the same center, but of

unequalradii (concentric circles).

84 . Draw two equalcircles just touching each other (tangent).Draw two unequalcircles tangent to each other.Within a large circle draw a smaller one tangent to it.

85 . Draw circles of equal radi i cutt ing each other. Drawintersecting circles of unequalradi i.

772 ARITHMETIC.

92 . Circumscribe a square about a circle . Anequ ilateraltriangle . A regular pentagon . A regular hexagon . A regular

octagon .

93 . Draw a pentagon . What is the smallest number of triangles into which a pentagon can be divided ? How manydegrees are there inallof these triangles ? How many degreesare there ineach angle Of a regular pentagon ?

94 . Draw a hexagon . What is the smallest number of tri

angles into which it can be divided ? How many degrees dothe six angles of a hexagoncontain ? How many degrees are

there ineach angle of a regular hexagon ?

95 . A quadrilateral is divisible into how fewtriangles ? A

pentagon into how few? A hexagon ? A heptagon ? AnoctagonThe smallest number of triangles into which a polygon is

div isible is how many less than the number of i ts sides ?

96 . How many degrees are there ineach angle Of a regularoctagon

97 . Using a protractor, construct a regular pentagon on a linetwo inches long .

98 . Ona three-inch line construct an equ ilateral triangle .

Onthe same line construct a square , a regular pentagon, a

regular hexagon , heptagon , octagon, nonagon, etc .

99 . Inscribe a square in a circle . Circumscribe a squareabout the same circle by drawing lines touching the four cornersof the inscribed square .

What is the ratio betweenthe areas of the two squares ?

100. Inscribe an equ ilateraltriangle ina circle . Circumscribean equ ilateral triangle about the same circle by drawing linestouching the vertex Of each of the three angles of the inscribedtriangle .

What is the ratio between a side of the inscribed triangle anda side of the circumscribed triangle ? What is the ratio betweenthe areas of the two triangles ?


PROBLEMS IN CONSTRUCTION.

Indrawing the following one hundred exercises, only the ruler

and the compasses are to be used . Use neither the protractornor the triangle.

1 . Draw a circle , radius an inch and a half. Outside of it,

and tangent to it , draw a second circle of aninch radius.

How far apart are the centers ?

2 . Draw two tangent circles having radu of an inch and ahalf and an inch , respectively , one within the other.How long is the line joining the centers ?

3 . With centers 3 inches apart draw two equal circles tangent to each other. How long is the radius of each ?

4 . With centers three inches apart draw two equalcircles of2 inches radius. Connect the centers.

Draw a line joining the two points inwhich the circles intersect . How does this line divide the line connecting the centers ?Draw radi i from each center to each point of intersection .

5 . Construct an isosceles triangle , base 3 inches, equalsides2 inches .

NOTE — Use circles or arcs where necessary .

6 . Construct an isosceles triangle , base 34} inches, equalsides4 inches.

Divide it into two equalparts. Do not locate the center ofthe base by measurements .

7 . Ona verticalline construct an isosceles triangle . Without measuring the length of the base draw a perpendicular tothe center of the base .

8 . Bisect a verticalline . Anoblique line .

Do not measure the length of the line.

9 . Construct anequilateraltriangle on a two-inch line .

10. Construct anequilateral triangle on a verticalline . On

an oblique line .

774 ARITHMETIO.

11 . Construct a scalene triangle .A triangle having sides measuring 1, 2 inches , respectively.

One whose sides measure 2, and 3 inches , respectively .

12 . Canyou construct anisosceles triangle whose base meas

ures 4 inches, equalsides 2 inches ?Try to construct a scalene triangle with sides measuring 1, 2,

and 3 inches, respectively .

13 . Draw a circle . Init draw a chord.

Bisect the chord , using as few lines and as short ones as youcan .

NOTE.— Do not use the ruler to ascerta inthe length of the chord before

bisecting i t.

14 . Divide a sector into two equalparts.

15 . Draw a circle . Draw a chord . Draw a radius throughthe center of the chord .

Is the radius perpendicular to the chord ? Why ?

16 . Bisect the arc of a circle and its chord .

Bisect the arc of a circle withou t drawing the chord .

17 . Draw a perpendicular to the middle point of a horizontalline . To the middle point of a verticalline . To the middlepoint of an oblique line .

18 . Draw ina circle two diameters perpendicular to eachother .

19 . Divide the circumference of a circle into four equalparts .

Into eight equalparts .

Inscribe a square ina circle .

20. Inscribe a regular octagon ina circle .

21 . Connect the opposite vertices of a regular Octagon inscribed ina circle by lines passing through the center of thecircle .

Lines connecting the opposite vertices of a polygonare called diagonals.

776 ARITHMETIC.

perpendicular at a point ina line betweenone end and the

center.

3 6 . Ona line 2 inches long , construct a square . Construct arectangle

,base 3 inches, altitude 2 inches.

3 7 . Onanoblique line 3 inches long, constru ct a square . On

anoblique line 3 inches long, construct a rectangle , altitude 2}inches.

3 8 . Construct a rhombus whose side is 3 inches, altitude 21}inches.

A rhombus whose side . is 3 inches, and which contains an

angle of

39 . Coi

nstruct a rhomboid whose adjacent sides measure 4inches and 3 inches

,respectively , altitude 2} inches.

40. Construct a right-angled isosceles triangle . An isoscelestriangle containing an angle of One containing an angleof

4 1 . Construct an isosceles triangle , base 3 inches, altitude 3inches. Canyou construct anequilateraltriangle whose altitudeshallbe 3 inches ?

42 . Construct a scalene triangle , base 3 inches, altitude 3inches. Anobtuse-angled triangle having a 3-inch base

,and the

altitude 3 inches.

43 . Construct a triangle whose base measures 5 inches, theother sides being 3 inches and 4 inches

,respectively . Draw its

altitude geometrically .

44 . Construct a triangle whose base measures 4 inches, theangles at the base measuring 120° and respectively . Drawits altitude geometrically .


45 . Draw a line . From a point above the line,let falla per

pendicular to the line .

46 . Inscribe a regular hexagon ina circle . Divide it into sixequilateraltriangles by diagonals .

How many degrees are there ineach of the six angles at thecenter of the circle ? How many degrees are there inthe are onwhich each angle at the center stands ?

47 . Inscribe an equ ilateraltriangle ina circle . Mark ineachangle the number of degrees it contains. Mark on each are thenumber Of degrees it contains.

Anangle at the circumference of a circle is measured by whatpart of the are on which it stands ?

48 . Do you know how many degrees there are ineach angleof a regular hexagon ?Construct a regular hexagon without drawing a circle and

without using triangles.

49 . Construct a regular octagon , each side one inch . Do notdraw a circle .

50. If an angle at the circumference is measured by onehalf of the arc on which it stands , how many degrees are conta ined between two lines forming a right angle whose vertex isat the circumference ?

51 . Inscribe several right angles ina circle , their sides intercepting the same arc .

52 . Inscribe ina circle a right-angled isosceles triangle .

53 . Construct a square whose diagonalmeasures 4 inches.

A rhombus whose side is 3 inches and which has one diagonal3 inches long.

54 . Inscribe ina circle a scalene triangle . How many degrees are there inthe sum of the arcs intercepted by the sidesforming the three angles ?Howmany degrees are there inthe sum of the three angles ?

778 ARITHMETIC.

55 . Draw two equilateral triangles having one side commonto both triangles. What figure have you drawn ?

56 . Construct two triangles, each having two sides measuring 2 inches and 2} inches, respectively , and the angle made bythese sides (the included angle) measuring 60

°

Cut out the triangles. Place one upon the other , and ascertainif the third side of one triangle is equalto the third side of theother . Are the two remaining angles of the first triangle equalto the corresponding angles of the second ?

57 . Construct a triangle having a base of 3 inches, and anglesat the base containing 60° and respectively . Construct asecond triangle having angles of 60

° and respectively,and

the included side 3 inches.

Cut both out Of paper , and test the equality of the corresponding parts of each .

58 . Construct a triangle whose angles measure andrespectively . Canyou construct another triangle having

angles equal to those inthe first triangle, but with sides notequalto those of the first triangle ?

59 . Construct a triangle having sides of 2,

and 3 inches,respectively .

Try to construct a second triangle having sides equal to thoseof the foregoing triangle , but having the corresponding anglesunequaL

60. Canyou construct a triangle containing three angles of

75° each ?

61 . Ina circle, inscribe a right-angled triangle whose hypot

enuse is 3 inches.

One whose hypotenuse is 3 inches,base 2 inches .

Hypotenuse 3 inches, perpendicular 11} inches.

What is the diameter of the circle ineach case ?

62 . Draw two equal circles. Mark off anare inone . Markoff anequalare inthe other.

780 ARITHMETIC.

inches, the angle formed by the perpendicular and the hypot

enuse being How long is the hypotenuse ?

7 1 . Draw ina circle whose radius is two inches, the chord of

an arc of Draw the chord of an arc of What is theratio between their respective lengths ?What is the ratio between the chord of an arc of 60° and thechord of an arc of

72 . In a circle of 2 inches radius, draw a chord . At one

extremity erect a perpendicular. From the other extremitydraw a diameter. Where will this diameter meet the perpendicular? Why ?

7 3 . Make a triangle of splints, fastening them by one tack ateach angle . Willthe triangle retain its shape under pressureMake a rectangle of splints . Fasten each corner by a single

tack . Willthe rectangle retain its shape under pressure ?

74 . Ina circle draw three parallel chords the same distanceapart .Draw three parallelchords not equ idistant.

75 . At the circumference of a circle , draw a line perpendicularto a radius. This perpendicular is tangent to the circumference .

A line that touches the circumference ina single point is called a tangent.A line that cuts the circumference is called a secant.

76 . Inscribe a circle ina square .

7 7 . Ina circle draw two chords not parallel. Bisect each bya perpendicular. Where willthese perpendiculars intersect ?

7 8 . Inscribe a circle inan equ ilateraltriangle . Inan isoscelestriangle . Ina scalene triangle .

79 . Circumscribe a circle about an equilateraltriangle. Aboutan isosceles triangle . About a scalene triangle .

80. Draw a circle , using a coin or something similar . Can

you find the center ?


8 1 . Draw an obtuse-angled triangle . Circumscribe a circleabout it .

82 . Without using the compasses, draw an arc. Canyou findthe center of the circle of which the are forms a part ?

83 . Construct a rhombus whose sides are 4 inches, the acuteangle being What is its altitude ?

84 . Construct a square on a line 3 inches long, using onlyarcs of 3 inches radius .

85 . Canyou construct a triangle with sides 2 , 4, and 6 inches?

86 . Draw a circle, radius 3 inches. Draw an equ ilateraltriangle around it .

87 Construct an equilateraltriangle , altitude 1} inches. Construct one having an altitude twice as great .

88 . Construct a triangle whose sides measure 1, and 2

inches,respectively . Construct a second having its correspond

ing sides twice as large as those of the first triangle .

Construct a triangle whose angles measure and

respectively . Try to construct a second triangle having itscorresponding angles twice as large as those of the first triangle .

89 . Construct a square which shallbe equal inarea to twosquares, one having a side of 2 inches; the other having a side of3 inches.

90. Canyou construct a square whose area willbe 13 squareinches

91 . Construct a square whose area will be equal to the difference of area betweentwo squares, one having a side of 3 inches,the other having a Side of 2 inches.

92 . Construct a square whose side is 3 inches. Constructanother having double the area .

93 . Construct an equ ilateral triangle , side 2 inches. Howmany 1-inch equ ilateraltriangles can be made from it ?

782 ARITEMETIO.

94 . Howmany l-inch equilateraltriangles canbemade fromanequilateraltriangle whose sides are 3 inches ?From one whose sides are 4 inches ? 5 inches ?

95 . Construct a triangle, sides 2, 3 , 4 inches. Divide it intofour equaltriangles. G ive the dimensions of each of the latter .

96 . Construct a triangle 1, 2 inches. Make a trianglenine times as large as the first by producing two of the sidesand drawing a fourth line . What are the dimensions of the

second triangle

97 . Drawa circle of 1 inch radius. Tangent to it , and enclosing it , draw one having four times the area Of the first.

98 . Can you draw a circle havin’g half the area of a circle

whose radius is 2 inches ?

99 . Draw an equ ilateraltriangle , side 2 inches . Constru ctanother of half the area of the first .

100. Canyou tellthe ratio between the area of an inscribedand that of a circumscribed hexagon ?

EQUAL TRIANGLES. EQUIVALENT TRIANGLES.

1271 . NOTE.-The protractor and the triangle may be used inthe

following exercises.

1 . Draw a rectangle , base 2} inches, altitude 2 inches. Drawa rhomboid

,base 2} inches, altitude 2 inches. Find the area of

each .

2 . With a base 2} inches, altitude 2 inches, draw

(a) A right-angled triangle .

(b) Anisosceles triangle .

(0) One or more acute-angled scalene triangles.

(d) One or more obtuse-angled triangles.

Calculate the area of each.

784 ARITEMETIO.

SIMILAR TRIANGLES.

Construct a right-angled tri

angle ABC. Make AB 4 inches, AC5 inches, BC 3 inches.

At the points n, g , and s, distant fromA one

,two

,and three inches , respec

tively ,erect perpendiculars. Measure

the length of each .

8 . An is one-fourth Of AB ; ascertain the betweenmnand CB .

B ABCCornare and CB see IfP xg

rug Ag

As is three-fourths of AB ; is rs QBCMeasure Am,Ax

,and Ar. How does each compare with

AC

Would the relations as found above , exist if mnwere not per

pendicular to AB ?Howdo the angles inthe triangle Amu compare with the

angles inthe triangle ABC ?

9 . Draw a triangle whose sides measure 4,5, and 6 inches,

respectively . Cu t out of paper a triangle whose sides measure2 ,

and 3 inches, respectively . Place anangle Of the smalltriangle on the corresponding angle of the large triangle

,and

compare their respective sizes.

How does the area of the smalltriangle compare with the areaOf the large triangle ?

10. Two angles of a triangle measure about 37° andrespectively . The sides opposite those angles measure 3 inchesand 4 inches, respectively . How many degrees does the thirdangle contain ? Calculate the length of the third side .

What will be the dimensions of a Similar triangle whose areais one-fourth that of the given triangle ? G ive the approximatenumber of degrees ineach angle of the smalltriangle .


11 . Draw a line DE,2} inches long . At any angle draw

DE. Commencing at D,mark Off on DE quarter-inch portions,

1,2,3,etc . , to 9. Join 9E

,

and with a triangle and aruler draw 8 b

, 7g , 6f, etc . ,parallelto 9E. Measure Eh .

Is it equal to kg, gfi fe, etc . ?D

a b c g h.

Why ?Into how many parts is the line ED divided ? What frac

tionof aninch does each part contain ?Inlocating the points 1, 2, 3 , 4 , etc .

,is it necessary to make the divisions

inch ? Would it be sufficient to use the compasses with the points anyconvenient distance apart ?

12 . Divide a line 23 inches long into 5 equalparts.

CALCULATING BEIGEIS AND DISTANCES.

1273 . To verify the results obtained by calculation, the pupilshouldmake diagrams, drawing the figures to a convenient scale.

1 . If AB ina right-angledtriangle measures 120 feet, anda perpendicular

, vw, erected 10feet from A measures 5 feet,calculate the length of BC.

s AB zzw s C ;

2 . A post 6 feet above ground throws a shadow Of 7} feet .Howhigh is a tree whose shadow measures 60 feet ?

786 ARITHMETIC.

3 . Wishing to ascertain thedistance between two houses, Rand S , on opposite sides of astream ,

I measure a line S V,

at right angles to SR, 200 feet.At T,

90 feet from S ,the per

pendicular TW measures 60

feet . Requ ired the distance SR.

VT= VS — ST.

4 . Beginning at B,100 feet from the bank of a river, a line ,

BC, is measured feet long . At D, distant from C 50

feet, the perpendicular DE is foundto measure 90 feet. What is thedistance from B to A

,a tree on

the Opposite bank ? How wide isthe river ?

5 . A boy,whose eye (E) is 4 feet

from the ground , can just see thetop (A) of a steeple when he stands back 3 feet from a fence

(CG) 6 feet high . The distance from the foot of the fence tothe center of the base of the steeple is 177 feet . Find theheight of the steeple AB .

788 ARITHMETIC.

9 . Wood-choppers, desiring toknow the height of a tree beforecutting it, sometimes make an isosceles right-angled triangle of woodor paper, and

“step off the distance

on level ground from the pointat which they find they can justsee the top of the tree looking alongthe hypotenuse of the triangle , the 0base being parallelto the ground .

How high is the tree AB , i f A C is 36 pacesfeet each , and the angle AOB is

10. B is a point on the bank of a streamdue east Of A on the other bank . A boywalks due south of A untilhe reaches a pointat which he finds

,from his pocket compass,

that he is directly south-west of B . If thedistance AC measures 119 yards, how wide isthe stream ?

1 . Calculate the length ininches of an arc Of the radiusof the circle being two inches. Calculate the length of an arc of

Of Of Of

2 . Calculate the length in inches Of a chord of 60’inthe

above circle . Of a chord of Of a chord of Of achord of Of a chord of

3 . Inthe parallelogram shown intheaccompanying diagram ,

the angle a measures and the angle 6 Howmanydegrees does the angle 0 contain ? The

angle d ? The angle 1 ? The angle 2 ?

4 . Inscribe a regular nonagon ina circle of 2 inches radius,

using the protractor.


5 . How many degrees does each angle of a regular nonagoncontain ? Draw a regular nonagon , each side measuring twoinches. (Use the protractor .)

6 . The distance around a polygon is called its perimeter.

What is the perimeter of a regular hexagon,inscribed ina circle

whose radius is 1 inch ?What is the circumference Of the circle ?

7 The distance from the center of a regular polygonto themiddle point of one side is called the ap othem .

Draw the apothem of a regular hexagoninscribed ina circleof 1 inch radius. About how long is it ?

8 . Cut a regular hexagon , side one inch , into six triangles.

Place three ina line , and fit in the other three so as to make arectangle . (Divide one of the triangles into two equalparts .)How long is the base of the rectangle ? What part Of the

perimeter ? About howlong is the perpendicular of the rectangle ?

9 . In a circle , radius 1 inch , inscribe a regular octagon.

Divide it into eight triangles, and make out of them a rectangle.

About what is the half perimeter of the octagon ? Its

apothem ? Its areaWhich has the greater perimeter , apothem , area, the hexagon

or the octagon ?

10. Find the approximate area of a regular hexagon,side 1

inch , apothem abou t inch .

1 1 . Find the approximate area of a regular octagon , side abou tinch , apothem aboutHe inch .

12 . If we inscribe ina circle a regular polygon of 16 sides,willits perimeter be greater or less than that of the octagon ?Which polygon willhave the greater apothem

13 . If we inscribe polygons of 32,64

, 128, etc . ,sides, what

will be the greatest perimeter we can have ina circle of 1 inchradius ? What willbe the greatest apothem ?

790 ARITHMETIC.

14 . Draw a rectangle that willbe about equalto a polygon Ofa million sides inscribed ina circle whose radius is 1 inch .

Mark upon i t the dimensions. Calculate the area.

15 . What is the area of a circle whose radius is 2 inches ?

16 . Find the area of a circle whose diameter is 10 inches.

17 Find the area of a circle whose circumference isinches.

18 . Calculate the area of a sector of a ci rcle whose radius is10 inches, the arc of the sector being

19 . How many square inches are there between the circumferences of two concentric circles whose radii measure 3 and

6 inches, respectively ?

SURFACES OF SOLIDS.

12 75. Prisms, Cylinders, Pyramids, Cones.

NOTE — The pupils should first examine a number of prisms and pyramids

, right and oblique, regular and irregular, triangular, quadrangular,pentagonal, etc. Right and Oblique cylinders and cones should also be at

hand .

A prism is a body bounded by plane faces,two of which are

equal and parallelpolygons, the remaining faces being parallelograms.

FIG . 1. FIG . 2. FIG . 3.

The two parallel faces of a prism are called its bases . The

remaining faces taken together constitute its convex surface.

792 ARITHMETIC.

u se. A pyramid is a body whoseconvex surface is made up of triangleshaving a common vertex, the base of

the pyramid being a polygon .

Pyramids are either right or oblique ; regF‘G 9° FIG 10

Ina right pyramid , each of the triangles that make up the

convex surface, is isosceles . When , in

addition , the pyramid is a regular one,these triangles will be equal to eachother.The altitude of any of these equal tri B

angles constitutes the slant height of a

right regular pyramid . The altitude Of

a pyramid is measured by a line drawnfrom the apex to the center of the base .

AG is the slant height of the square pyramid, Fig. 11. AF is its altitude.

FIG . 11.

1281 . The cone has a singlecircular base ; its convex surfaceis curved , sloping to the apex .

Inthe right cone, Fig. 12, HI is the

slant height , and HE is the altitude.

LO is the altitude Of the Oblique cone,FIG . 12.

Fig 13 .

12 82 . Surfaces of Prism, Cylinder, Pyramid, Cone.

1 . Draw the developed convex surface of a squareheight 3 inches, one side of base 1 inch .

2 . Draw the developed (entire) surface of a triangularheight 3 inches

,each side of base 2 inches.

3 . Draw the developed convex su rface of a prism 3

high,each base being a triangle having sides of 1

,

inches, respectively .


4 . Find the convex surface of each prism,and show that the

convex surface is found by multiplying the perimeter Of the baseby the altitude .

5 . Show that the convex surface of a cylinder is found bymultiplying the circumference of the base by the altitude .

6 . Draw the developed convex surface Of a square pyramid,slant' height 3 inches, each side of base 2 inches.

Cut it out of paper , leaving on one edge a smallstrip for gumming . Fold into a hollow pyramid,and measure its altitude .

7 . Show that the convex surface of a regular pyramid isfound by multiplying the perimeter of the base by one-half theslant height .

8 . Make out of paper a hollow square pyramid whose altitude shallbe 3 inches, each side of base 2 inches.

The developed convex surface of a cone is a

sector, ABDC

Howmany inches does the arc EDOmeasure whentheB

diameter of the base of the cone is 2 inches ?The slant height of the cone is the radius of the circle

of which the sector is a part. AB,AD

, or AC is the slant height.

9 . Cut out of paper a semicircle , radius 3 inches, adding anarrow strip for gumming

,and fold into a cone .

What is the slant height of the cone ? The diameter of thebase ? The radius of the base ? The circumference of the base ?What is the ratio between the radius of the base and the slant

height ? Between the diameter Of the base and the slant height ?

10. Calculate the convex surface of the above cone , and Showthat it is equal to the circumference of the base multiplied byone-half the slant height.

11 . Find the diameter Of the base of a cone made by foldinga paper sector of 90 degrees (quadrant), the radius of the sectorbeing 3 inches. What is its slant height ?

794 ARITHMETIC.

When a sector of 60 degrees is used , what willbe the di ameterand the slant height , the radius of the sector being 3 inches ?

(Make no allowance for overlapping.)

12 . If you wish to make a hollow paper cone whose slantheight shallbe 5 inches , and the diameter of whose base shallbe3 inches, howmany degrees should the arc Of the sector contain ?

13 . Draw the development of a right pyramid 4 inches inaltitude , whose base is a rectangle 3 inches by 2 inches. Is thealtitude (slant height) Of each Of the four triangles the same ?

14 . Calculate the Slant height of each convex face of a rectangular pyramid whose altitude is 12 inches, and whose basemeasures 10 inches by 18 inches.

Find the convex surface .

1284. Surface of Frustum of Pyramid and Cons.

When the upper part of a pyramid or of a cone is cut off bya plane parallelto the base , the remaining partis called a frustum.

15 . Draw one face of the frustum of a squarepyramid . Of the frustum of a triangular pyramid . What figure have you drawn ?Calculate its surface when the length of the

upper side is 4 inches, that of the lower side is 8 inches, and theslant height Of the frustum is 10 inches.

16 . Draw the developed convex surface of the frustum of a

regular triangular pyramid , each side of the upper base measuring 1 inch , of the lower base 2 inches, the Slant height being 2inches.

SUGGESTION. Locate the apex of thewhole pyramid ofwh ich the givenfrustum forms a part.

17 . Find the convex surface of the frustum of a squarepyramid , one side of the upper base measuring 2 feet

, of thelower base 3 feet, and having a slant height of 4 feet.Find the entire surface .

796 ARITHMETIC.

24 . Find the convex surface of the whole cone, ECB , and theconvex surface of the part cut Off, BAB .

A sp here is a body all points onwhose surface are equally distantfrom the center .The distance from the center to

the surface is called the radius

of the sphere . The diameter is aline running between two points onthe su rface and passing throughthe center.

CG ,CK , CD, CF, CA , and CI are rad i i ; AD and FG are diameters

1287 . If a sphere be cut through at any part, the cut surfacewill be a circle . When the cutting plane passes through the

center Of the Sphere, the circle is called a great circle ; other circles are called smallcircles .

FX GC is a great circle ; HYIB and ILEZ are smallcircles.

25 . Find the length of anarc Of 60°of a great circle of a

Sphere whose circumference is miles.

26 . Calling the arc AI inthe preceding figure, the angleBOI willmeasure Calculate the radius BI of the smallcircle when the radius CI Of the large circle is miles.

(IAH'

: arc Of IH= chord Of

27 If I is 60°from G ,

a point on the equator, find the lengthof the circumference of the smallcircle EYI, assuming the circumference Of a great circle to be miles.

28 . What is the ratio between the length of a degree on thesmallcircle E YI, and the length Of a degree of a great circle ?

29 . Calculate the radius of a smallcircle formed by passing aplane parallel to COEF through a point on GA 45 degreesfrom G .

ELEMENTARY GEOMETRY . 797

1288 . Surface of Sphere.

We have seen (Art . 1151)that it may be experimentally shownthat the surface Of a sphere is equalto thesurface of four of its great circles.

Calling the radius of the sphere R,its

su rface is 4 TR’.

30. Find the surface of a sphere whosediameter is 6 inches.

3 1 . How many square inches are thereinthe convex surface of a hemisphere whose radius is 3 inches ?What is the area Of the great circle that forms the base of the

hemisphereFind the entire surface of the hemisphere .

32 . IS there any difference between the convex surface of aSphere and its entire surface ? Why ?

VOLUMES.

1289 . Prisms and Cylinders.

1 . How many one-inch cubes will cover the base of a box 4inches by 3 inches ? If the box is 2 inches deep , how many one

inch cubes will it contain? How many cubicinches are there inthe volume of a right prismwhose base is a rectangle measuring 4 inches by3 inches

,and whose altitude is 6 inches ?

2 . If the above hollow prism were dividedinto two equalparts by a thinpartition extending from a verticaledge to one diagonally opposite

,how many cubic inches of sand would each

part contain ?

3 . How many cubic inches are there in thevolume Of a prism whose base is a right-angledtriangle 3 by 4 by 5 inches, and whose altitudeis 6 inches ?

798 ARITHMETIC.

4 . Find the volume of a triangular prism , the area of the

base being 6 square inches, and the altitude 6 inches.

Find the volume of a triangular prism , each sideof whose base measures 6 inches, its altitude being8 inches.

5 . What are the solid contents of a pentagonalprism formed by fastening together three triangularprisms whose bases contain , respectively , 12, 16, and18 square inches, the altitude Of each being 15 inches ?

6 . If a very great number of triangular prisms of

the same height are united so as to form a cylinderwhose base contains square inches

,and whose altitude

measures 5 inches , what are the solid contents ?

1290. Pyramids and Cones.

With a center at C, and a convenient radius,describe anarc AB. Mark

06'

four equalportions v, w, x, and g ; and drawthe equalchords . Cuttingou t CAvwxy, with anadditional narrowstrip along Cg for gumming, and creasingalong the lines Cv, Cw, Cx,

and Cg,we can

fold the paper into a square pyramid .

Measure its altitude, and make a square

prism of equalaltitude and with anequal

base.

Filling the pyramidwith sand , and pouring the sand into the prism,

itwillbe foundthat the latterwillcontainthe contents of the former three times ; that is,the volume of a square pyramid is one-th ird that Of a square prism havinganequalbase and anequalaltitude.

The same ratio willbe found true inthe case of a triangular, or anyother pyramid, as compared wi th the corresponding prism,

and of the coneas compared with a cylinder.

1291 . The volume of a pyramid or of a cone is equal,there

fore , to the area of the base multiplied by one-third of the

altitude .

800 ARITHMETIC.

11 . Find the totalvolume of three square pyramids, the alti

tude of each being 12 inches, and the areas Of their bases being 25sq . in. ,

225 sq . in.,and 75 sq . in.

, respectively .

12 . Find the number Of cubic feet ina block of stone whoseshape is that of a frustum Of a square pyramid 4 feet high

,each

side of the upper base measuring 3 feet, and each side of thelower base 5 feet .

1293 . The volume of the frustum of a pyramid is equalto the sum of

the volumes of three pyramids, each hav ing analtitude equalto that of thefrustum ; the base Of one of them being equal inarea to that of the lowerbase of the frustum, the base of a second being equalinarea to that of theupper base of the frustum , and the base of a th ird being a meanproportionalbetweenthe area of the other two .

Base of first = 3 x 3 sq . ft of second,5x 5 sq . ft. ; of th ird , V9 x 25

sq . ft. 15 sq . ft.

NOTE — The meanproportionalbetweentwo numbers is equal to the

square root of their product .

13 . Find the volume of the frustum of a square pyramid , itsupper base containing 64 square inches, and its lower base 196square inches, its altitude being 18 inches.

1294. Note that the meanproportional between64 and 196 is 8 x 14,or 112. Since each is multiplied by one-th ird of the altitude, the opera

tionis shortened by adding together the three areas , 64 , 196, and 112, andmultiplying their sum by one-th ird of 18 .

Calling the alti tude A ,the side of the large square S, of smallsquare 3 ,

the volume V, we have

ELEMENTARY GEOMETRY . 801

The volume of the frustum of a cone is found inthe samewayas that of the frustum of a pyramid .

altitude (area upper base area lower base area meanproportional).

Calling the radius of the upper base r, and that of the lower base R,the

area of the upper basewillbe arr”,of the lower base xR’

,of meanpropor

tionalrrR.

V==}A(rr’

IR“ 1 7 12)

Since 1 (or is a commonfactor,we cansave time by first addingr’,R’

,and rR, and thenmultiplying by r .

3

14 . The diameters of the bases of the frustum of a cone measure 8 and 15 inches, respectively ; the altitude is 18 inches.

Find the volume .

15 . How many cubic inches of water willa panhold , whoselower base is 12 inches indiameter , whose upper base is 16 inchesindiameter , and whose depth is 6 inches ? How many gallons ?

The pupils should make a frustum of a square pyramid Of convenient Size

,and the three corresponding pyramids, as

given inthe rule . Fillthe latterwith sand , and pour the contentsOf allthree into the frustum .

To make the frustum , drawtwo concentric circles. Lay of

equal arcs, AB ,BD

, DE,EF .

Draw the chords and radii from theextremities Of each chord . Drawthe chords ab, bd , de, and ef . Cut

out , after constructing a square foreither the upper or the lower base ,and taking care to provide flapsfor pasting .

802 ARITHMETIC.

To get a mean proportionalbetween ab and AB for one side ofthe base of the third pyramid , lay Off a line If equalinlength toab AB . On this line construct a semicircle .

Make IX equalto AB , and at K erect a perpendicular K ill. KMis a mean proportional betweenab and AB .

1297 . Oblique Prisms.

We have seen that a rectangle, ABCD, and a parallelogram ,

EFCII, are equal in area when thebases, AB and EF , and the altitudes,AD and HX , are equal, each to each .

This can be shown by cutting bothout Of paper , and by shifting the tri A B E

angle HEX to the right Side of the parallelogram .

12 98 . In a somewhat similar way, we can show that anOblique prism is equal to aright prism that has anequalbase and an equalaltitude .Make from a potato or a tur

nip an Oblique prism havingrectangular bases, and change it to a right prism of the same

height by cutting and shifting a portion .

1299. The volume of any prism (or cylinder) is found bymultiplying the area of the base by the altitude .

1300. Inthe same way it can be shown that the volume of

any pyramid or cone is equalto the product of the area of thebase by one-third the altitude .

1301 . The Sphere.

A Sphere may be considered as made up of a great number ofpyramids whose bases together make the surface of the sphere

,

TABLES

1303 . Measures of Extension.

LONGMEASURE. DRYMEASURE.

12 inches (in.) 1 foot (a.) 2 pints (Pt ) 1 quart (qt )3 feet 1 yard (yd .) 8 quarts 1 Peck (Pk )5} yards 1 rod (rd .) 4 pecks 1 bushel(bu .)320 rods 1 mile (

mi ) LIQUIDMEASURE.

A furlong (fun) 1mile'

2 pints (pt .) 1 quart (qt .)

Sum m ons,MEASURE.

4 quarts 1 gallon (gal.)A gillis pint.

Inches 1 huk The capacity of tanks, vats, etc .,

100 links 1 chain (ch .) is frequently expressed inbarrels of

80 chains 1 mile 31} gallons.

1 chaina 4 rods 66 feet. 1 qt. dry measure 67} cu . in.

1 qt . liqu id measure 57} cu . in.

SQUAREMEASURE.

144 sq . in 1 square foot9 sq . ft . 1 square yard30} sq . yd . 1 square rod160 sq . rd . 1 acre640 acres 1 square mile

10 square chains 1 acre.

CURIoMEASURE.

1728 cu . in. 1 cubic foot27 cu . ft 1 cubic yard

1 cord 128 cu . ft.

1 bushel cu . in.

1 gallon 231 cu . in.

TROY WEIGHT.

24 grains (gr.) I pennyweight

(pwt.)20 pennyweights, 1 ounce (oz .)12 ounces 1 pound (lb .)Troy weight is used inweighing

gold, silver, precious stones, etc .

APOTHEOARIES’

WEIGHT.

20 grains (gr) 1 scruple (29)3 scruples 1 dram (3)8 drama 1 ounce ( 3 )12 ounces 1 pound (lb)

806 ARITHMETIO.

Apothecaries' weight is used in

prescriptions. Drugs are bought andsold by avo irdupo is weight. The

grain, the ounce, and the pound

apothecaries’ weight are the same

as the corresponding denominationsof troy weight.

AVOIRDUPOIS WEIGHT.

16 ounces (oz .) 1 pound (lb .)2000 pounds 1 ton (T.)

1lb . avoirdupois 7000grains.

1 lb . troy 5760

loz . avoirdupois= 437}1 oz . troy 480

Inweigh ing ores and coalat the 60 seconds 1 minutemines and incalculating duties at 60minutes 0 1 degree (

0

)the U. S. custom houses, the follow 360 degrees 1 circumferenceing table is used

0

28 pounds 1 quarter (qr.)4 quarters, 1 hundredweight (cwt.)20 hundredweight 1 ton(T.)1 cwt. 112 lb. 1 T. 2240lb .

1 dime 10s.

1 eagle $10.

ENGLISHMONEY.

12 pence (d .) 1 shilling (s)20 shillings 1 pound

1 farthing penny .

£1

The Canadian dollar is equalinvalue to that of the UnitedStates, and is also divided into100 cents.

The French franc (fr.)19.3f, is divided into 100 centimes

The G erman mark (reichsmark) (M. ,

m .) is di

vided into 100pfennigs

60seconds (sec.) 1minute (min.)60minutes .1 hour (hr.)24 hours 1 day (da .)7 days 1week (wk .)365 days 1 common year (yr.)366 days 1 leap year

Years divisible by 4 and notby 100 are leap years.

1892, 1896 are leap years.

Years divisible by 100 butnot by 400 are common years.

1700, 1800, 1900 are common

years 1600and 2000are leap years .

808 ARITHMETIC.

The following table Shows the rates of interest allowed by law

inthe several states and territories, the first column giving therate allowed where none is specified inthe note or other docu

ment. Any rate up to the one given inthe second column is

permitted when anagreement is made inwriting .

States and Territories.States and Terri tories. m awt

Alabama

Arizona .

Arkansas

California .

Colorado

ConnecticutDelaware .

District ofColumbiaFlorida

G eorgia

Idaho

Illinois

Indiana .

IndianTerritoryIowa

Kansas

KentuckyLouisiana

MaineMarylandMassachusettsMich iganMinnesotaMississippiMissouri

MontanaNebraska

Nevada

NewHampshire

NewJersey .

NewMexicoNewYork

North Carolina

North DakotaOh io

Oklahoma .

Oregon

Pennsylvania

Rhode Island .

South Carolina

Sou th Dakota

Tennessee

Texas

Utah

VermontV irginia

Wash ingtonWest V irginia

W isconsin

Wyoming

APPENDIX .

PARTIAL PA YMENTS.

End the amount qf theprincip alto the time whenthe payment,or the sum qf two or more p ayments equals or exceeds the interest,

if such time is one year or more from the time the interest com

menced . From this amount deduct the p ayment or sum q)"

p ayments . Use the balance thendue as a new princip al, and

proceed as before.

If, however, any p ayments exceeding the interest thendue, aremade within a year from the time from which the interest is

reckoned,the amount qf the p rincip alis calculated for one year,

and from this amount are deducted the amounts q)"such p ayments

from the time made untilthe end of the year. Whenthe settlement is made inless than one year from the time from whichinterest is reckoned , the amount qf the princip aland those qf the

payments are calculated to the date qf settlement.

1 . NORWICH, CONN J an. 5, 1896.

Two years after date,we, jointly and severally , promise to pay

to Emerson W . Keyes, or order, Three Hundred Dollars, valuereceived , with interest at 6 per cent . JOSIAH H . PITTS ,

CHARLES W . FIELD.

Endorsements : May 20, 1896, $ 100 ; Nov . 2 , 1896, $ 100 ;March 17,1897

, $50.

How much is due J an. 5, 1898 ?

Amount Of $ 300 to J an. 5,1897, 1 year

Amount of $100fromMay 20,1896 to J an. 5, 1897, 7}mo .

Amount of $ 100 from Nov . 2,1896 to J an. 5, 1897 , 2

mo . 3 da.

Balance J an. 5,1897

Interest on$ 113 .20 to J an. 5, 1898

Amount .

Amount of $50 fromMarch to Jan. 5, 1898,

9 mo . 18 da.

Due Jan. 5, 1898

810 ARITHMETIC.

2. DANIELSONvILLE, CONN ,April15, 1896.

Ondemand , Ipromise to pay to the order of Robert P . Webb,

Two Thousand Dollars, value received , with interest .

$200010fi . JOHN J . BARNIOLE.

On this note are endorsed the following payments : $ 50,

Sept. 20, 1896 ; $ 100,May 26, 1897 ; June 20,1898 .

How much is due Dec . 27, 1899 ?

Face of noteAs the first payment, $50, is not large enough to meet the interest thendue, the interest on 2

,000 is calculated to dateof second payment,May 26, 1897, 1 yr. 1 mo. 11 da

Amount dueMay 26, 1897Less payments, $50 and 3 100

BalanceMay 26, 1897 3Interest on to June 20, 1898, 1 yr. 24 da.

Amount due June 20, 1898Less third paymentBalance June 20, 1898

Interest on to Dec . 27 , 1899, 1 yr. 6 mo. 7 da.

Due Dec. 27, 1899

3 . Find the amount due April16, 1898, on an interest-bearingnote for drawn J une 1, 1896 ; on which were pai d 150

,

Sept. 16, 1896 ; and $50, Sept. 16, 1897.

4. NEW HAVEN, CONN ,July 25, 1896.

Ondemand , Ipromise to pay to the order of Frank Regan,

Jr. , Five Hundred Dollars, with interest atJAMES W . NAUGHTON.

Endorsements : Sept . 18, 1896, $100 ; Feb. 5, 1897, $200.

How much is due April1, 1897 ?

5. HARTFORD, CONN J an. 2

,1894.

Three years after date, we promise to pay to the order ofG eorge G . Brown , Eight Hundred Seventy and 1

559, Dollars ,

value received , with interest.

WHITLOCK, TULLY , a Go .

812 ARITHMETIC.

7 . Hiram Stevens, of Hallowell,Me. , borrowed of John Karst ,June 15, 1895, seven hundred dollars, agreeing to repay it on

demand with six per cent per annum,and six per cent on the

deferred payments Of annual interest . The following paymentswere made : Nov . 15, 1895, $ 20 ; Feb . 15, 1897 , $ 80 ; Sept . 15,1898 , 15. Find the amount due Oct . 15, 1899.

1309 . NewHampshireRule.

InNewHampshire , partialpayments on a note wi th interest

annually are subj ect to the following rule

If the p ayments during a year equalor exceed the interest due

at the end of the year, the amount qf the payments to that date is

deductedfrom the amount qf the p rincipal.

If the p ayments are less than the interest due,the amount

qf these p ayments is used (1) to cancel the interest due onthe

annual interest, and (2) to redu ce the annual interest thendue.

But if no interest is du e excep t what is accru ing during the year,only the p rincip alqf the p ayments is thus app lied , no interest

being added .

NOTE — Whena payment is less thanthe interest due onthe annualinterest, the remaining simple interest is not subject to interest.

3 PORTSMOUTH, N.H., June

Ondemand I promise to pay to the order of W .M. DavisSix Hundred and 1

9090Dollars, value received , with interest

annually , at six per cent.

$ 60019096 . HENRY C. MOLEAN .

Endorsements : Feb . 1,1896, $ 30 ; July 1

, 1896, 100 ;

Oct . 1,1897 , $ 200. How much is due Oct . 1, 1898 ?

Principal

Annualinterest due June 1, 1896Payment Feb. 1

,1896

Balance of interest due June 1, 1896Annualinterest due June 1, 1897Interest on 6 for 1 year .36

Amount due June 1 , 1897

APPENDIX . 813

BroughtforwardAmount of $ 100 July 1, 1896 to June 1, 1897

NewprincipalJune 1, 1897Interest on to June 1, 1898

Amount due June 1,1898

Amount of $200Oct. 1, 1897 to June 1, 1898

NewprincipalJune 1, 1898Interest on to Oct. 1, 1898

Due Oct. 1,1898

9. CONCORD, N.H May 1, 1896.

Ipromise to pay Albert S . Caswell, on demand , Two ThousandFive Hundred Dollars

,with interest annually .

$ 25OOIQGQ0" WALTER B. GOODNOUGH .

Endorsements : Oct . June 1, 1897 , Nov .

1,1898 , 50. Find the amount due Oct . 1

,1899.

10. How much is dueMarch 11 , 1899, upon a note forwith annual interest , drawn J an. 3

,1895, on which the follow

ing payments have been made : June 1, 1895, $ 10 ;March 14,1896

,10 ; Sept . 30, 1898 , $500?

NOTE— Ordinary interest-bearing notes onwh ich partial paymentshave beenmade, are subject to the US . Rule.

1310. Vermont Rule

Partialpayments on annualinterest notes are used inVermont (1) to cancelthe interest due on the annualinterest ; (2) toreduce the annualinterest then due ; (3) to reduce the principal.

If thep ayments during a year equalor exceed the interest due

at the end of the year, the totalof the amounts of thep ayments is

dedu ctedfrom the amount of thep rincip al.If the totalpayments are less thanthe interest due, the amounts

of thesep ayments are used (1) to cancel the interest due on the

annualinterest, and (2) to redu ce the annualinterest thendue.

814 ARITHMETIC.

NOTE.-Whenthe amount of a payment is not sufi cient to cancelallthe

interest due onthe annualinterest, the balance of th is simple interest isnotsubject to interest.

11. BURLINGTON, VT , June

Due Halsey Clark , Six Hundred Dollars, payable on demand ,with interest at six per cent annually .

CHARLES HERMAN .

Endorsements : Feb . 1, 1896, $ 30 ; July 1, 1896, $ 100 ; Oct .

1, 1897 , $200. How much is due Oct. 1, 1898 ?

Principal

Annualinterest due June 1, 1896Amount of $ 30 Feb . 1, 1896 to June 1, 1896

Balance of interest due June 1, 1896Annualinterest due June 1, 1897Interest on$5.40 for one year

Amount due JuneAmount of 100 July to June

NewprincipalJune 1, 1897Interest on$ 536.22 to June 1

,1898

Amount due JuneAmount of $ 200Oct. 1, 1897 to June

NewprincipalJunc 1, 1898Interest on to Oct. 1, 1898

Due Oct. 1, 1898

12 . RUTLAND, VT. , June 1, 1895.

Ipromise to pay Ahern Bro . ,on demand , Two Thousand

Five Hundred Dollars, with interest annually .

$ 2500fi °c~ MAGNUS SCHULER.

Endorsements : Oct. 1 , 1896, $ 100 ; June 1, 1897,Nov . 1, 1898 , $ 50. Find the amount due Oct . 1, 1899.

13 . Find the amount dueMarch 11, 1899, upon a note forwith annualinterest, drawnJ an. 3 , 1895, onwhich the

following payments have been made : June 1, 1895, $ 10 ;March 14, 1896, $ 10 ; Sept. 30, 1898 , $500.

16 ARITHMETIC.

Realand personalproperty is placed inthe town“ list at1% of its appraised value .The grand list comprises the totalof the polllist and 1%

of the assessed (appraised) value of the real and the personalproperty

,and is the amount on which alltaxes are levied .

3 . The realestate of a towninwhich a tax Of is to beraised , is appraised at and the personal property at

There are 100 taxable polls. What is John G riffin ’

s

tax,who pays two polls, the appraised value of his realproperty

being and of his personalproperty ,

appraised value Of real

and personalproperty .

Th of a l of appraised value.

2 x 100 $ 200, the polllist.200 the grand list.

rate on 1 Of the grand list.

[1 of 2 X 2) 104, JohnGrifin

’

s grand list.x 104 156

, JohnGrifin’

s towntax .

4 . Find the city tax of Parker P . Simmons, whose property ,realand personal, is assessed at and who pays one poll

,

the rate being

5 . How many taxable polls are there ina town inwhich theappraised value of the real and personal property isthe tax rate necessary to raise being $2 ?Let x the number of polls ; 2 x polllist.

6 . Henry W . Hallock ’

s taxes on real and personalproperty,

appraised at amount to $ 252, including 3 poll taxes.

The totalamount levied is Find the appraised value of

all'

the property,the number of taxable polls being 200.

13 13 . Values of Foreign Coins.

The following estimate by the Director Of theMint , of thevalues of foreign coins, has been proclaimed by the Secretaryof the Treasury , January 1, 1897 :

APPENDIX .

VALUES OF FOREIGN COINS.

COUNTRY.

Argentine RepublicAustria-HungaryBelgiumBoliviaBraz il

Bri tish Possessions N . A . (exce

pt Newfoundland)

Centre Amer. States

Cuba

G ermanEmpire

Standard .

G old and silverG oldG old and silverS ilver

Silver

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

0 0 0 0 0 0 0 0 0 0 0 0 0 0

Monetary uni t.

Pound (100 plasters)

Pound sterlingDrachma

0 0 0 0 0

817

Angles, acu te, 755, 757 , 759, 763 .

A d jacent, 755, 758, 759.

Construction of, 757 , 758, 759

760, 761, 762, 763 , 764, 765

767, 775, 776, 777 , 778, 779

781, 783 .

Designationof, 756.

Exterior, 763 .

Formed by parallel lines, 761,762, 763 , 788 .

Kinds of, 755 .

Measurement of, 756, 757, 758759, 760, 761, 762, 763 , 764

765, 767 , 775, 776, 777, 778 .

Oblique, 765 .

Obtuse, 755, 759, 764 .

Of polygon, 772 , 777 .

Of triangle, 762 , 763 , 777 .

Opposite, 755, 759 .

Righ t , 755, 757 , 758, 759 .

Vertical, 755, 759 .

Annualinterest , 719, xv i i i .Part ialpayments onannualinterest notes, 811, 812 .

NewHampshire rule, 812,

813 .

Vermont rule, 813 , 814, 815.

Antecedent , 610, xxi .Apothecaries’ weigh t, 499 , 805 .

Apo them of regular hexagon, 789.

Of regular octagon, 789 .

Applicat ions of algebra to ari th

metical problems, 461, 467

486, 514, 515, 519, 534 , 551

558, 584, 585, 660.

To geometrical problems,753 .

Of square root , 634, 650, 656665, 667 , 668 , 672, 690, 694

711, 712 , 721, 722 .

Approximations ( see Manual),483 , 562 .

Arabic notation, i .Arc , 695, 770, 775, 778 .

Length of, 578, 579, 770, 775,778 , 788 , 796.

Area of circle, 601, 653 , 667, 695,

Of ring, 696, 790.

Of sector, 669, 695, 790.

Of segment, 695, 704,

INDEX .

Area of rectangle, 487 , 496, 502508 , 514, 515, 516, 539, 541

551, 552, 557, 572 , 595, 597

601, 621, 622, 633 , 675, 717

722,748, 753 , 782.

Of regular polygon, 789.

Of hexagon, 668 , 789.

Of octagon, 789.

Of rhomboid and rhombus , 516,517, 667, 668 , 768 .

Of7

trapez ium ,517 , 652 , 668 , 678 ,

69.

Of trapezo id , 516, 551, 643 , 644 ,668 , 769 .

Of triangle , oblique-angled , 516538 , 601, 652, 653 , 667, 668

678, 754 .

Righ t-angled , 515, 550, 643 ,653 , 666, 668 , 722 , 749, 753 .

Assessed value ( see Taxes), 463475, 501, 511, 549, 564, 566

574, 593 , 673 , 680, 698, 815

816, xv i .

Assets, xx i i .

Avails ( see Proceeds), 489, 572,599, 683 .

Average of accounts ( seeEquationof payments), 657 .

Term of cred i t , 658 .

Rules,etc .

,xxi .

Avoirdupo is weigh t, 499 , 806Long ton, 806.

Balance,x i i .

Bank ch eck ( seeManual), xxv i .Bank d iscount ,definitions ,etc .

,xix .

Of interest-bearing no tes, 503 ,504

, 614 , 639 .

Of non-interest-bearing notes,489, 490, 491, 513 , 514 , 515,526, 527 , 528 , 533 , 535 , 571,572, 583 , 594 , 596, 599 , 606,615, 63 1, 636, 651, 656, 670,676, 683 , 684 , 686, 688 , 702 ,703 , 710, 718 .

Days of grace, 489, 579, 807 .

Proceed s (Avails), 489, xix .

To find face of note, etc . ,

534

535, 593 , 597 , 598 , 615, 636

639, 647 , 686, 711, 714.

Barrel, 805.

INDEX .

Base, bonds, 564, xxv .

Commerc iald iscount,xiv .

Commission, 564, xv .

Duties, xv i .Insurance, 564, xv .

Line (U . S . lands), 720.

Of isosceles triangle, 765.

Of prism , 790.

Percentage, 461, xi i i .Profit and loss

,464, 465, xiv .

Stocks, 564 , xxv .

Taxes, 564 , xv i .

Bill ( seeManual), 561, 598, 624,648, 671 .

Definitions, etc . ,xi i .

Howreceipted , x i i .

Of exchange, 576, 585, xxvi .Bins and tanks, ca

gac ity of

,467 ,

474, 488 , 500, 13 , 550, 552 ,568 , 584 , 600, 601, 604 , 630,650

, 689, 723 , 727 .

Bisectionof angle , 775.

Of arc , chord ,ah d sector,7 5 .

Of line , 773 .

Board foo t , 692 .

Measure, 692 , 693 , 694 , 721, 727 .

Bonds and stocks, 466, 502 , 562

563 , 564 , 568 , 575, 596 , 600

638 , 639, 652 , 670, 683 , 691

699 , 704 , 711, 714 .

Brokerage, 564 , 638 , 639, 652,695, 713 , xxv .

Defini t ions, etc . ,xxv .

Broker,xxv .

Brokerage ( see Bonds), 594 , 638,xxv .

Bus iness forms, 472 , 489, 492 , 503 ,561, 576, 579, 585, 588, 610,655, 677 .

Definit ions, xx iv .

Calculation of h eigh ts and d is

tances, 584 , 606, 785, 786, 787 ,788 .

Canad ianmoney, 806.

Cancellat ion( seeManual), 508 .

Definitions, etc ., v .

Capacity of b ins and tanks , 467474 , 488 , 500, 513 , 550, 552

568 , 584 , 600, 601, 604 , 630

650, 689, 723 , 727 .

821

Cap ital, xxi i .

Stock , xx iv .

Carpeting, 467 , 529, 542 , 551, 597 ,602, 671, 688 , 7 17 , 727 .

Cause

53

and efl ect ( see Analysis),5

Cha in ( see Surveyors’ measure),635, 644 .

Check seeManual), xxv .

Chord,695

, 770.

Length of, 770, 780, 788, 796.

Circle,area of

, 601, 653 , 667, 695,722, 790.

Circumference of,539, 578, 612 ,

613,653 , 667 , 711, 727 , 769.

Circumscribed , 668 , 780.

Concentric,696, 770, 790.

Diameter of , 539, 578, 613 , 653 ,667, 668 , 704 , 769 .

G reat , 697 , 796.

Inscribed , 780.

Rad ius of, 653 , 667 , 769.

Small, 796.

Tangent , 770, 782 .

Circularmeasure, 578 , 756, 806.

Circumference of c ircle, 539, 578

612 , 613 , 653 , 667 , 711, 727769 .

Circumscribed circle, 668 , 780.

Pol

l

ggon, 771, 772 , 775, 780, 781,2 .

Clearing of fractions, 451 .

Coeffic ient, 728 .

Commercial d iscount , 509, 510

511, 529, 530, 531, 561, 586

592, 627, 628, 629, 662, 666

669, 677, 686.

Definitions, xiv .

Comm ission, 462 , 463 , 474, 475

483 , 495, 498 , 533 , 539, 545

548 , 568, 571, 572, 577 , 591

593 , 598, 614, 670, 672, 686

687 , 713 .

Agent , xiv .

Defini tions, etc . ,xiv .

Commondenominator, ix .

Factor, v .

Fraction, vi i .Mult iple, v i .Prime factor, v .

Complex decimal, 584, x .

822 INDEX .

Complex fraction, 474, 475, 476604, 632, 656, 704, 706, 707

709, 712, 714 , 715, vi i i .Reductionof, 474, 475, 476,

604, 632, 656, x.

Composite factor, v .

Number, v .

Compound add i tion, 499, 544, 577,594, 632, 665, 726.

Dec imal, x .

Denom inate number, x i i .Division, 475, 480, 500, 503 , 507513 , 514, 546, 549, 550, 574

582, 600, 630, 631, 632, 645

646, 648, 671, 680, 685, 687694 , 717 , 726 .

Fraction, 475 , 476, 479, 529, 688704 706, 709, 712, 714, 715

v i i i

Interest, 565, 566, 591, 594 , 606,xvi i i .Multiplicat ion, 477 , 499, 500

504 , 506, 507 , 508, 550, 561

578 , 582 , 663 , 677 , 726.

Numbers, reduct ionof, 504, 560,632, 645, 647

,649, 656, 665,

671 .

Subtraction, 507 , 544 , 546, 584,594, 632 , 665, 691.

Concentric c ircle,696, 770, 790.

Concrete number, i .Cone, defini tions, etc .

, 792 .

Surface of

,682

, 683 , 703 , 793 ,96.

Developed surface of,682,

793 , 794 .

V olume of,689, 690, 798, 802 .

Surface of frustum , 795.

Volume of, 801 .

Connecticut Rule, partial payments ( seeManual), 809, 810,811.

Consequent, 610, xxi .Consignee, xv .

Consignment , xv .

Consignor, xv .

Construct ionof angles, 757, 758759, 760, 761, 762 , 763 , 764

765 , 767 , 775, 776.

Of parallellines, 760, 761, 762 .

Of parallelogram , 767 .

Constructionofrectangle, 768, 769,776, 782, 790.

Of regular polygon, 772, 774,775, 776, 777, 788 , 789.

Of rhombo id , 768 , 776.

Of rhombus, 767, 776, 777 , 781.

Of square, 767, 771, 772, 774,775, 776, 777 , 781.

Of trapez ium, 768 , 769.

Of trapezoid , 767 , 768, 769.

Of triangle, acute-angled , 769,782.

Equilateral, 764, 765, 772,773 , 775, 777 , 778, 780, 781, 782.

Isosceles, 765, 773 , 774 , 776

777, 780, 782.

Obtuse-angled , 769, 781, 782 .

Scalene, 762 , 774, 776, 777 ,778, 780, 782, 783 , 784.

Righ t-angled , 764, 765, 776,777, 778, 779 , 782 , 784.

Convex surface, of cone, 682, 793,

796.

Of cylinder, 681, 698, 791, 793 .

Of hem isphere, 797 .

Of frustum , 794, 795.

Of prism , 681, 682, 790, 793 .

Of pyram id , 682, 690, 793 , 794 .

Of solids, development of, 681,682 , 683 , 792, 793 , 794 , 795.

Couplet , xxi .

Cred itor, x i .

Cross add ition,528, 573 , 605, 680,

698.

Subtraction, 485.

Cube ( th ird power), 699.

Defini tions, .xxi i i .

Root,699

, 700, 710, 711, 712 , 7 16.

Of dec imals, 700, 710, 712 .

Of fractions, 700, 712 .

Cub ic measure , 488, 518 , 805.

Customs ( see Duties), xvi .Cylinder, definit ions, etc .

, 791.

Surface of, 681, 682 , 698 , 712,791, 793 .

Volume of, 689, 797 , 802 .

Dates, t ime between, 807, x i i i .Days of grace ( seeManual), 489,

579, 807 .

Debtor, xi .

824 INDEX .

Definit ions, etc . , Continued .

Concr’

ete number, i .Cone

, 792 .

Consequent, 610.

Consignee, xv .

Consignment, xv .

Consignor, xv.

Convex surface, 681, 790.

Couplet , xxi .

Cred itor, x i .

Cube ( th ird power), 699, xxi i i .Root, 699, xx i i i .

Cylinder, 791.Debtor, x i .Decagon, 771.Dec imalx .

Frac tion, x .

Po int, x .

Demand note, xxv .

Denom inate fract ion, x i i .Number

,x i i .

Uni t , xi i .Denom inator, vu .

Diagonal, 774Diameter of c ircle , 769 .

Di fference , subtract ion, i i .Percentage, xi i i .

Discount , stocks, xx iv .

Dividend , d ivision, iv .

Stocks, 562 , xxv .

Division,iv .

Div isor, iv.

Domestic exchange, xx.

Draft,576, xxvi .

Drawee, xxvi .Drawer, xxvi .Du ties, xv i .Endorser, xxv .

Equaltriangles, 783 .

Equated t ime, 658 , xx .

Equat ionof payments, 658 , xx .

Equ ilateraltriangles , 764 .

Equ ivalent triangles, 783 .

Evennumber, v .

Evolut ion, xx i i i .Exact d iv isor, v .

Interest, 648 .

Exchange, domest ic , xx .

Foreign, xx .

Exponent, 745, xx i i i .Exterior angle, 763 .

Defini tions, etc . , Continued .

Extremes,625.

Face of note, xxv .

Factor, v .

Factoring, v .

F ire insurance, xv .

Fore ignbills of exchange, 576.

Exchange, xx.

Fract ion,v i i .

Fractionalam it , vi i .Frustum , 794 .

G reat c ircle, 697 , 796G reatest commond iv i sor, v.

Factor, vMeasure , vv .

Hep tagon, 771.

Hexagon, 771.

Hogshead , 529 .

Horizontalline , 758 .

Improper frac t ion, v .

Indorser, xxv .

Insurance,xv .

Insured,xv .

Insurer,xv.

Interest,xv i .

Interest-bearing note , xxv .

Invo ice , x i i .Involu t ion

,xxi i i .

Isosceles triangle , 764.

J o int and severalno te, xxv i .J o int note, xxv .

Least commondenom inator, ix .Mult iple, v i i .Legalrate of interest , xv i .Liab ilities

, xxi i .

Life insurance, xv .

Like numbers, i1 .Maker of note,xxv .Marine insurance

,xv .Market value of stock , etc xxiv .Maturi ty of no te

, xxv i .Means,625,Measure, x i i .Minuend , i i .Mixed dec imal

,x .

Number, vi i i .Mult iple , vi .Mult iplicand , in.Mult iplicat ion, i i i .Multiplier, i i i .Negotiable note, xxvi .

INDEX .

Definitions, etc .,Continu ed .

Net price, x iv .

Proceeds, xv .

Nonagon, 77 1 .

Non-negotiable note, xxvi .Notat ion, i .Arab ic , i .

Roman,i .

Number, i .

Numerat ion, i .Numerator, vi i .

Oblique angle, 755.

Oblique-angled triangle, 763 .

Oblique line, 758 .

Solid , 791 .

Oblong, 766.

Obtuse angle, 755.

Obtuse-angled triangle, 763 .

Oc tagon, 771 .

Odd number, v .

Opposite angles, 755.

Orders of units , i .Parallellines , 760.

Parallelogram , 766.

Parallelop ipedon, 791 . ii i

Partialpayments, xix .

Partners, xx i i .Partnersh ip , xxi i .Par value, xxi v .

Payee , xxv .,xxvi .

Pentagon, 77 1.

Pentagonalprism , 791.

Per cent, x i i i .Percentage, x11i .Period , i .

Perpend icularlines, 759 .

Personalproperty, xvi .Tax ,

xv i .

Place o f figure, i .

Policy, xv .

Polltax ,xvi .

Polygon, 77 1.

Power,xx i i i .

Prem ium ,insurance, xv .

Stocks, etc . ,xxv .

Present worth , 592 .

Prime factor, v .

Numbers , v .

Princ ipal, xv i .Principalmerid ian, 720.

Prism , 790.

825

i i i

Definitions,etc . ,

Continu ed .

Proceeds,xix .

Product,i i i .

Profit or loss,x iv.

Prom issory note , xxv .

Proper fraction, vi i .Proportion, 624 .

Property, xv i .

Tax, xvi .

Protractor, 757.

Pyram id , 792 .

Quadrangular prism, 791.

Quadrant , 770.

Quadrilateral, 766.

Quotient, iv .

Rad ius, 769.

Range (U. S . lands), 720.

Rate , 461, xi ii , xv i .Rat io

,610

,xxi .

Realproperty, xv i .Receipt , xi i .Rec iprocalof a fract ion, vi i i .Of a number, v i i i .

Rectangle , 766.

Reduc tionascend ing, x ii i .Descend ing, xi i .Of denom inate numbers, xi i .Of fract ions, v i i i .

Regular polygon, 771.

Solid , 791.

Remainder,i i .

Rhombo id, 766.

Rhombus,766.

Righ t angle, 644 , 755.

Righ t-angled triangle , 763 .

Righ t solid , 791.

Root , xxi i i .Romanno tat ion

,i .

Scalene triangle, 764 .

Secant line, 780.

Second power, xx i i i .Sect ion(U . S . lands), 720.

Sector, 669, 770.

Segment , 695, 770.

Sextant, 770.

Share (stock), xx iv .

S igh t draft , xxvi .

S imple denom inate number, x i i.Frac tion

,v i i i .

Slant h e igh t , 792 .

Smallc ircle, 796.

826 INDEX .

Definitions , etc ., Continued .

Spec ific du ty, xvi .

Sph ere , 796.

Square , 766.

(Second power), 607 , xx i i i .Root, xx i i i .

Stock , xxiv .

S tockholder,xxiv.

Subtraction, i i .Subtrahend , i i .Sum ,

i i .

Discounted , xix.

Tangent c ircles, 770.

Line, 780.

Taxes,xvi .

Term of d iscount , 490, x ix .

Terms of fraction, vi i .Th ird power, xx i i i .Time draft, xxvi .

Note , xxv .

Townsh ip (U. S . lands), 720.

Transposing, 455.

Trapez ium , 766 .

Trapezo id , 766.

Triangular prism , 791.

True d iscount, 592 .

Underwriter, xv .

Uni t, i .Of fraction, vn.

Unlike numbers, i .Value of fraction, v i i .Verticalangles, 755.

Line, 758.

Degrees of longitude, length of,

578, 579, 582, 676, 796 .

Demand note, 472, 588 , 651, 653 ,810, 811, 812 , 813 , 814, 815, xxv.

Denom inate fraction, 471, 479,550, 632, 688, 706, 709, x i 1.

Denom inate numbers, add itionof

499,544 , 577 , 594, 632 , 665

726.

Definitions, etc ., xi i .

Divisionof, 475, 480, 500, 503507, 513 , 514, 546, 549, 550574, 582, 600, 630, 63 1. 632

645, 646, 648, 671, 680, 685

687 , 694 , 717 , 726.Metric system , 721—725, 727 .Miscellaneous, 474 , 475, 482

483 , 494, 495, 496, 499 , 500

Denom inate numbers, Continued .

501, 503 , 506, 508, 512, 529,538, 542, 546, 550, 556, 560,562, 568, 570, 577, 578, 579,582

, 585, 597, 601, 604, 630,63 1, 632, 633 , 645, 646, 649,650, 656, 663 , 664, 665, 666,670, 671, 677, 680, 684, 686,687, 694, 706, 715, 716, 717 ,718.Mult iplicat ionof, 477 , 499, 500504, 506, 507 , 508 , 550, 561

578, 582, 663 , 677 , 726 .

Reductionof, 494, 504, 539, 560,564, 565, 596, 621, 632 , 645,647, 649, 656, 665, 671, 672 .

Sub traction of, 506, 544 , 546,584, 594, 632, 665, 686, 691.

Denom inate uni t , x i i .Dengm inator, 705, 706, 707 , 709,41.

Least common, 707 , 709 .

Developed surface o f cone,682

,

683 , 793 , 794 .

Of frustum of cone, 794 , 795Of

0

frustum of pyram id , 194 ,8 1.

Of prism (see Manual), 467,486, 681, 792 .

Of

(pyram id , 682 , 793 , 794 , 798,8 l.

Diagonalof polygon, 774 , 777 .

Of rectangle, 644, 656, 672, 711.

Of rhomboid, 779.

Of rhombus, 653 , 668 .

Of square, 667 668 .

Of trapez ium , 517 , 652, 678Of trapezoid , 517 .

Diameter of c ircle, 539, 578, 612

613, 653 , 667 , 668 , 696, 704

722, 727 , 769, 790.

Of sphere, 697 , 698 , 701, 702 ,703 , 795, 797, 803 .

Difierence, betweendates, 807,x i i i .Subtract ion

, i i .Percentage , x i i i x iv . ,

xv . , xxv .

Discount, bank , 489, 490, 491, 503504, 513 , 514 , 515, 526 , 527528, 533 , 534, 535, 57 1, 572583 , 593 , 594 , 596, 597 , 598599, 606, 614, 615, 631, 636

828 INDEX .

Examinationquest ions, Continued .

546, 547 , 548 , 556, 567 , 568

570, 573 , 574 , 575, 583 , 590

591, 593 , 595, 597, 599, 603

604, 605, 613 , 619, 623 , 633

636, 639, 640, 647, 649, 651

656, 660, 661, 662, 663 , 665

669, 670, 671, 672, 683 ,686, 690, 694 , 698, 702, 704

711, 712 , 713 , 714, 715, 717718 , 726.

Exch ange, domestic, 575, 576, 577 ,579, 580, 634, 636, 704 .

Foreign, 585, 586, 587 .

Definit ions, etc .,xx.

Exponent, 745, xx i i i .

Face of bill of exchange, to find,

586.

Of draft, 576, 577, 580, 634,xx .

Face o fnote, xxv .

To find , 534 , 535, 593 , 597 , 598615, 636, 639, 686, 688, 711

714 , 718, x ix .

Factors, 715, v .

Prime, 656, v .

Federalmoney, 806 .

Fractionalparts of a dollar(oral)482 , 483 , 494 , 538 , 583 , 596

621, 622, 661, 669, 676, 684

685, 713 .

Fence, area of,468 , 486, 512, 552 .

Fire insurance (see Insurance), xv .

Foreignco ins,values of, 817 .

Foreignexchange , 585, 586, 587,xx .

Fractionaluni t, v i i .Fractions, add it ion of, 507 , 549

573 , 605, 632, 641, 647 , 694

704, 706, 707 , 709 .

Clearing of, 451.

Complex ,474 , 475, 476, 604, 632

656, 704 , 706, 707 , 709, 712

714, 715.

Compound , 475, 476, 479, 5296

8g,704, 706, 709, 712, 714

7Cube root of, 700, 712, xxiv .

Dec imal( see Dec imals), x .

Definit ions , princ iples, andrules, vi i . , v i i i . , ix. , x . , x11.

Fractions, denom inate, 471, 479,550, 632, 688, 706, 709.

Divisionof ( see Complex fract ions), 479, 507 , 529, 550, 567 ,583 , 600, 604 , 63 1, 706.

Miscellaneous, 507 , 512, 536, 549,583 , 600, 604 , 619, 631, 633 , 712 .

Multiplicat ionof (see Compoundfractions), 484, 493 , 616, 650,661.

Reductionof,474, 475, 549, 624,

632, 650, 656, 666, 707, 7 14 .

Square roo t of, 618 , 663 , xxiv .

Subtraction of, 549, 704, 706,707, 718 .

French money, 586, 806.

Frustum of cone, 794 .

Developed surface, 795.

Surface, 795.

Volume, 801.

Frustum of pyram id , 794, 795.

Developed surface, 794 , 801 .

Surface, 794, 795 .

Volume, 799 , 800.

Furlong ( see Long measure),604.

G eometricalexercises, 755—803 .

Angles, 755, 756, 757, 758 , 759760, 761, 762 , 763 , 764 , 765

767, 775, 776, 777, 778 , 779

781, 783 , 788 .

Calculating heigh ts and d is

tances, 785, 786, 787, 788 .

Circles, 769, 770, 771, 772 , 773

774, 775, 777, 778, 780, 781

782, 788 , 789, 790, 796 .

Construct ionproblems, 773- 782 .Mensurat ion, 788-803 .

Parallels, 760, 761, 762, 763 , 785,786, 787 , 788 .

Polygons, 771, 772, 774, 775, 777 ,782, 789, 790.

Quadrilaterals, 766, 767, 768 , 769772, 775, 776, 777, 778 , 779

780, 781, 782 , 788 .

Triangles, 762 , 763 , 764 , 765

768 , 771, 772 , 773 , 774, 775776, 777, 778, 779, 780, 781

382, 783 , 784, 785, 786, 787

88 .

INDEX .

G ermanmoney, 586, 806.

G overnment lands, 720, 721, 727 .

G race,days of, 489, 579, 807 .

G reat circle, 697 , 796 .

G reatest commond ivisor, 502, 549550

,.

557, 575, 670, 705, 719

v ., v i .

Fac tor, v .Measure

,v .

Heigh ts, calculationo f, 584 , 606,785, 786, 787 , 788 .

Hep tagon, 771, 772.

Hexagon, 771.

Regular, apo th em of, 789 .

A rea of, 668 , 789 .

Constru ct ionof, 771, 772 , 777 ,

Perimeter of, 789 .

Hogshead , 547 .

Horizontalline,758.

Hypotenuse, 635,643 , 648 , 656

662,663 , 665, 672 , 675, 694

702, 711, 712 , 721, 753 , 754

778, 779, 794 .

Improper fract ion, v .

Indorsement ( see Endorsement),X IX .

Indorser ( see Endorser), xxv .

Inscribed polygon, 771, 772 , 774,775

,777

, 782 , 789.

Square , 772 , 774, 775.

Triangle, equ i lateral, 772 , 775,777

,780.

Righ t-angled , 777, 778, 780.

Scalene , 777 , 780, 781.

Insurance , 463 , 464 , 488, 495, 498548 , 564, 570, 597 , 613 , 650

651, 665, 669, 673 , 686, 714 .

Definit ions,etc .

,xv .

Interest, 462 , 464 , 466, 471, 472473 , 474 , 475, 480, 483 , 488491

,494 , 496, 497 , 511, 512

513, 514, 515, 519, 520, 521

522 , 523 , 524 , 525, 526, 527

533 , 536, 545, 546, 547 , 548

557, 558, 564, 568 , 571, 572

574, 575, 583 , 585, 588 , 589

590, 592 , 593 , 594 , 596, 604

605, 606, 614 , 615, 624, 626

829

Interest , Continued .

630, 636, 637 , 646, 647, 648649, 650, 651, 653 , 654, 655

656, 657 , 658 , 659, 661, 663

669, 671, 672

713 , 714 , 716

Accurate, 511, 636,

711.

Annual, 719, 811, 812 , 813 , xv i i i .Compound , 565 , 591, 606, 63 7,646, 647 , 714, xv i i i .

Definitions, etc ., xvi . , xv i i . ,

xvi i i .,x ix .

Exact , 511, 636, 648, 649, 669,711

,xvi i i .

G eneralmethod of compu ting,471, xv i i .

Legalrates of, 808 .Method by aliquot parts, 523 ,614, 615, xvi i .

Notes bearing, 472, 496, 497, 503 ,542 , 585, xxv .

Part ial payments, 588, 589 ,590, 592, 614, 636, 651, 653 ,654 , 809 , 810, 811, 812 , 813 ,814, 815, x ix .

S ix per cent method,xvi i i .

To find princ ipal, rate, or t ime519, 583 , 594 , 604 , 630, 649

650,661, 663 , 664, 665, 671

713 , 716, xv i i .

Invo ice,492

, 561, 610, 677 .

Definit ion, xi i .Involu tion

, 607 , 699 .

Definit ions, etc ., xxu i .

Isosceles triangle, altitude of, 652

$53 , 682 , 765, 773, 776, 782

92 .

Area of, 652 , 653 , 667, 682 .

Base of, 765.

Construction of, 765, 773 , 774,776, 777 , 780, 782 .

Definit ion, 764 .

J o int and severalnote, 809, xxv i .J o int note, xxv .

Leap year, 806 .

Least commondenom inator, 709 .

Defini t ions, etc . , ix

830 INDEX .

Least commonmultiple, 502, 549,550

,557 , 575, 670, 719 .

Defini tions, etc . ,v i .

,vi i .

E dgeraccount ( seeManual), 655,664, 692 , xi i .

Legalrate of interest , 608 , xvi .Liabilities, xxi i .

Liab ility of indorser, xxv .

Life insurance, xv.

Like numbers, i .Lines, hori zontal, 758, 759, 760,

761, 774, 779.

Oblique, 758, 759, 760, 761, 765,773 , 774 , 775, 776, 779.

Parallel, 760, 761, 762 , 774 .

Perpend icular, 759, 760, 775, 777 .

Tangent, 780.

Secant, 780.

Vert ical, 758, 759, 760, 765, 773 ,774, 775.

Link ( see Surveyors’ measure),635.

Liqu id measure , table, 488, 805 .

Longitude, and t ime,580, 581

582, 584, 593 , 616, 617 , 618

$3

2,671, 685, 703 , 704, 712

2

Length of degrees of longitude ,578, 579, 582 , 676 .

Long measure, 504, 564, 565, 645,647 , 805.

Long ton, 506, 507 , 529, 539, 600,604, 633 , 664, 691, 806.

Lowest terms, 549, 632, 650, 666,705, vi i i .

Lumber measure, 692 , 693 , 694,721, 727.

Market value of stocks and bonds,xx iv .Maturi ty of note ( see Days of

grace), xxv i .Means,625.Measure, x i i .Measurements

,miscellaneous, 467

468,480, 486, 487, 488, 500

518, 551, 552, 553 , 569, 578

601, 630, 634 , 644 , 646, 661

665, 666, 667 , 671, 672, 676

$81, 682 , 690, 695, 696, 72425 .

Measurements, Continued .

Of arcs, 578 , 579, 770, 775, 778,788, 796.

Of angles, 756, 757, 758 , 759760, 761, 762, 763 , 764, 765

767, 775, 776, 777 , 778 .

Of b ins, 474 , 488 , 513 , 552 , 584 ,601

, 727 .

Of board s, 692 , 693 , 694 , 727 .

Of carpeting, 467 , 529, 542, 551,597, 602 , 671, 688 , 717, 727 .

Of chords, 770, 780, 788 , 796.

Of circles, 601, 653 , 667, 695,722, 790.

Of circular rings, 696, 790.

Of circumferences, 539, 578, 612 ,613 , 653 , 667 , 711, 727 .

Of

7

cones, 682 , 683 , 689, 690, 703 ,

93 .

Of cylinders, 681, 689, 698 , 712,627, 798 .

Of degrees of longitude, 578, 579,582 , 676, 796 .

Of fenc ing, 468 , 486 , 487, 512,

$39, 547 , 552, 558 , 567 , 570,21.

Of frustums,794 , 795, 799, 800,

801.

Of hexagon, 668 , 789.

Of hypotenuse, 635, 643 , 648656, 662, 663 , 665, 672 , 675694

,702

,711

,712

, 721, 753

754, 778, 779, 794 .

Of lumber,693 , 694

Of octagon, 789 .

Of parallelopipedon, 468 , 517518, 551, 582 , 593 , 597, 601

602, 664, 687 , 723 .

Of prism , 681, 682 , 689, 711, 793 ,797 , 798.

Of pyram id , 682, 689, 690, 793 ,794

, 795.

Of rectangle, 487 , 496, 502 , 508514 , 515, 516, 539, 541, 551

552 , 557, 572, 595, 597 , 601621, 622, 633 , 675, 717 , 722748, 753 , 782 .

Of regular polygon, 789.

Of rhombo id , 516, 517, 667,

782 .

Of rhombus, 517, 668.

Percentage, duties, 462 , 508, 539547

, 548, 561, 570, 594 , 633

664, 693 , 705, xvi .

Insurance, 463 , 464, 495,498 , 533 , 547 , 548 , 564, 570,593 , 596, 597, 613 , 650, 651,665, 669, 673 , 684, 685, 686,714, 716, xv .

Profit and loss,464, 465, 466,

469,470, 479, 480,

496, 497, 498, 501,508 , 510, 513 ,546

,548

,549

,

567, 568, 569,573 , 575, 591,

533,

557 ,570

,

592,

604 ,648 ,

494 ,502,541,558 ,571,594,605,

495,503 ,545.566

,

572,

595,63 1,

649,686

, 687 ,710, 712 ,

714,716

,xiv .

Stocks, 466, 502 , 511, 562

563 , 564, 566, 575, 595, 596

638 , 639, 652 , 663 , 691, 704

711, xx iv . , xxv .

Taxes,463 , 475, 501, 511

549, 564, 566, 574 , 593 , 673

680, 686, 698 , 815, 816, xv i .

Definit ions and rules, xi i i . , xiv .

To find the base or the rate,

461,462

, 714 , x i i i . , x iv .

Perpend icular lines, 759, 760, 775,777 .

Policy, insurance , 548 , 570, 593 ,669

,xv .

Polltax,549

,xv i .

Pq‘lygon (see Areas,Measure

ments, etc . 771.

Angles of regular, 771, 772 .

Constructionofregular, 771, 772.

Powers, 607 , 699 .

Definit ions,etc .

,xxi i i .

Prem ium,x .

Exchange , 576, 577 , 579, 580,634 , xx .

Insurance,463 , 495, 533 , 564

596,613 , 650, 665, 669 , 7 14

xv .

Stocks and bonds, xxv .

Present worth , 592 , 593 , 597 , 600,604 605, 619, 672 , 695, 7 12,715

650,688,713

,

Prime factors, 656.

Numbers, 556, 685.

Definit ions,etc . , v .

Princ ipal see Interest), xv i .Princ iples, add i tion, i .Cancellat ion, v .

Dec imals, x .

Division, 1v .

Fractions, v 1u .

Add it ionof,ix .

Subtract ionof,ix .

G reatest commond iv isor, v i .Least commonmult iple, v i i .Multiplicat ion, i i i .Proport ion, xx i i .Ratio

,xx i .

Subtract ion, 11.Prisms

,k ind s, etc .

, 790, 791.

Surface of, 681, 682 ,

Developed surface of ( seeManual),Volume of, 689,

Problems ininterest, 519, 583 , 594604 , 630, 649, 650, 661, 663

664, 665 , 67 1, 713 , 7 16, xv i i .

Inpercentage ( see Percentage,

Proceeds ( see Bank d iscount), x ix.

Product , i ii .

Profit and loss,464

,465

,466

,469

470, 479, 480, 494, 495, 496

497, 498, 501, 502 , 503 , 508

510,

533 , 541, 545, 546

548 , 557 , 558 , 566, 567568 , 570, 571, 572 , 573

575, 592 , 594 , 595, 596

605, 63 1, 633

649, 650, 669

684, , 686, 687, 688 , 694

712 , 713 , 714

716Defini t ions, etc . ,

465, x iv .

Proper fract ion, v i i .Proport ion ( see Analysis, Cause

and effect,etc . 624

,625 , 626,

627 , 636, 637 673 , 709 , 712,785, 786, 787

Definit ions, etc .

,xx i . , xx i i .

Pro tractor, 756 .

Pyram id , definit ions, etc ., 792 .

Surface of,682 , 690, 793 , 794.

834

Pyram id , developed surface, 682,793 , 901.

Volume of, 689, 799, 802 .

Surface of frus tum , 794 , 795 .

Developed surface, 794 , 801 .

Volume of, 799, 800, 802 .

Quadrilateral ( see Rectangle ,Rh ombo id , 766 .

Construct ion of ( see Square

767 , 768 , 769, 772 , 775

776, 777 , 778, 779, 780, 781

782, 788 .

Quotient , 480, 497, 507 , 598 , 600,670, 705, iv .

Rad ius of c ircle, 613 , 653 , 667 , 669,695, 696, 698 , 7 11, 769, 790.

Of Sphere , 697 , 698, 701, 702,796, 797 , 803 .

Range (U . S . lands), 720.

Rate ( see Discount , Interest , Percentage), 461, 471, 519, 534 ,xi i i . , xv i . , xvi i .

Ratio , 610, 611, 612 , 613 , 624, 648

701, 702 , 703 , 705, 715, 780

782, 793 .

Definitions, etc ., 610, xx i .

Rectangle, area of,487, 496, 502

508,514 , 515, 516, 539, 541

551, 552, 557 , 572 , 595, 597

601, 621, 622 , 633 , 675, 717722 , 748 , 753 , 782 .

Definition, 766.

Diagonalof, 644 , 656, 672 .

Reduct ionof dec imals, to commonfract ions, 495, 497 , 54 1, 559 ,560, 584 , 717 , x i .

To per cents, 497 .

Of denom inatenumbers, ascending, 504 , 560, 564, 565, 597621, 632 , 645, 647 , 649, 669

671, 672 , 715, x i i i .

Descend ing, 479, 501, 504508

,560

, 596, 621, 656, 688

706, x i i .

Of fractions to commondenom inator, 694 , 707 , 709, ix .

To dec imals, 497 , 501, 507560, 569, 573 , 583 , 584 , 669

687, 706, 707 , 709, xi .

INDEX .

Red uctionof fract ions, to h igher

terms, 459, v i i i .

To lowest terms, 507 , 529,549

,632

, 650, 666, v i i i .

To per cents,459

,497 , 569,

583 .

Of complex fract ions to

simplest form ,474, 475,

549,550

, 567, 604, 63 1,656, 664, 705, 712 , 714 , x .

Of compound fract ions, 475,549

, 624, 706, 707 .

Of per cents to dec imals, 510.

To commonfractions,459

,

501, 510, 583 .

Regularpolygon, area of ( see Hex

agon), 789 .

Constru ct ionof ( see Pentagon,771, 772 .

Defini tion, 771.

Regular prism , 791.

Pyram id , 792 .

Remainder,480

,507 , 600, 670, 7 18,

w.

Rev iew exerc ises ( see Examinat ionquest ions,

Bank d iscount, 514 , 526, 535,

615.

Bonds and stocks, 638 .

Com

énerc ial d iscount

,511

, 529 ,62

Cube root, 710.

Decimals, 559, 603 .

Denom inate numbers,499

,544

,

594 .

Domest ic exchange, 634 .

Frac

tions, 484, 493 , 543 , 549,

63

Fundamental processes, 484485

, 486, 542, 543 , 544, 594595.

Interest,simple , 491, 514, 520,

523, 614 .

Compound , 637.

Longitude and t ime, 616.Measurements, 467 , 486, 504 ,515, 550, 655 , 667 , 678 .Miscellaneous, 499

, 504 , 545,

566, 577 , 582 , 621, 644, 678 ,695.

Percentage , 469.

836 INDEX .

Rules, Continued .

Cylinder, surface of, 793 .

Volume of, 689, 802 .

Dates,t ime between, 807, x i i i .

Dec imals,add i tionof

,x i .

Div isionof, xi .Multiplicat ionof, x i .Reductionto commondenom

inator, x i .To commonfract ions

,x i .

Sub tractionof,x i .

To read , x.

To wri te, x .

Denom inate numbers, redu ct ionascend ing, x i i i .

Reductiondescend ing, x i 1.Denom inator, least common

,o f

fractions, ix . i i i

Of dec imals,x i .

Diameter of c ircle, 578, 667, 697 .

Difference between dates, 807 ,

m u .

Discount,bank

,489, 490, x ix .

Of interest-bearing notes,503 .

Proceeds, 489.

To find face o fCommerc ial

,509.

True,593 .

Div idends, stock , 638 .

Div isionof dec imals,x i .

Of fractions,x .

Of integers, iv .

Draft, sigh t , cost of, 576, 577 , xx .

Face of,xx .

Time,cost of

,579

,xx .

Face of,xx .

Equated time,xx i .

Exact interest , 648 , xv i i i .Face o f b illof exchange , 586.

Of draft,xx .

Of note, 534 .

Factors,commonprime

, v i .

Prime,v .

Fract ions,add i tionof

,ix .

Clearing of, 452 .

Complex ,reductionof

,ix .

Cube root of, xx iv .

Div isionof, ix .

Improper, redu ctiono f,ix .

Mixed numbers, reduc t ion, ix .

Rules, Continued .

Fract ions,multiplicat ionof, ix .

Reduc tionof,to h igherterms

,

V111.

To least common denom inator, ix .

To lowest terms, v i i i .

Square root of,xxiv .

Sub tractionof,ix .

Frustum of cone, surface of, 795.

Volume of, 801.

Of pyram id , surface of,795.

Volume of, 800.

G reatest commond ivisor, v i .

Hypotenu se, length of,635 .

Improperfractions, reductionof ,i x .

Interest,accurate , 648 , xv i i i .

Annual,719

,xv i i i .

Compound , 568 , 637 , xvu i .Exact

, 648, xv i i i .

S imple , generalmeth od,471

,

xvi i .

Method by aliqu ot parts,523 , xv i i .

S ix per cent m eth od , xvi i i .

Amount , 472.

To find th e princ ipal, 519 , xvn.

Ra te,519

,xv i i .

Time,519, xv i i .

Least commondenom inator,ix .

Multiple , v i i .Longitude and t ime

,616 .

Lowest terms,v i i i .Merchants’ rule

, part ial payments

,654

,x ix .Mixed numbers

,redu ct ionof

,ix .Multi ple , least common

, v i i .Multiplicat ion, algebra ic quant i ties, 747

Decimals,xi .

Fractions,x .

Sh ort m eth ods,484 .

Integers, i i i .Short meth ods

,542 .

By 10, 100, etc .,iv .

Mixed numbers,short meth

ods, 484 , 493 .

New Hampsh ire rule, partial

payments, 812 .

Notat ionof dec imals,x .

Rules,Continued .

Nota t ionof integers, 1.English method

,591 .

Numerationof decimals, x .

Of integers, i i .Oblique solids, volume of

, 802 .

Part ial payments, Connec ticutrule, 809.Merchants’ rule, 654, x ix .

Uni ted States rule, 588 , 589,x1x.

Annualinterest notes, 811.

NewHampsh ire rule, 812 .

Vermont rule, 813 .

Partnersh ip ( seeManual), xx i i .Percentage, 461, x i i i .To find base, 462, x i i i . , x iv .

Ra te,462

,x i i i .

Perpend icular, righ t-angled tri

angle , 635.

Power, second , 607 , xxi i i .Th ird

,699, xx i i i .

Prime factors, v .

Common, v i .Princ ipal, to find , 519, xvn.

Prism,convex surface , 681 .

Volume, 802 .

Proceeds, 489 .

Profit and loss, 465.

Proport ion, 625, xxi i i .Pyram id , convex surface, 793 .

Frustum of,convex surface ,

795.

Volume, 800.

Volume of,802 .

Ra d ius of c ircle, 697 .

Of sph ere, 697 , 797 .

Rate, d iscount , 534 .

Interest , 519, xvi i .Percentage, 461, x i i i .

Rat io,610.

Reduct ion of dec imals to com

mondenom inator, x i .To commonfractions, x i .

Of denom inate numbers , as

cend ing, x i i i .Descend ing, x i i .

Of fract ions to common de

nom inator, ix.

To dec imals, xi .

To h igher terms, vi ii .

Rules, Continued .

Reductionto lowest terms,V i i1 .

To simplest form ,ix .

Of improperfractionto m ixednumber

,ix .

Of m ixed number to improperfraction

,ix .

Roo t,cube

,700, 710, xx iv .

Square, 608 , 609, xxi i i . , xxiv.Sph ere, surface, 697 .

Volume,701 .

Square root of fractions, xxiv.

Of integers, 608 , 629, xx i ii .Subtract ion

,algebra ic , 731.

Of dec imals, x i .Of fract ions, ix .

Of integers, i i . , in.

Surface,of cone

,793 .

Of cylinder, 793 .

Of frustum o f cone,795.

Of pyram id , 795.

Of prism , 681.

Of pyram id , 793 .

Of Sphere, 697, 701, 803 .

Taxes, 8 15 .

V ermont meth od, 815.

Term of d iscount,x ix

,xx .

Time,betweendates

, 807 , x i11.

Princ ipal, interest, and ra te

given, 519, xv i i .Discount

, 534 .

Draft , cost of, 579, xx .

Face of,xx .

Transposing, 455.

True d iscount, 593 .

Triangle, area , 652 .

Volume of cone, 687 , 798, 802 .

Of cylinder, 689 , 802 .

Of frustum of cone, 801 .

Of frustum of pyram id , 800.

Of prism , 689, 802 .

Of pyram id , 689, 798 , 802 .

Of sph ere, 701, 803 .

Scalene triangle, 764.

Alt itude of,668 , 678, 754, 767 ,

776, 782 .

Construct ionof, 762 , 774, 776,

777 , 778 , 780, 782, 783 , 784 .

Secant , 780.

Second power, 607 , xxi i i .

838

Sect ion (U. S . lands), 720, 721,727 .

Sector, area of, 669, 695, 790.

Definition, 669, 770.

Segment, area of, 695, 704 .

Definit ion,695, 770.

Sextant, 770.

Share,562

,xxiv .

Sh ort meth od s, add it ionand sub

trac t ion, 543 , 544 , 595, 622 .

Add i t ionof, 99, 999, etc .,481 .

Bank d iscount , 526.

Commerc iald iscount, 509, 529 ,530.

Compound interest, 565, 566,

637 .

Div ision, 544 , 609.

Div isors : 125, 25, 75, etc .,

481, 53 7 , 620, 674 .

Multiplicat ion, 542, 543 , 616 .

Fract ionalmult ipliers, 484 ,537 , 616.

Mult iplier a m ixed number,484, 493 , 543 , 616, 620, 674.

99, 999, etc .,484 , 537 , 543 ,

616, 620.

125, 25, 75, etc . ,484 , 543 ,

616.

S imple interest , 474 , 525.

Subtraction and add ition,543 ,

544,595, 622.

Of 99, 999, etc . ,481.

S igh t drafts, 576 .

Cost of,576, 577, 580, 634, xx .

Face of, 576, 577 , 634 , xx.

Defini t ions,xxv i .

S im ilar triangles, 784 .

Slant heigh t of cone, 683 , 793 ,794 .

Of regular pyram id , 682 , 792 ,794 .

Of irregular pyram id , 794 .

Small

] circle,

circumference o f,

96.

Rad ius of, 796.

Solid contents ( see V olumes).

Spec ialdrills, 481, 537 , 620, 674 .

Sphere, great circle of, 697, 796.

Smallc ircle of, 796 .

Surface of, 697, 701, 702, 703 ,

797 , 803 .

INDEX.

Sphere, volume of, 701, 702, 703 ,704, 803 .

Square, area of, 482, 601, 665, 667,668 , 695, 704.

Construct ion of, 767, 771, 772,

774,775, 776, 777, 781.

Defini t ion,766 .

Diagonal of, 644 , 648 , 650, 665,667

, 668 , 695, 704 .

S ide o f,487 , 539, 552 , 634 , 665,

666,672

,696.

Square measure, 805 .

Square roo t of algebraic quanti ties,

Of

7

decimals,629, 630, 640, 665 ,

17 .

Of fractions, 618 , 663 .

Of integers, 539, 607 , 608 , 609,629, 634 , 640, 676, 716, 717 .

Of m ixed numbers, 618 .

Definitions and rules, xx i i i .,

xxiv .

Stocks and bonds,466, 502, 562 ,

563 , 564, 568 , 575, 596, 600,638, 639, 652, 670, 683 , 691,699, 704, 711, 714 .

Definitions, xxv .

Sub traction, algebraic quant it ies,730, 731, 732 .

Removing parentheses, 732.

Dec imals, 477, 478 , 532 , 557, 575,688 , 708 .

Denom inate numbers, 506, 544,546, 584, 594, 632 , 665, 686,691

Fract i ons,536, 549, 704, 706, 707,

718 .

Integers, 541, 543 , 595.

Cross subtract ion, 485.

Defini t ions,i i .

Subtrah end , 731, i i .Successive d iscounts ( see Commer

ciald iscount).Sum ,

729,i i .

Surface of cone,682, 683 , 703 , 793 ,

796.

Of cylinder, 681, 682, 698 , 712 ,

Of fence, 468 , 486, 512, 552 .

Of frustum of cone, 795 .

Of frustum of pyramid , 794 , 795 .

840 INDEX .

Volume of frustum of pyram id , Volume of prism , 689, 797, 798 .

799, 800. Of pyram id , 689, 690, 798, 799,Of parallelop ipedon, 468 , 517 802 .

518, 551, 582 , 593 , 601, 602 Of sphere , 701, 702, 703 , 704,664, 687, 723 . 903 .

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Ease. Our round vertical script canbe eas ily wri tten. Eu

gravers oftenproduce gracefulforms and comb inations , bu t such as

one cannot reproduce easily with the pen. Every form and comb ina

tioninour system has beenthoroughly tested to avoid such d ifficulties .

Rapidity . Many of the letter-forms at first considered becausethey were artistic and graceful, after having beenpu t to the test were

d iscarded because they did not perm it rapid execu tion.

Educative The c0pies in the primary numbers are large andare illustrated wi th tasty ou tline drawings , stimulating interest inthewri ting and correlating read ing , number, nature study , and spellingwith the spec ialwriting lesson. So far as practicable the correlation

of stud ies has beencarried throughout the series . The size of the

letter forms is gradually reduced inthe first fournumbers .

Economy . Such facilities have beensecured for theirmanufac

ture , that books of the very b est quality will be furnished at the verylowest prices .

Descriptive circular and sample comes sent onrequest.

D. C. HEATH d: Publis hers

BOSTON NEW vomc CHICAGO LONDON

Ma Teema Tics F or Comm on Schools Part III Higher Arithmetic

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Transcript of Ma Teema Tics F or Comm on Schools Part III Higher Arithmetic