Fundamentals of Calculations - JBLearning

Chapter 1

1

Fundamentals of CalculationsO B J E C T I V E S

Upon completion of this chapter, the technician student willbe able to:

• Convert Roman numerals to Arabic numbers and convertArabic numbers to Roman numerals.

• Reduce fractions to their lowest terms.

• Convert fractions into whole numbers and mixed numbers,and convert whole numbers and mixed numbers intofractions.

• Correctly add, subtract, multiply, and divide fractions,mixed numbers, and improper fractions.

• Correctly add, subtract, multiply, and divide decimal fractions.

• Correctly round decimals to a given place.

• Correctly convert fractions to decimals and decimals tofractions.

• Correctly change decimals and fractions to percents.

• Demonstrate an understanding of significant figures.

• Define ratio and proportion and calculate problems for a missing term using ratio and proportion.

TERMS• Arabic

numbers• Common

fraction• Decimal

fraction• Denominator• Improper

fraction• Number• Numeral• Numerator• Percent• Proportion• Ratio• Reciprocal• Roman

numerals• Significant

figure

4 5 6 X

7 8 9 —

% C

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© Jones & Bartlett Learning, LLCNOT FOR SALE OR DISTRIBUTION




















This chapter focuses on a basic review of numbers, fractions, decimals, and theirmathematical operations, as well as calculating basic ratio and proportion problems.The student will also be introduced briefly to dimensional analysis.

Numbers and NumeralsA number indicates the total quantity of units. A numeral is a word, sign, or group ofwords or signs expressing a number. For example 3, 6, and 48 are Arabic numeralsexpressing numbers that are, respectively, 3 times, 6 times, and 48 times the unit 1.

Many symbols in mathematics and science are used to provide instructions fora specific calculation or that indicate relative value. Some of the common symbolsof arithmetic are presented in Table 1.1.

Kinds of NumbersIn arithmetic, the science of calculating with positive real numbers, the number isusually (a) a natural or whole number (integer), such as 549; (b) a fraction, or sub-division, of a whole number, such as 4⁄7; or (c) a mixed number, consisting of a wholenumber plus a fraction, or part, such as 37⁄8.

number A total quantity or

amount of units.

numeral A word, symbol,

or group of words or symbols

that expresses a number.

Pharmaceutical Calculations for the Pharmacy Technician2

Table 1.1 Some Arithmetic Symbols Used in Pharmacy

Symbol Meaning

% Percent; parts per hundred

+ Plus, add; positive

− Minus, subtract; negative

± Add or subtract; plus or minus

÷ Divided by

/ Divided by

× Times; multiply by

< Is less than

= Is equal to; equals

> Is greater than

≠ Is not equal to; does not equal

≈ Is approximately equal to

≤ Is less than or equal to

≥ Is greater than or equal to

. Decimal point

Adapted with permission from Tapson F. Barron’s Mathematics Study Dictionary. Hauppauge, NY: Barron’s EducationalSeries, 1998.

Many other symbols are used in pharmacy, as in the metric and apothecaries systems of weights and measures, inprescription writing, in physical pharmacy, and in many other areas. Many of these symbols are included and definedelsewhere in this text.

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A number such as 4, 8, or 12, taken by itself without a label to indicate distinc-tion, is called an abstract or pure number. It merely designates how many times theunit 1 is represented; it does not imply anything else about what is being counted ormeasured. An abstract number may be added to, subtracted from, multiplied by, ordivided by any other abstract number. The result of any of these operations alwaysresults in an abstract number designating a new total of units.

A number that designates a quantity of objects or units of measure, such as 4 g,8 mL, or 12 oz, is called a concrete or denominate number. It designates the totalquantity of whatever is being measured. A denominate number has a label and indi-cates precisely what is to be counted or measured. A denominate number may beadded to or subtracted from any other number of the same denomination, but adenominate number may be multiplied or divided only by a pure number. The resultof these operations is always the same denomination.

CHAPTER 1 • Fundamentals of Calculations 3

Examples:

12 3 4oz oz÷ =

4 2 8apples apples× =

10 5 5g g g− =

4 6 10apples apples apples+ =

4 5 6 X

1 2 3 –

0 • = +

7 8 9 —

% C

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Differing Denominations Rule

Numbers of different denominations cannot be added or subtracted from oneanother. For example, it is not possible to add four apples and three oranges; a com-mon denominator is necessary. Four pieces of fruit can be added to three pieces offruit to get seven pieces of fruit. A denominate number can be multiplied or dividedby a different denomination, in fact the multiplier or divisor is an abstract number.For example, if 1 oz costs $0.05, to find the cost of 12 oz, one multiplies $0.05 not by12 oz but by the abstract number 12 to get the cost of $0.60 for 12 oz.

Arabic NumbersThe Arabic system of notation is the one we are most familiar with. It uses the Arabicnumbers 0, 1, 2, 3, 4, 5, 6, 7, 8, 9. These numbers can be written as fractionsor decimals.

Roman NumeralsThe Roman numeral system of notation dates to the ancient days of Rome and usesletters to designate amounts. Roman numerals merely record quantities; they are ofno use in computations. Roman numerals were used exclusively in the apothecaries’

Arabic numbers The

standard set of symbols,

1, 2, 3, and so on, to designate

units. Arabic numbers are

used in fractions and decimals.

Roman numerals A num-

bering system using the letters

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system of measure. We will see this number system used in prescription and med-ication orders to indicate doses.

To express quantities in the Roman system, eight letters of fixed values are used,as shown in Table 1.2.

When Roman numerals are written in lowercase letters i, vi, xii, they may betopped by a horizontal line to help avoid errors. In the case of i the dot is abovethe line.


Table 1.2 Roman Numerals

Roman Numeral Numeric Value

ss 1⁄2

I or i 1

V or v 5

X or x 10

L or l 50

C or c 100

D or d 500

M or m 1000

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Rules to Apply to Roman Numerals

• When a Roman numeral of equal or lesser value is placed after one of equal orgreater value, the value of the numerals is added. A numeral is not repeatedmore than three times.

• When a Roman numeral of lesser value is placed before a numeral of greatervalue, the value of the first numeral is subtracted from the numeral of greatervalue.

I, V, X, L, C, D, and M to des-

ignate units. This system dates

to ancient Rome and is used

in the apothecary system of

measure.

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Review Set 1.1: Arabic and Roman NumeralsWrite the Roman numerals for the following:

1. 42. 83. 1⁄24. 185. 646. 377. 998. 489. 20

10. 45

Write the Arabic numbers for the following:

11. IV12. XXIV13. xix14. XC15. xlv16. xii17. MDCCCXIV18. ii19. CCLVII20. CCIV

Interpret the quantity in each of the phrases taken from prescriptions:

21. Capsules no. xlv22. Drops ii23. Tablets no. XLVIII24. Ounces no. lxiv25. Lozenges no. xvi26. Transdermal patches no. LXXXIV27. Tablets no. xxiv28. Ounces no. viii29. Capsules no. C30. Troches no. xxxv

Common Fractions, Decimal Fractions,and Percents

The arithmetic of pharmacy requires expertise in the handling of common fractionsand decimal fractions. Even if the student technician already has a good workingknowledge of their use, the following brief review of certain principles and rules

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should be helpful, and the practice problems should provide a means of gainingaccuracy and speed in their manipulations.

FractionsFractions are an expression of parts of a whole number. A fraction contains twoparts: a numerator, the top number, expresses the number of parts in question; thedenominator, the bottom number, expresses the total parts of the whole number(Fig. 1.1).

In a common fraction, also called a proper fraction, the numerator is less thanthe denominator. The value of a common or proper fraction is less than one. (Thevalue of a fraction is the numerator divided by the denominator.) Examples of com-mon fractions include 1⁄8, 3⁄16, 1⁄2, and 2⁄3.

In an improper fraction the numerator is larger than the denominator, and thevalue is greater than 1. Examples include 4⁄2, 6⁄3, 5⁄4, and 12⁄8.

A mixed number is a whole number and a fraction, such as 11⁄2 or 35⁄8. Mixednumbers can easily be converted into improper fractions by following the principlesgoverning arithmetical operations for fractions.

numerator The top number

of a fraction, which indicates

the number of parts in question.

For example, in 1⁄3, the whole is

three parts; the number of parts

in question is 1.

denominator The bottom

number in a fraction; it indi-

cates the number of parts in

the whole. For example, in 1⁄3,

the number of parts in the

whole is 3.


NumeratorDenominator

13

1/3

Figure 1.1 Numerator and denominator.

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Fundamental Rules of Fractions

• Multiplying or dividing the numerator and denominator of a fraction by thesame number does not change the value of the fraction.

• To change a fraction to its lowest terms, divide its numerator and its denominatorby the largest whole number that will go into both evenly.

• The lowest common denominator is the smallest whole number that can bedivided evenly by all of the denominators in a problem.

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common fraction A part

of a whole number, sometimes

termed a proper fraction

(e.g., 1⁄2); the numerator is less

than the denominator. The

value of a common or proper

fraction is less than 1.

improper fraction A

fraction with the numerator

(top number) larger than the

denominator (bottom number).

The value of an improper

fraction is greater than 1.


Example of multiplying or dividing the numerator and denominator of afraction by the same number:

Therefore, the lowest term for 4⁄16 is 1⁄4. Figure 1.2 illustrates this point.

Example of changing a fraction to its lowest terms:The first fundamental rule of fractions allows us to reduce fractions to the lowestcommon denominator, which is the smallest number divisible by all of the othergiven denominators.Reduce 36⁄2880 to its lowest terms.

Example of finding the lowest common denominator:Reduce the fractions 3⁄4, 4⁄5, and 1⁄3 to a common denominator.By testing successive multiples of 5, we discover that 60 is the smallest numberdivisible by 4, 5, and 3; 4 is contained 15 times in 60; 5, 12 times; and 3, 20 times.

34

34

4560

45

45

4860

13

12

= =

= =

=

× 15× 15× 12× 12× 20× 20

== 2060

362880

36 362880 36

180

= ÷÷

= The largest common diivisor is 36.

14

14

28

2 416

4 216 2

28

2 28 2

14

= = = =

÷÷

= = ÷÷

=

× 2× 2

× 28 × 8

14

28

416

==Figure 1.2 Fractions.

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The first rule for adding fractions states that it is necessary to convert fractionsto the lowest common denominator. This is accomplished using the third funda-mental rule of fractions, remembering that the lowest common denominator is thesmallest number into which all of the denominators will divide evenly.

Examples:

Reduce by dividing both the numerator and denominator by 2:

Find the lowest common denominator.The lowest common denominator is 12.

14

16+ =

24

24

22

12= ÷ = , answer

14

14

24+ =

14

312

16

212

312

212

512

= =

+ =

;

, answer

15

35

45+ = , answer


4 5 6 X

1 2 3 –

0 • = +

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% C

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Rules Governing Arithmetical Operations of Fractions

• Reduce every mixed and improper fraction.

• Express whole numbers as a fraction having 1 for its denominator.

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Rules for Adding Fractions

Step 1. Convert fractions to lowest common denominator if necessary.

Step 2. Add numerators; place sum over denominator.

Step 3. Reduce to lowest terms.

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Convert the mixed number into an improper fraction:

Reduce:

In preparing batches of a formula, a pharmacist used 1⁄4 oz, 1⁄12 oz, 1⁄8 oz, and 1⁄6 ozof a chemical. Calculate the total quantity used.The lowest common denominator is 24.

Reduce:

1524

58= oz, answer

1 14

14+ =

14

624

112

224

18

324

16

424

624

224

32

= = = =

+ +

, , , and

444

2415

24+ =oz oz

64

24

121 1= = , answer

54

14

64+ =


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1 2 3 –

0 • = +

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Rules for Subtracting Fractions

Step 1. Convert fractions to lowest common denominator if necessary.

Step 2. Subtract the numerators and place the amount over the denominator.


Examples:

Lowest common denominator is 6.

Reduce:

26

13= , answer

56

36

26− =

56

12− =

34

24

14− =

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Convert mixed numbers into improper fractions:

Reduce:

A hospitalized patient received 7⁄12 L of a prescribed intravenous infusion. If he hadnot received the final 1⁄8 L, what fraction of a liter would he have received?The lowest common denominator is 24.

712

1424

18

324

1424

324

1124

= =

− =

;

L, answer

83

232= , answer

163

83

83− =

5 213

23− =


4 5 6 X

1 2 3 –

0 • = +

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Rules for Multiplying Fractions

Step 1. Convert mixed numbers into improper fractions.

Step 2. Multiply the numerators to get a new numerator.

Step 3. Multiply the denominators to get a new denominator.


Examples:

Reduce:

14

12

18× = , answer

13 , answer

12

23

26× =

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Convert mixed numbers to improper fractions:

Reduce:

If the adult dose of a medication is 2 teaspoonfuls and the child dose is 1⁄4 the adultdose, calculate the dose for a child.

Reduce:

2 13

12× 1 =

12 teaspoonful, answer

14

21

24× =

3 336

12= , answer

73

32

216× =

reciprocal 1 divided by the

number in question.


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Rules for Dividing Fractions

Step 1. Convert mixed numbers into improper fractions.

Step 2. Find the reciprocal of the divisor (the secondfraction) (invert the second fraction) and multiply.


Examples:

Reduce:

Reduce:

348

28

144 4= = , answer

2 18

12

178

12

178

21

348÷ = ÷ = =×

1 12 , answer

34

12

34

21

64÷ = =×

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If 1⁄2 oz is divided into 4 equal parts, how much will each part contain?

A manufacturer wishes to prepare samples of an ointment in sealed foil envelopes,each containing 1⁄32 oz of ointment. How many samples may be prepared from 1 lb(16 oz) of ointment?

If a child’s dose of a cough syrup is 3⁄4 teaspoonful and that is 1⁄4 of the adult dose,what is the adult dose?

34

14

34

41

124 3tsp tsp tsp,÷ = = =× answer

161

132

161

321

5121 512÷ = = =× samples, answer

12

41

12

14

18oz oz÷ = =× , answer


Review Set 1.2: FractionsIndicate which fraction is the smallest:

1. 1⁄16, 1⁄8, 1⁄32, 2⁄16

2. 1⁄4, 1⁄3, 1⁄6, 1⁄2

3. 5⁄9, 4⁄8, 1⁄2, 3⁄16

Indicate which fraction is the largest:

4. 1⁄4, 1⁄3, 1⁄6, 1⁄2

5. 2⁄9, 5⁄8, 2⁄5

6. 1⁄12, 6⁄36, 5⁄20

Reduce the following to the lowest terms:

7. 5⁄10

8. 20⁄45

9. 16⁄32

10. 21⁄93

11. 12⁄144

12. 36⁄27

13. 22⁄55

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14. 75⁄60

15. 21⁄51

16. 150⁄20

Add the following:

17. 1⁄3 + 1⁄2

18. 21⁄2 + 41⁄4

19. 5⁄8 + 9⁄32 + 1⁄4

20. 1⁄150 + 1⁄200 + 1⁄100

21. 1⁄60 + 1⁄20 + 1⁄16 + 1⁄32

Subtract the following:

22. 3⁄4 − 1⁄2 =

23. 23⁄8 − 11⁄3

24. 31⁄2 − 15⁄64

25. 1⁄30 − 1⁄40

26. 21⁄3 − 11⁄2

27. 1⁄150 − 1⁄400

Find the product of the following:

28. 1⁄2 × 1⁄3

29. 2⁄3 × 3⁄4

30. 25⁄8 × 1⁄2

31. 1⁄25 × 3

32. 1⁄5 × 3⁄5

33. 7⁄8 × 1⁄4

34. 5 × 1⁄150

35. 6 × 2⁄3

36. 31⁄3 × 41⁄2

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37. 2⁄3 × 1⁄3

38. 1⁄3 × 1⁄20 × 1⁄4

39. 1⁄4 × 1⁄2 × 1⁄8

40. 30⁄75 × 15⁄32 × 25

41. 21⁄2 × 12 × 7⁄8

42. 1⁄125 × 9⁄20

What is the reciprocal of each of the following?

43. 1⁄10

44. 31⁄3

45. 21⁄1

46. 3⁄2

47. 17⁄8

48. 1⁄64

Find the quotient of each of the following:

49. 2 ÷ 1⁄2

50. 1⁄8 ÷ 1⁄4

51. 3⁄16 ÷ 3⁄16

52. 25 ÷ 1⁄2

53. 5⁄15 ÷ 5

54. 2 ÷ 1⁄8

55. 4⁄5 ÷ 2⁄3

56. 31⁄3 ÷ 45⁄8

57. 50 ÷ 1⁄2

58. 2⁄3 ÷ 1⁄24

59. 1⁄5000 ÷ 12


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Decimal FractionsDecimal fractions are fractions with a denominator of 10 or any multiple of 10; forexample 100, 1000, 10000. The denominator of a decimal fraction is not writtenbecause the decimal point indicates the place value of the numerals. The numericvalue of a decimal fraction is always less than one (Table 1.3).

All operations with decimal fractions are carried out in the same manner asthose with whole numbers, but care is needed when putting the decimal point in itsproper place.


60. 61⁄4 ÷ 1⁄2

61. 1⁄150 ÷ 1⁄2

62. A cookie mix makes 36 cookies. The day care provider gives each child 3 cookies. What fractional part of the batch did each child receive?

63. A bottle of Tylenol® children’s liquid contains 24 doses, each measuring 5 mL. If one child receives 4 doses, what fractional part of the bottle does the child receive?

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Multiplication of Decimals

Multiply as with whole numbers. In the final answer, make sure the decimal point isplaced correctly. If the final digit after the decimal is a zero, it should be eliminated.

decimal fraction Frac-

tion with the denominator

being 10 or any multiple of

10, usually expressed without

the denominator and with a

decimal point: 0.14 = 14⁄100.

The numeric value of a frac-

tion is always less than one.

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Examples:

1 45 2 9. . ,÷ =0.5 answer

16 2 8 1. . ,÷ =2 answer

Examples:

4 55 18 2. . ,× 4 = answer

1 5 3 6. . ,× 2.4 = answer


Table 1.3 Decimal Place Values

Number of Places Representative Number

6 Hundreds of thousands (100,000)

5 Tens of thousands (10,000)

4 Thousands (1000)

3 Hundreds (100)

2 Tens (10)

1 Ones (units)

. Decimal point

1 Tenths (0.1)

2 Hundredths (0.01)

3 Thousandths (0.001)

4 Ten-thousandths (0.0001)

5 Hundred-thousandths (0.00001)

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Division of Decimals

• When a decimal is divided by a whole number, the decimal point remains in thesame place as in the dividend.

• When a decimal or whole number is divided by a decimal, the decimal point inthe divisor is moved to the right to produce a whole number. The decimal pointin the dividend is moved the same number of places to the right, and the num-bers are divided as with whole numbers.

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4 5 6 X

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Other Rules Regarding Decimals

Familiar operations involving decimals that are worth recalling are as follows:

• As a direct consequence of the place value in decimal notation, moving the deci-mal point one place to the right multiplies a number by 10; moving it two placesto the right multiplies it by 100, and so on. Likewise, moving the decimal point oneplace to the left divides a number by 10; moving it two places to the left divides itby 100, and so on. It is extremely important that decimal places be accurate in cal-culations of medication doses. Giving a patient 10 times the ordered dose can belethal, and underdosing patients can have a similar disastrous effect.

• A decimal fraction may be changed to a common fraction by writing thenumerator over the denominator and (if desired) reducing to the lowest terms: 0.125 = 125⁄1000 = 1⁄8.

• A common fraction may be changed to a decimal by dividing the numeratorby the denominator (the result may be a repeating or endless decimal fraction):3⁄8 = 3 ÷ 8 = 0.375; 1⁄3 = 1 ÷ 3 = 0.333333. . . .

• Sometimes decimals are rounded to a predetermined decimal place. The samegeneral rules apply as for any rounding: 5 and above go to the next digit; 4 andbelow are dropped.

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Rules for Writing Decimals

• Always place a 0 to the left of the decimal point if there is no whole number there.This draws attention to the decimal point and helps eliminate errors.

Incorrect

(continued)

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Incorrect

Correct

Correct

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Rules for Writing Decimals (continued)

• Never follow a whole number with a decimal point and a 0. This may cause a medication error because the order may be misinterpreted if the decimal point is not noticed.

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Critical Thinking 1.1

Frank Smith presented the following prescription forCoumadin® to the pharmacy. The pharmacy technicianentered the prescription information into the computer.

The order was misinterpreted as 10 mg of Coumadin® daily.How could this have been avoided?

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Review Set 1.3: Decimal FractionsExpress the following in decimals:

1. one-hundredth

2. three-tenths

3. twenty-five thousandths

4. one and one-hundredth

5. forty-six thousandths

Round the following to the nearest hundredth:

6. 1.106

7. 0.211

8. 4.23891

9. 0.4912

10. 42.882

Rx:

Sig:

Dr. Debra Lawson888 NW 27th Ave., Miami, FL 98885

247-555-6613

678 Apple St.Virginia Millhouse

Dispense as written May substitute

Dr. Debra Lawson

Coumadin 1.0 mgi po daily

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PercentThe term percent means parts per hundred. So 50% means 50 parts in each 100 partsof the same item.

Common fractions may be converted to percents by dividing the numerator bythe denominator and multiplying by 100 (parts per hundred) or by simply movingthe decimal point two places to the right. See Table 1.4 for equivalencies of com-mon fractions, decimal fractions, and percents.

percent Parts per hundred.


Round to the nearest tenth:

11. 0.1234

12. 1.992

13. 89.17

14. 0.878

15. 0.333

Add the following:

16. 3.5 + 4.25

17. 0.04 + 0.612

18. 45.6 + 12.4

Subtract the following:

19. 2.5 − 0.75

20. 46.459 − 1.411

21. 0.04 − 0.025

Multiply the following:

22. 2.14 × 0.012

23. 2.5 × 10

24. 16.12 × 0.5

Find the quotient of the following:

25. 2 ÷ 0.05

26. 6.5 ÷ 1.5

27. 25.4 ÷ 6

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Examples:Convert 3⁄8 to a percent.

Convert 0.125 to a percent.

0 125 12 5 12 5. . % . %× 100 = =or 0.125

38 37 5× 100 = . %


Table 1.4 Equivalencies of Common Fractions, Decimal Fractions, and Percents

Common Decimal Common DecimalFraction Fraction Percent (%) Fraction Fraction Percent (%)

1⁄1000 0.001 0.010 1⁄5 0.200 20.0

1⁄500 0.002 0.20 1⁄4 0.250 25.0

1⁄100 0.010 1.00 1⁄3 0.333 33.3

1⁄50 0.020 2.00 3⁄8 0.375 37.5

1⁄40 0.025 2.50 2⁄5 0.400 40.0

1⁄30 0.033 3.30 1⁄2 0.500 50.0

1⁄25 0.040 4.00 3⁄5 0.600 60.0

1⁄15 0.067 6.70 5⁄8 0.625 62.5

1⁄10 0.100 10.00 2⁄3 0.667 66.7

1⁄9 0.111 11.10 3⁄4 0.750 75.0

1⁄8 0.125 12.50 4⁄5 0.800 80.0

1⁄7 0.143 14.30 7⁄8 0.875 87.5

1⁄6 0.167 16.70 8⁄9 0.889 88.9

Review Set 1.4: PercentConvert the following percents to fractions and decimals:

1. 25%

2. 48%

3. 2%

4. 15%

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Convert the following decimals to fractions and percents:

5. 0.35

6. 0.15

7. 0.16

8. 0.04

Convert the following fractions to percents and decimals:

9. 1⁄4

10. 1⁄32

11. 7⁄8

12. 3⁄20

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..

Ratio Division Rule

If two terms of a ratio are multiplied or divided by the same number, the value isunchanged. The value is the quotient.

Ratio and ProportionRatioThe relative magnitude of two like quantities is called their ratio. Ratio is sometimesdefined as the quotient of two like numbers. This quotient is always expressed as anoperation, not a result; in other words, it is expressed as a fraction, and the fractionis interpreted as indicating the operation of dividing the numerator by the denomi-nator. Thus, a ratio presents us with the concept of a common fraction as expressingthe relation of its two numbers.

The ratio of 20 and 10, for example, is not expressed as 2 (that is the quotientof 20 divided by 10), but as a fraction 20⁄10. Similarly, when the fraction 1⁄2 is to beinterpreted as a ratio, it is traditionally written as 1:2, and it is read not as one-halfbut as 1 to 2.

All of the rules governing common fractions equally apply to a ratio. The fol-lowing principle is particularly important.

ratio The relationship of one

quantity to another quantity.

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For example the ratio 20:4, or 20⁄4, has a value of 5. If both terms are divided by 2,the ratio becomes 10:2 or 10⁄2. Again, the value is 5.

When two ratios have the same value, they are equivalent.


4 5 6 X

1 2 3 –

0 • = +

7 8 9 —

% C

..

Equivalent Ratio Rule

In an equivalent ratio the product of the numerator of the one and the denominatorof the other always equals the product of the denominator of the one and thenumerator of the other; that is, the cross-products are equal.

Because 2⁄4 = 4⁄8

2 × 8 (or 16) = 4 × 4 (or 16)

4 5 6 X

1 2 3 –

0 • = +

7 8 9 —

% C

..

Equal Ratios Rule

If two ratios are equal, their reciprocals are equal.

Because 2⁄4 = 4⁄8, 4⁄2 = 8⁄4

4 5 6 X

1 2 3 –

0 • = +

7 8 9 —

% C

..

Numerator Equivalence Rule

The numerator of the one fraction equals the product of its denominator and theother fraction.

If 615

25

25Then 6 15 or

15 25

6

And 2

=

=( )

=

=

× ×

55 615

2615× ×

or( )

=

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An extremely useful practical application of these facts is found in proportion.

ProportionA proportion is the expression of equality of two ratios. It may be written in any oneof three standard forms:

• a:b = c:d

• a:b :: c:d

•

In any proportion it is possible to find the value of the missing term using cross-multiplication. The numerator of the first ratio multiplied by the denominator of thesecond ratio is equal to the numerator of the second ratio times the denominator ofthe first ratio. It is necessary simply to solve the equation to find the missing term.

Using this information, it is possible to derive the following fractional equations:

It is helpful to the technician student to set up the equation with the unknownratio second, putting the unknown in the fourth position in the formula.

Very few arithmetical problems cannot be solved most directly by proportion.Given correct interpretation of the relationships implied by the data and any threeterms of a proportion, it is easy to calculate the value of the fourth.

Examples:

Cross-multiply 100 × X = 25 × 2 = 100 × X = 50

X

X answer

=

=

50100

0 5. ,

If then

and

ab

cd

a bcd

b adc

c add

d bca

=

= = = =

,

, ,

1002

25=X

ab

cd

=


4 5 6 X

1 2 3 –

0 • = +

7 8 9 —

% C

..

Denominator Equivalence Rule

The denominator of the one equals the quotient of its numerator divided by theother fraction.

15 6 or 6 15

And 5 2 or 2

25

52

615

156

= ÷ ( ) =

= ÷

×

×(( ) = 5

proportion The expression

of equality between two ratios.

For example, 2:5 = 4:10.

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For an order for 400 mg of a certain drug that comes in 100-mg tablets, the ratio is

Cross-multiply 100 mg × X tablets = 1 tablet × 400 mg

If 3 tablets contain 975 mg of aspirin, how much in milligrams should be containedin 12 tablets?

Cross-multiply:

If 3 tablets contain 975 mg of aspirin, how many tablets should contain 3900 mg?

If 12 tablets contain 3900 mg of aspirin, how much in milligrams should 3 tabletscontain?

If 12 tablets contain 3900 mg of aspirin, how many tablets should contain 975 mg?

Pharmacy technicians often set up mixed ratios in their proportions, invoking theprinciple that if the ratios are regarded as abstract numbers, the means or theextremes may be interchanged without destroying the validity of the equation.Cross-multiplying the proportion a:b = c:d versus a:c = b:d yields a × d = b × c inboth instances. This is useful for performing pharmaceutical calculations. If 3 tablets

12 3900975

tabletstablets

mgmg

975 t

X

X

=

= 12 × aablets 3 tablets,3900

= answer

123

3900

3

tabletstablets

mgmg

mg

=

=

X

X × 3900122

= 975 mg

3 9753900

3

tabletstablets

mgmg

ta

X

X

=

= × 3900 bblets tablets,975

= 12 answer

X answer= =12 9753

3900× mg

mg,

312

975tabletstablets

mgmg

=X

X

X

tabletstablet mg

mg

tablets,

=

=

1 400100

4

×

answer

100400

1mgmg

tablettablets

=X


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contain 975 mg of aspirin, then 12 tablets contain _______ (3900 mg). Pharmacytechnicians must always remember to put the ratios in the same order, normallystrength per volume or milligrams per tablet or capsule or milligrams per milliliter.

The third set is the form normally used to solve pharmaceutical calculations,remembering to put the unknown proportion last. (Weight or quantity of drug per vol-ume of drug.) The technician student must be careful to keep these ratios in the sameorder; for example, 250 mg/tablet. How many tablets give 750 mg? It is essential toset up the ratio on both sides of the equation to be tablets per milligram or milligramsper tablet and not have one side of the equation be tablets per milligram and the other,milligrams per tablet.

Correctmg

1 tabmg

mg mg

250 750

750

=

=X

X × 250 × 11

× 1

tab

mg tab250 mg

tabs

X

X

=

=

750

3

If tabletstablets

mgmg

Then

312

9753900

3

=

ttabletsmg

tabletsmg

Thenm

975123900

3900

=

ggtablets

mgtablets

Thenmg

129753

3900975

=

mgtabletstablets

= 123


Critical Thinking 1.2

An order is received in the pharmacy for 125 mcg of a drug.The drug is available in 0.25-mg scored tablets. The phar-macy technician calculates that the patient should receivetwo tablets per dose. The drug label indicates that 0.25 mgis equivalent to 250 mcg. Is the patient to receive twotablets? If not, how many tablets should the patient receive?What mistake did the pharmacy technician make in calcu-lating the dose?

4 5 6 X

1 2 3 –

0 • = +

7 8 9 —

% C

..?

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Proportions need not contain whole numbers. If common or decimal fractionsare supplied in the data, they may be included in the proportion without changingthe method.

Because calculating with common fractions is more complicated than withwhole numbers or decimal fractions, however, it is useful to know and wheneverpossible to apply these two facts:

• Two fractions having a common denomination are directly proportional to theirnumerators.

• Two fractions having a common numerator are inversely proportional to theirdenominators.

Examples:If 30 mL is 1⁄6 of the volume of a prescription, how much in milliliters is in 1⁄4 of thevolume?

Most pharmaceutical calculations deal with simple, direct relationships; twice thecause, double the effect, and so on. Occasionally they deal with inverse relationships,twice the cause, half the effect and so on, as when you decrease the strength of a solu-tion by increasing the amount of diluent. Here is a typical example:

If 10 pints of a 5% solution is diluted to 40 pints, what is the percentage ofstrength of the dilution?

1040 5

10 1 25

pintspints

×

× 5%40

X

X answe

%%

. %,= = rr

16

14

64

30volumevolume

mLmilliliters

Or

( )( ) =

X

==

= =

30

6 45

mLmL

mL4

mL

X

X × 30

23

27

73

23

27

23

72

73

=

÷ = =Proof: ×

60100

50100

60100

50100

60100

1005

=

÷ =

6050

×Proof: 00 = 6050


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Review Set 1.5: Ratio and ProportionSolve for X:

1.

2.

3.

4.

5.

6.

7.

8.

9. Make valid ratios between these familiar quantities:a. 3 gal and 2 qtb. 1 yard and 2 feetc. 4 hours and 120 minutesd. 2 feet and 6 inches

Solve by proportion:

10. A recipe for chocolate chunk cookies calls for 3 eggs and makes 3 dozencookies. You need 12 dozen cookies for a family reunion. How many eggswill you need?

1100

1150

× 4 = X

2 000 000400 000

6, ,

,× = X

2005

10× = X

4 652 4

8 4..

.=X

14

23

250=

X

1255

375=X

60 255X

=

5001 5

125.

=X

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Dimensional AnalysisDimensional analysis is an alternative method of performing pharmaceutical calcu-lations that will be demonstrated periodically throughout this text. This method ofproblem solving entails the logical sequencing and placement into an equation of allof the arithmetical terms (both quantities and units) involved in the problem so thatall of the units cancel out except the unit or units of the desired answer (e.g., milli-liters, milligrams per milliliter). By this method, ratios of the data are used, conver-sion factors added as necessary, and individual terms inverted to their reciprocals topermit the cancellation of like units in the numerator and denominator, leaving onlythe desired term of the answer. An advantage to the use of dimensional analysis is theconsolidation of multiple arithmetical steps into a single expression. Pharmacy tech-nician students with a strong chemistry background may be very familiar with thismethod of problem solving and choose to use it rather than ratio-proportion. Alter-native examples using this method of calculation appear throughout this text.


11. You are going to faux-paint your living room. The directions on the glazedirect you to mix the glaze and paint in a 1:4 mixture. You have one quart of paint; how much glaze do you need?

GLAZEGLAZEGLAZE

12. If 250 lb of a chemical costs $480, what is the cost of 135 lb?

13. In a clinical study, the drug triazolam produced drowsiness in 30 of the 1500 patients studied. A certain pharmacy has a patient count of 100. Howmany of them can be expected to feel similar effects?

14. A formula for 1250 tablets contains 3.25 g of diazepam. How many grams ofdiazepam should be used in preparing 350 tablets?

15. If 100 capsules contain 500 mg of an active ingredient, how much in milligramsof the ingredient do 48 capsules contain?

16. If 450 lb of green soap costs $310.50, what does 33 lb cost?

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Examples:How many fluidounces are in 2.5 L if there are 1000 mL in 1 L and 29.57 mL in 1 fl oz?Solve by ratio and proportion:

Step 1. 2.5 L1 L

mL1000 mL

mL

Step 2.

= =X X 2500

11 fl ozfl oz

mL2500 mL

fl ozX

X= =29 57 84 5. . ,, answer


Solve by Dimensional Analysis

129 57

fl ozmL

1000 mL1 L

L 84.5 fl o.

× × 2.5 = zz, answer

Solve by Dimensional Analysis

Terms should have drops in the numerator and minutes in the denominator toresult in the desired answer in drops per minute.

10 drops

1mL

1

480 minutes1000 mL 21drops p× × = eer minute, answer

A medication order calls for 1000 mL of a dextrose intravenous infusion to beadministered over an 8-hour period. Using an intravenous set that delivers 10 dropsper milliliter, how many drops per minute should be delivered to the patient?

Solve by ratio and proportion:

X = 21 drops per minute, answer

Step 1. hours minutes

mLmL

m

8 480

1000

=

= 480 X

iinutesminute

mL/minute,

Step

1 X answer= 2 1.

22. mL/minute1 mL

drops/minutedrops

2 110

. = X

The use of ratio-proportion predominates in this text, but there are examplesusing the dimensional analysis method of solving pharmaceutical calculations.

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Review Set 1.6: Dimensional AnalysisSolve the following using dimensional analysis.

1. How many grains are in a 325-mg aspirin tablet?

2. If a patient is to receive 300 mg/kg of a drug every day, how much in grams ofdrug should a 176-lb patient receive in 10 days?

3. How many teaspoon doses are available in a 4-oz bottle of Tylenol® elixir?

4. How many grams of a medication are contained in a 6-oz bottle containing125 mg/5 mg?

5. A patient is to receive 2 L of IV fluid over 24 hours from an IV set that deliv-ers 10 gtt/mL. How many drops per minute should be delivered to the patient?


Significant FiguresWhen objects can be counted accurately, every figure represents an object. These fig-ures must be taken at face value and are known as absolute figures. When a measure-ment—be it volume, weight, or length—is recorded, the last figure on the right mustbe taken as an approximate figure, meaning that it is at the limit of possible precisionor accuracy with the device used for measuring. The number of significant figuresand the degree of deviation depend on the sensitivity of the measuring device. Anexample of an absolute number is counting tablets or capsules: the countexpresses the number of objects. An example of a significant figure is in weighing325 mg. The 3 means exactly 300 mg, neither more nor less, and the 2 means exactly20 mg more, but the 5 means approximately 5 mg (plus or minus some fraction of amilligram).

It is necessary to distinguish significant figures from decimal places. When ameasurement is recorded, the number of decimal places indicates the degree of pre-cision with which the measurement has been made, whereas the number of signifi-cant figures retained indicates the degree of accuracy that is significant for a givenpurpose. For prescription information the assumption is that everything is inter-preted to the same degree of accuracy.

4 5 6 X

1 2 3 –

0 • = +

7 8 9 —

% C

..

Rules for Determining Significant Figures

1. Digits other than zero are always significant.

2. Final zeros after a decimal point are always significant.

3. Zeros between two other significant digits are always significant.

4. Zeros used only to space the decimal are never significant.

significant figure Consec-

utive figures that express the

value of a number accurately

enough for a given purpose.

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Dr. Doug Griffin

PRACTICE PROBLEMS

1. Interpret the quantities in this prescription:


Review Set 1.7: Significant Figures1. State the number of significant figures in each of the italicized quantities:

a. 1 fl oz equals 29.57 mLb. 1 L equals 1000 mLc. 1 inch equals 2.54 cmd. The chemical costs $1.05 per pounde. 1 g equals 1 million mcgf. 1 mcg equals 0.001 mg

2. Round off each of the following to three significant figures:a. 32.75b. 200.39c. 0.03629d. 21.635e. 1.0751f. 0.86249g. 1.00595632

3. Round off each of the following to three decimal places:a. 0.00083b. 34.795c. 0.00494d. 6.12963e. 14.8997f. 1.00595632

Dr. Doug Griffin7611 147th Terrace, Monterey, CA 38411

705-555-6644

Name: 474 Hurricane AlleyAndrea Kelly

Rx:

Disp:

Dispense as written May substitute

zinc oxide parts v

Wool fat parts xv

Petrolatum parts lxxx

ounces iv

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2. What fractional parta. of 64 is 2?b. of 1⁄16 is 1⁄20?c. of 1⁄32 is 2?

3. What decimal fractiona. of 18 is 21⁄4?b. of 25 is 0.005?c. of 7000 is 437.5?

4. Write the following as decimals and add: 3⁄1000, 75⁄100, 3⁄20, 5⁄8, 13⁄25

5. Write the following as decimals and add: 3⁄5, 1⁄20, 65⁄1000, 19⁄40, 3⁄8

6. Perform the following functions and round your answer to the nearest hundredth:a. 6.39 − 0.008b. 24 × 0.25 gc. 0.720 × 0.095 graind. 56.824 ÷ 0.0905e. 71.455 ÷ 0.512

7. Perform the following functions and retain only significant figures in youranswer:a. 6.39 − 0.008b. 7.01 − 6.0c. 24 × 0.25 gd. 5.0 × 48.3 g

8. A clinical study of a new drug demonstrated that the drug met the effectivenesscriteria in 646 of the 942 patients enrolled in the study. Express these results asa decimal fraction and as a percent.

9. A pharmacist had 3 oz of hydromorphone hydrochloride. He used the following:1⁄8 oz1⁄4 oz11⁄2 oz

How many ounces of hydromorphone hydrochloride were left?

10. A pharmacist had 5 g of codeine sulfate. He used it in preparing the following:8 capsules each containing 0.0325 g12 capsules each containing 0.015 g18 capsules each containing 0.0008 g

How many grams of codeine sulfate were left after he prepared the capsules?

11. The literature for a pharmaceutical product states that 26 patients of the 2103enrolled in a clinical study reported headache after taking the product. Cal-culate (a) the decimal fraction and (b) the percent of patients reporting thisadverse response.


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12. Solve each of the following:a. (1⁄120 ÷ 1⁄150) × 50 = X

b.

c. 3⁄4 × X = 48

d.

Solve by proportion:

13. If a cough syrup contains 2 mg of brompheniramine maleate in each 5-mLdose, how much brompheniramine maleate in milligrams does 120 mL ofsyrup contain?

14. If 24 lb of a chemical cost $46.80, how much in pounds can be bought for $78?

15. If 15 gal of a certain liquid costs $36.25, how much will 4 gal cost?

16. If 125 gal of a mouth rinse contains 20 g of a coloring agent, how much ingrams does 160 gal contain?

17. If 50 tablets contain 1.5 g of active ingredient, how much of the ingredient ingrams do 1375 tablets contain?

18. If a diarrhea remedy contains 2.7 mL of paregoric in each 30 mL of mixture,how much paregoric in milliliters does 1 tsp (5 mL) of the mixture contain?

19. A metered dose inhaler contains 225 mg of metaproterenol sulfate, which issufficient for 300 inhalations. How much metaproterenol sulfate in milligramsdoes each inhalation administer?

20. How much of a substance in milligrams is needed for 350 tablets if 75 tabletscontain 3 mg of the substance?

21. Ipecac syrup contains the equivalent of 32 grains of ipecac in each fluidounce(480 minims) of the syrup. How much in minims provides the equivalent of 20 grains of ipecac?

22. A pediatric vitamin product contains the equivalent of 0.5 mg of fluoride ion ineach milliliter. How much fluoride ion in milligrams does a dropper that deliv-ers 0.6 mL provide?

23. If a pediatric vitamin product contains 1500 U of vitamin A per milliliter ofsolution, how many units of vitamin A are administered to a child given 2 dropsof the solution from a dropper calibrated to deliver 20 drops per milliliter ofsolution?

24. An elixir of aprobarbital contains 40 mg of aprobarbital in each 5 mL. Howmuch aprobarbital in milligrams is in 4000 mL of the elixir?

1500 55

× X =

1 12

100× 1000 = X


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25. An elixir of ferrous sulfate contains 220 mg of ferrous sulfate in each 5 mL.If each milligram of ferrous sulfate contains the equivalent of 0.2 mg of ele-mental iron, how much elemental iron in milligrams does each 5 mL of theelixir contain?

26. At a constant temperature, the volume of a gas varies inversely with the pres-sure. If a gas occupies a volume of 1000 mL at a pressure of 760 mm, what isits volume at a pressure of 570 mm?

27. If an ophthalmic solution contains 1 mg of dexamethasone phosphate permilliliter of solution, how much solution in milligrams is needed to deliver0.15 mg of dexamethasone phosphate?

28. How many 0.1-mg tablets will yield the same amount of drug as 50 tablets, eachcontaining 0.025 mg of drug?

29. A 15-mm package of nasal spray delivers 20 sprays per milliliter of solution,with each spray containing 1.5 mg of drug. (a) How many total sprays will thepackage deliver? (b) How much drug in milligrams does the 15-mL package ofthe spray contain?

30. A penicillin V potassium preparation provides 400,000 U of activity in each250-mg tablet. How many total units of activity are provided if the patient takes4 tablets a day for 10 days?

31. A solution of digitoxin contains 0.2 mg/mL. How much in milliliters contains0.03 mg of digitoxin?

32. A pharmacist prepared a solution containing 5 million units of penicillin per 10 mL. How many units of penicillin does 0.25 mL contain?

33. If a 5-g packet of a potassium supplement provides 20 mEq of potassium ionand 3.35 mEq of chloride ion, (a) how much of the powder in grams would pro-vide 6 mEq of potassium ion? (b) How much chloride ion in milliequivalentsdoes this amount of powder provide?

34. If an intravenous fluid delivery system is adjusted to deliver 15 mg of medica-tion per hour, how much in milligrams is delivered per minute?

35. If a potassium chloride elixir contains 20 mEq of potassium ion in each 15 mLof elixir, how much in milliliters provides 25 mEq of potassium ion to thepatient?

36. The blood serum concentration of the antibacterial drug ciprofloxacin increasesproportionately with the dose of drug administered. If a 250-mg dose of thedrug results in a serum concentration of 1.2 mcg of drug per milliliter of theserum, how much drug in micrograms per milliliter of serum would be expectedfollowing a dose of 500 mg of drug?

37. If a syringe contains 5 mg of medication in each 10 mL of solution, how muchin milligrams is administered when 4 mL of solution is injected?


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38. The dose of the drug Mintezol® is determined in direct proportion to a patient’sweight. If the dose of the drug for a patient weighing 150 lb is 1.5 g, whatshould be the dose for a patient weighing 110 lb?

39. If 0.5 mL of a mumps virus vaccine contains 5000 U of antigen and the 0.5 mLof vaccine is diluted to 2 mL with water for injection, how many units does eachmilliliter contain?

40. A sample of Oriental ginseng contains 0.4 mg of active constituents in each100 mg of powdered plant. How much active constituent in milligrams does15 mg of powdered plant contain?


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Fundamentals of Calculations - JBLearning

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