1 chapter review of basic math...

Review of Basic Math FundamentalsPayal Agarwal, PhD

LeARning OBjectives

After successful completion of this chapter, the student should be able to:

1. Recognize Arabic and Roman numerals2. Explain and differentiate between common and decimal fractions3. Solve problems based on ratio and proportion4. Describe significant figures and explain the concept’s importance in pharmacy

Key teRMs

Arabic numeralsCommon fractionDecimal fraction

ProportionRatioRoman numerals

1.1 intRODuctiOn tO ARABic AnD ROMAn nuMeRALs

The most common form of numbers that we use today is Arabic numerals. The origin of these numbers is not very well known. The use of Roman numerals can be seen

in some older prescriptions. In our current system of prescription writing, use of Roman numerals is minimal, but it can still be seen occasionally on current prescriptions. Roman numerals are used either to specify the quantity of the ingredient in the apothecary system or units of dosage to be dispensed. They are also used to represent different schedules of controlled substances (for example, Schedule I, II, III, IV, and V).

A mathematical algebraic number (currently described as an Arabic number) in the Roman system of numbers is designated by a letter, as given in Table 1.1. This table shows various stem numbers that are used in the Roman system of number writing. Other numbers in the Roman system are generated from these stem numbers; for example, 15 is expressed as XV (10 1 5 5 15).

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The following section demonstrates how to read Roman numerals.

1.1.1 How to Read Roman numeralsRule 1: AdditionWhen a lower number is right after a higher number, add all the numbers together:

XII 5 10 1 1 1 1 5 12Similarly, VI 5 5 1 1 5 6.

Rule 2: Subtraction When a lower number precedes a higher number, the lower number is subtracted from the higher number:

IV 5 1 and 5. Because 1 , 5 and 1 precedes 5, 1 is subtracted from 5. So, IV 5 4; similarly, IX 5 9.

Rule 3: Smaller number between two larger numbers XIX represents 19; X 5 10, I 5 1, and X 5 10. Because I precedes X, it is calculated as 10 2 1 5 9 and 10 1 9 5 19.

It can be easily misunderstood as 21, which is 11 1 10 5 21. However, that is incorrect. Twenty-one in Roman form is expressed as XXI.

Rule 4: Avoid the repetition of more than three occurrences of the same letterFor example, 7 is written as VII, not as IIIIIII.

Rule 5: Use the largest value numeral In accordance with Rule 4, when required, the largest value numeral should be used. For example, 99 is written as XCIX, not as XXXXXXXXXIX.

Rule 6: Power of 10 The smaller number must be a power of 10, and cannot precede a number more than 10 times its value. For example, 99 cannot be expressed as IC 5 100 2 1 5 99. The cor-rect way is XCIX. Similarly, 49 is correctly expressed as 40 1 9 5 XLIX and not as IL.

Rule 7: Use only one preceding smaller number There cannot be more than one small number in front of a larger number. For example, IIX is an invalid Roman numeral. It should be VIII.

Rule 8: Using a barA bar placed on top of a letter increases the numeral’s value 1,000 times (XII 5 12, XII 5 12,000).

tABLe 1.1 Roman system of numbers

Arabic Number Roman Designation

1 I or i

5 V or v

10 X or x

50 L

100 C

500 D

1,000 M

2 Chapter 1 / Review of Basic Math Fundamentals


test yourself 1.1

1. Express the following numerals in Arabic numbers:a. XXIVb. MDXLIIIc. XCIId. LXXIe. MMXX

2. Express the following numbers in Roman numerals:a. 35b. 407c. 198d. 1,020e. 2,487

1.2 cOMMOn AnD DeciMAL FRActiOns

Common fractions (CF) are proportions of whole numbers. These are expressed as 4/3, 1/2, 3/7, and so on. In contrast, a decimal fraction (DF) is a fraction whose denominator is 10n, and n ≥ 1. A decimal fraction is expressed in terms of decimals and not as a CF; for example, 4/100 is expressed decimally as 0.04 and 55/10 as 5.5. Rules for how to per-form mathematical operations, such as addition, subtraction, multiplication, and division, in common fractions along with the conversion rules are demonstrated in Table 1.2.

solved examples

1. Convert the following common fractions to decimal fractions:a. 2/4

2/4 5 1/2 5 0.5

b. 1/101/10 5 0.1

c. 24/3024/30 5 4/5 5 0.8

d. 6/126/12 5 1/2 5 0.5

e. 18/518/5 5 3.6

2. Convert the following decimal fractions to the lowest common fractions:a. 0.234

0.234 5 234/1000 5 117/500

b. 62.562.5 5 625/10 5 125/2

c. 10.510.5 5 105/10 5 21/2

d. 25.225.2 5 252/10 5 126/5

e. 0.0010.001 5 1/1000

1.2 / Common and Decimal Fractions 3


Rule Examples

Conversion

Common fraction to decimal fraction Divide the denominator into the numerator. 5 5

5

2040

24

0.5

125

0.04

Decimal fraction to common fraction Express the decimal as a ratio and reduce it to the lowest possible common fraction.

5

5 5

0.257257

1000

1.51510

32

Addition

Fractions with the same denominator Simply add the numerators and divide the result by the denominator.Reduce the fraction to its simplest form.

1 114

54

34

1 15

1 5 34

94

9/4 5 Lowest fraction

Fractions with different denominators Find the least common multiple (LCM) for the denominator.Convert the individual fractions to reflect the LCM denominator.Add the numerators.Reduce the fraction to its simplest form.

1 126

31

72

LCM 5 62/6 5 2/63/1 5 18/67/2 5 21/6

1 15

2 18 216

416


Subtraction

Fractions with the same denominator Simply subtract the numerators and divide the result by the denominator.Reduce the fraction to its simplest form.

25/7 2 4/72

525 4

7217

5 5217

31

Lowest fraction

Fractions with different denominators Find the LCM for the denominator.Convert the individual fractions to reflect the LCM denominator.Subtract the numerators.Reduce the fraction to its simplest form.

231

26

LCM 5 63/1 5 18/62/6 5 2/6

2 5 5186

26

166

83


Multiplication

Multiply the numerators.Multiply the denominators.Write the new fraction.Reduce it to its simplest form.

3 546

25

830

5830

415


Division

Invert the divisor.Apply the rules for multiplication. 4

92

34

Divisor 5 3/4Invert the divisor 5 4/3

3 592

43

366

36/6 5 6/1 5 Lowest fraction

tABLe 1.2 Rules for performing mathematical operations with common fractions and decimal fractions



3. Perform the necessary mathematical operations and report the final answer.a. 14/5 1 6/3 1 10/9

Find the LCM for the denominator: 45.Convert each fraction to have a common denominator:

53

35

53

35

53

35

14

5

14 9

5 9

126

45

6

3

6 15

3 15

90

45

10

9

10 5

9 5

50

45

Perform the operation:

1 1 5126

45

90

45

50

45

266

45

The lowest fraction is 266/45.

b. 2/3 2 1/3The denominators are the same, so

25

2 1

3

1

3

c. 14/5 2 4/2This has different denominators.Find the LCM for the denominator: 10.Convert each fraction to have a common denominator:

53

35

53

35

14

5

14 2

5 2

28

10

4

2

4 5

2 5

20

10

Perform the operation:

2 528

10

20

10

8

10

Reduce to the lowest fraction:

58

10

4

5

Answer 5 4/5

d. 6/5 3 5/23

35

6 5

5 2

30

10


530

10

3

1

Answer 5 3/1

1.2 / Common and Decimal Fractions 5


e. 15/1 ÷ 3/53

35

15 5

1 3

75

3


575

3

25

1

Answer 5 25/1

test yourself 1.2

Express the results of the following problems in common fractions and decimal fractions:

1. 1 14

2

2

91

2. 22

8

1

11

3. 26

2

1

2

4. 33

5

1

8

1.3 RAtiO AnD PROPORtiOn

Ratio can be defined as the relationship of two or more things in a quantitative manner. For example, Jen has $1,000 to spend on a night out, whereas Emily has $200. Therefore, we can say that Jen has 5 times more dollars than Emily to spend for the night.

In terms of ratio, 1000 dollars/200 dollars 5 5/1 5 5 to 1 or 5:1.A ratio can be expressed in different ways, for example, a / b, a:b, or a to b, where a

and b can be whole numbers, decimals, fractions, or a combination of any of them.When two or more ratios can be expressed with an equals sign, then the two ratios are

said to be in proportion. For example, a:b 5 c:d, where a, b, c, and d can be whole num-bers, decimals, fractions, or a combination of any of them.

As another example, a / b 5 c / d, where a and d are extreme values and b and c are mean values. In any given proportion, when two ratios are equal, the product of extreme values is equal to the product of mean values, which means if a / b 5 c / d, then a 3 d 5 b 3 c.

Therefore, using this rule, an unknown value can be determined if any of the three values are known in a given proportion.

Rules to Remember About numerators and Denominators

■■ Numerators should have the same units.■■ Denominators should be of same units.■■ Three out of four variables should be known.



1.3.1 setting up the Right Ratio and ProportionIf the cost of 10 prescription bottles for Drug A is $25, what would be the cost of 6 pre-scription bottles for the same drug?

To use a ratio, we have to establish a relationship between prescription bottles and dollars. As mentioned in the problem statement:

10 bottles 5 25 dollars6 bottles 5 ? dollarsSet the ratio:

510 bottles

25 dollars

6 bottles

dollarsx

According to the rule of proportion, 10 3 x 5 25 3 6.x 5 150 / 10 5 15 dollarsRatio and proportion have a wide range of applications. You will use them often.

1.3.2 Dimensional AnalysisDimensional analysis, also known as unit factor method, is another way to solve prob-lems. This system is very useful when one unit of measurement has to be converted into a different unit of measurement. This method can also be easily used in place of ratio and proportion to solve problems. Dimensional analysis is based on the fact that any number or unit can be multiplied by one without changing its value. Use the following steps to perform dimensional analysis:

1. Write the desired unit on the right side of the page and line up all the available data with the units on the left side of the page.

2. Include the conversions that pertain to the units given in the data.

3. Invert the dimensions as needed.

4. Cross out the units that appear in the numerator and denominator of the dimensions, leaving only the desired unit.

5. Check: If all the other units are crossed out, leaving the desired unit, it means the prob-lem has been set up with correct dimensions.

6. Perform the calculation.

solved examples

4. Convert 2 lb to g.

1. Write the desired unit on the right side and line up all the available data with the units on the left side.

2 lbg

2. and 3. Include the conversions that pertain to the units given in the data, and invert if required.

2 lb1 kg 1,000 g

2.2 lb 1 kg

1.3 / Ratio and Proportion 7



2 lb1 kg 1,000 g

2.2 lb 1 kg

5. Check: If all the other units are crossed out, leaving the desired unit, it means the problem has been set up with correct dimensions.

Yes, every other unit is crossed out, leaving g.


3 35

2 1 1000

2.2909.1g

5. Convert 1 week into seconds.

1. Write the desired unit on the right side and line up all the available data with the units on the left side.

1 weekseconds

2. and 3. Include the conversions that pertain to the units given in the data and invert if required.

1 week7 days 24 hrs 60 min 60 sec1 week 1 day 1 hr 1 min


1 week7 days 24 hrs 60 min 60 sec

1 week 1 day 1 hr 1 min

5. Check: If all the other units are crossed out, leaving the desired unit, it means the problem has been set up with correct dimensions.

Yes, every other unit is crossed out, leaving seconds.


3 3 35

7 24 60 60

1604,800 seconds

6. Convert 2.6 meters to centimeters.

2.6 m100 cm

5 260 cm1 m

Or,

3 52.6 m100 cm

1 m260 cm.

Or, 2.6 m 5 260 cm.



7. Convert 2.6 km to mm.

2.6 km1,000 m 100 cm 10 mm

1 km 1 m 1 cm

Or,

3 3 3 52.6 km 1000 m

1 km

100 cm

1 m

10 mm

1 cm 2,600,000 mm.

Or, 2.6 km 5 2,600,000 mm.

8. Convert 110 liters to milliliters.

110 L1,000 mL

1 L

35

110 1000 mL

1110,000 mL

Or, 110 L 5 110,000 mL.

9. Determine the ratio of the following:a. 8 to 24

8 to 24 5 8/24 5 1/3, which means 8 is one third of 24.

b. 21 to 721 to 7 5 21/7 5 3/1, which means 21 is 3 times 7.

10. Find the unknown value in the given proportion:a. 2/3 5 x/10

a/b 5 c/da 3 d 5 c 3 b2 3 10 5 x 3 320 5 3 3 xOr, x 5 20/3 5 6.66.

b. 5/25 5 10/xa/b 5 c/da 3 d 5 c 3 b5 3 x 5 25 3 105 3 x 5 250Or, x 5 250/5 5 50.

11. Drug A is available in both generic and brand forms. The generic version costs three fourths of the branded product. If the price of the generic form is $16, calculate the price of the branded product.Generic product 5 16 dollarsGeneric product 5 3/4 of branded productLet the cost of branded product 5 $xTherefore, cost of generic product 5 3/4 of x

1.3 / Ratio and Proportion 9


16 dollars 5 3/4 of x

516 3/4

1x

516

3/4x

x 5 21.33 dollars

test yourself 1.3

1. If 1 tablet of acetaminophen contains 250 mg of the active ingredient, how many tablets can be prepared from 1,500 mg of the drug?

2. If 2.6 g of dye is required to prepare 3 liters of a solution, what quantity of the dye will be required for preparing 1 liter of the same solution?

3. A vial of drug X contains 30 mg of drug in 1 mL of the solution. In order to dispense 15 mg of the drug, what volume of the solution will be required?

4. A patient purchases a prescription drug bottle of 200 tablets for $65. What would be the cost for 70 tablets?

1.4 signiFicAnt FiguRes

The number of significant figures in a result is simply the number of figures that are known with some degree of reliability.

1.4.1 Rules for Deciding the number of significant Figures in a Measured QuantityTable 1.3 shows rules along with examples to determine the number of significant figures in any given number. To express and handle numbers that are very large or very small, expo-nential notation can be used. The powers of 10 are used to express numbers exponentially. For example, a very large number such as 1,004,567,000 can be represented as 1.0 3 109, 14,600 can be represented as 1.46 3 104, and 23,506,001 as 2.35 3 107. Similarly, small numbers such as 0.0045 can be written as 4.5 3 1023 or 0.0000611 as 6.11 3 1025.

Rule Examples

All nonzero digits are significant. a. 1.234 g has 4 significant figures.b. 1.2 g has 2 significant figures.

Any zeroes between nonzero digits are significant. a. 1,002 kg has 4 significant figures.b. 3.07 mL has 3 significant figures.

Leading zeros to the left of the first nonzero digits are not significant; such zeroes indicate the position of the decimal point.

a. 0.0043 has 2 significant figures.b. 0.01 has 1 significant figure.

Trailing zeroes that are also to the right of a decimal point in a number are significant.

a. 0.0230 mL has 3 significant figures.b. 0.20 g has 2 significant figures.

When a number ends in zeroes that are not to the right of a decimal point, the zeroes are not necessarily significant.

To be explicit about the number of significant figures in this situation, it is a good practice to express the number in terms of scientific notation.

a. 230 miles may be 2 or 3 significant figures.b. 21,400 calories may be 3, 4, or 5 significant figures.

For example:a. 230 5 23 3 101 5 2 significant figuresb. 21,400 5 214 3 102 5 3 significant figuresc. 21,400 5 2.14 3 104 5 3 significant figuresd. 2.140 3 104 5 4 significant figures

tABLe 1.3 Rules for deciding the number of significant figures in a measured quantity



1.4.2 Rules for Rounding Off numbersTable 1.4 shows rules to be used for rounding off numbers.

1.4.3 Rules for Mathematical OperationsIn carrying out calculations, the general rule is that the accuracy of a calculated result is limited by the least accurate measurement involved in the calculation. This is explained with examples in Table 1.5.

solved examples

12. Determine the number of significant figures in the following:a. 2.567

2.567 5 4

b. 7,0027,002 5 4

c. 6,4006,400 5 2, 3, or 4

d. 92.5092.50 5 4

e. 0.2110.211 5 3

tABLe 1.4 Rules for rounding off numbers

Digit to Be Dropped Rule Example

Digit to be dropped . 5 The last retained digit is increased by 1.

39.6 is rounded to 40.

Digit to be dropped , 5 The last retained digit is not changed.

39.4 is rounded to 39.

Digit to be dropped 5 5 If the digit following 5 is not zero, the last retained digit is increased by 1.

39.351 is rounded to 39.4.12.2502 is rounded to 12.3.

If all the digits following 5 are zero, the last retained digit is unchanged.

39.500 is rounded to 39.40.125 is rounded to 40.12.

tABLe 1.5 Rules for performing mathematical operations and reporting the end value with the correct number of significant figures

Addition and Subtraction

Report the final value with the same number of decimal places as the measurement with the least number of decimal places.

104.3 (1 decimal place) 1 23.643 (3 decimal places)Calculator answer 5 127.943Correct answer 5 127.9 (round to 1 decimal place)

Multiplication and Division

When two or more numbers are approximate numbers

Report the final value with the same number of significant figures as in the component with the least number of significant figures.

3.0 (2 significant figures) 3 12.60 (4 significant figures)Calculator answer 5 37.8000Correct answer 5 38 (2 significant figures)

When one of the numbers is an absolute number

Report the final value with the same number of significant figures as in the approximate number.

21 (absolute number, with 2 significant figures) 3 3.542 (approximate number with 4 significant figures)Calculator answer 5 74.382Correct answer 5 74.38 (4 significant figures)

1.4 / Significant Figures 11


f. 0.00140.0014 5 2

g. 1.01.0 5 2

h. 0.01 3 102

0.01 3 102 5 1

i. 0.0010.001 5 1

13. Give the exact numerical result for the following, and then express it with the correct number of significant figures:a. 47.76 1 1.507 1 103.2

b. 25 2 6.23 1 3.19

c. 3.05 3 5.0

d. 400.0 / 3.2002

e. (2.0)3

f. 12 1 0.5

g. 38 1 2.125

h. 48 3 25

i. 12 1 111.5

j. 10.01 1 4.0552

Solutions

Problem Calculator AnswerAnswer with the Correct

Number of Significant Figures

47.76 1 1.507 1 103.2 152.467 152.5

25 2 6.23 1 3.19 21.96 22

3.05 3 5.0 15.25 15

400.0 / 3.2002 124.992188 125.0

(2.0)3 8 8.0

12 1 0.5 12.5 12

38 1 2.125 40.125 40

48 3 25 1,200 12 3 102

12 1 111.5 123.5 123

10.01 1 4.0552 14.0652 14.07

test yourself 1.4

1. Determine the number of significant figures in the following:a. 2.366b. 1.002c. 2.00d. 3.45e. 0.15 3 103

f. 0.0050g. 890



2. Give the exact numerical result for the following, and then express it with the correct number of significant figures:a. 23.04 1 11 1 0.002 5b. 527.15 2 2.614 5c. 34 3 2 5d. 12 2 3.03 1 0.19 5e. 3.05 3 5.0 5f. 400.0 / 3.2002 5g. 2.0132 3 198 5h. (5.0)3 5i. 0.326 3 (72 2 12.4) 5j. Find the average of 0.23, 0.260, 0.1155, 1.242, and 2.1.

1.5 APPLicAtiOn OF MAtH FunDAMentALs tO PHARMAceuticAL cALcuLAtiOns

As mentioned earlier in this chapter, Roman numerals were used in ancient prescription writing, and their use is still evident in many current prescriptions. This section will pro-vide some examples related to compounding and dispensing of pharmaceutical products covering the concepts discussed in previous sections of this chapter.

Use the information you have learned throughout this chapter to work on the follow-ing prescriptions.

solved examples

14. Interpret the correct quantities of different ingredients in the following prescriptions:

a.

Name John SmithAddress 51 Broadway Blvd Date 1/25/14

Refills M.D.

DOB 6/7/69

0 B. Pajamo

Acetaminophen 65 mgLactose q.s. ad 300 mgDisp: Capsule XXXSig: take 1 cap po bid

B. Pajamo, M.D.4701 Main St.

Baltimore, MD 12345

The formula is given for one capsule; dispense XXX means dispense 30 such cap-sules. Also calculate the amount of lactose required for 30 capsules.Quantity of acetaminophen required for 30 capsules: 65 3 30 5 1,950 mgLactose q.s. ad means the quantity of lactose sufficient to make the total weight of the individual capsule equal to 300 mg. (Refer to Chapter 2.)Amount of lactose in one capsule 5 300 mg 2 65 mg5 235 mg.For 30 capsules 5 235 3 30 5 7,050 mg.

1.5 / Application of Math Fundamentals to Pharmaceutical Calculations 13


15. Calculate the quantity of drug X in 30 mL of the following prescribed solution:

Name Laya Griffin

Address 51 Broadway Blvd Date 6/14/14

Refills M.D.

DOB 8/20/88

0 B. Pajamo

Drug X 2.5 gPurified water qs 100 mLDisp: 30 mLSig: Take 2 tsp qid for 5 days


Baltimore, MD 12345

2.5 g of drug is required to make 100 mL of solution. For 30 mL of solution, set the ratio and use the proportion:

52.5 g of drug X

100 mL of solution

g of drug X

30 mL of solution

x

0.75 g of drug X is present in 30 mL of the solution.

AnsweRs tO test yOuRseLF

test yourself 1.1

1. a. 24b. 1,543c. 92d. 71e. 2,020

2. a. XXXVb. CDVIIc. CXCVIIId. MXXe. MMCDLXXXVII

test yourself 1.2

1. 29/9, 3.2

2. 7/44, 0.159

3. 5/2, 2.5

4. 3/40, 0.075

b.

Name John Smith


Refills M.D.

DOB 6/7/69

0 B. Pajamo

Zinc oxide parts viCamphor parts iiWhite petrolatum parts liiiDisp: ounces xvSig: Apply on the affectedarea as directed bid


Baltimore, MD 12345

Zinc oxide: 6 partsCamphor: 2 partsWhite petrolatum: 53 partsDispense: 15 ounces1 ounce 5 30 g (Refer to Chapter 3.)So, 15 ounces 5 450 g.Use the concepts of ratio and proportion.Add the total parts of the individual ingredients in the prescription.6 parts 1 2 parts 1 53 parts 5 61 partsTo prepare 61 g of ointment:6 g of zinc oxide2 g of camphor53 g of white petrolatumTo prepare 450 g of ointment:

56 g of zinc oxide

61 g of ointment

g of zinc oxide

450 g of ointment

x

x 5 44.26 g of zinc oxide

52 g of camphor

61 g of ointment

g of camphor

450 g of ointment

x

x 5 14.75 g of camphor

553 g of white petrolatum

61 g of ointment

g of white petrolatum

450 g of ointment

x

x 5 391 g of white petrolatum



15. Calculate the quantity of drug X in 30 mL of the following prescribed solution:

Name Laya Griffin


Refills M.D.

DOB 8/20/88

0 B. Pajamo

Drug X 2.5 gPurified water qs 100 mLDisp: 30 mLSig: Take 2 tsp qid for 5 days


Baltimore, MD 12345

2.5 g of drug is required to make 100 mL of solution. For 30 mL of solution, set the ratio and use the proportion:

52.5 g of drug X

100 mL of solution

g of drug X

30 mL of solution

x

0.75 g of drug X is present in 30 mL of the solution.

AnsweRs tO test yOuRseLF

test yourself 1.1

1. a. 24b. 1,543c. 92d. 71e. 2,020

2. a. XXXVb. CDVIIc. CXCVIIId. MXXe. MMCDLXXXVII

test yourself 1.2

1. 29/9, 3.2

2. 7/44, 0.159

3. 5/2, 2.5

4. 3/40, 0.075

1.5 / Application of Math Fundamentals to Pharmaceutical Calculations 15


test yourself 1.3

1. 6

2. 0.866 g

3. 0.5 mL

4. $22.75

test yourself 1.4

1. a. 4b. 4c. 3d. 3e. 2f. 2g. 2 or 3

2. a. 34.042, 34b. 524.536, 524.54c. 68, 68d. 9.16, 9e. 15.25, 15f. 124.992188, 125.0g. 398.6136, 398.61h. 125, 1.3 3 102

i. 19.4296, 19.4j. 0.7895, 0.78950

ReFeRences1. Ansel CH. Pharmaceutical Calculations. 13th ed. Philadelphia, PA: Lippincott Williams & Wilkins; 2009.

2. Khan MA, Reddy IK. Pharmaceutical and Clinical Calculations. 2nd ed. New York, NY: CRC Press; 2000.



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