Chapter 3 Dosages and Calculations

© 2012 The McGraw-Hill Companies, Inc. All rights reserved.

Chapter 3Relationships of Quantities: Percents, Ratios, and Proportions

PowerPoint® Presentation to accompany:

Math and Dosage Calculations for Healthcare ProfessionalsFourth Edition

Booth, Whaley, Sienkiewicz, and Palmunen

3-2


Learning Outcomes

3-1 Convert values to and from a percent.

3-2 Convert values to and from a ratio.

3-3 Write proportions.

3-4 Use proportions to solve for an unknown quantity.

3-3


Key Terms Cross-multiplying

Means and extremes

Percent

Proportion

Ratio

3-4


Introduction For dosage calculation you must:

understand percents, ratios, and proportions ;

be able to find a unknown quantity in a proportion.

3-5


Percents

Percents provide a way to express the relationship of parts to a whole. Percent is indicated by the symbol %.

Percent means “per 100” or “divided by 100.”

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Percents (cont.)

A number < 1 is expressed as less than 100 percent.

A number > 1 is expressed as greater than 100 percent.

Any expression of one equals 100 percent.

55

1.0 = = 100 percent

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Converting Values To and From a Percent

Rule 3-1Rule 3-1 To convert a percent to a decimal, remove the percent symbol. Then divide the remaining number by 100.

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Converting Values To and From a Percent (cont.)

Convert 42% to a decimal:

Move the decimal point two places to the left.

Insert the zero before the decimal point for clarity.

42% = 42.% = .42. = 0.42

ExampleExample

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Rule 3-2Rule 3-2To convert a decimal to a percent, multiply

the decimal by 100. Then add the percent symbol.

Converting Values To and from a Percent (cont.)

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Working with Percents (cont.)

Convert 0.02 to a percent:

Multiply by 100%.

0.02 x 100% =2.00% = 2%

ExampleExample

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Rule 3-3Rule 3-3To convert a percent to an equivalent fraction, write the value of the percent as the numerator and 100 as the denominator.

Then reduce the fraction to its lowest term.

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Convert 8% to an equivalent fraction.

8% =

ExampleExample

252

1008

1008

2

25

3-13



Rule 3-4 Rule 3-4 To convert a fraction to a percent: 1. convert the fraction to a decimal;

2. round to the nearest hundredth;

3. then follow the rule for converting a decimal to a percent.

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Convert 2/3 to a percent.

Convert 2/3 to a decimal and round to the nearest hundredth.

2/3 = 2 divided by 3 = 0.666 = 0.67

0.67 x 100% = 67%

ExampleExample

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PracticeConvert the following percents to decimals:

Convert the following fractions to percents:

Answer = 0.14

300% Answer = 3.00

14%6/8

4/5 Answer = 80%

Answer = 75%

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Ratios

The relationship of a part to the whole

relates a quantity of liquid drug to a quantity of solution;

is used to calculate dosages of dry medication such as tablets.

3-17


Ratios (cont.)

Like a fraction, a ratio has two parts.

The first part = numerator.

The second part = denominator.

The two parts are separated by a colon.

3-18


Converting Values To and From a Ratio

Rule 3-5Rule 3-5Reduce a ratio as you would a fraction.

Reduce 2:12 to its lowest terms.

Both values 2 and 12 are divisible by 2.

2:12 is written 1:6

ExampleExample

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Converting Values To and From a Ratio. (cont.)

Rule 3-6Rule 3-6To convert a ratio to a fraction, write value A (1st

number) as the numerator and value B (2nd number) as the denominator, so that A:B =

Convert the following ratio to a fraction:

4:5 =

BA

54

ExampleExample

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Converting Values To and From a Ratio.(cont.)

Rule 3-7 Rule 3-7 To convert a fraction to a ratio, write the

numerator as the 1st value A and the denominator as the 2nd value B.

= A:B

BA

3-21


Converting Values To and From a Ratio (con’t)

Convert the following into a ratio:

= 7:12127

1211

3 = 47:12

ExampleExample

1247

3-22


Converting Values To and From a Ratio. (cont.)

Rule 3-8Rule 3-8 To convert a ratio to a decimal:1. write the ratio as a fraction;

2. convert the fraction to a decimal.

3-23


Converting Values To and From a Ratio (cont.)

Convert the ratio 1:10 to a decimal.

1. Write the ratio as a fraction.

1:10 = 10

1

ExampleExample

2. Convert the fraction to a decimal.

101

= 1 divided by 10 = 0.1

Thus, 1:10 = 101

= 0.1

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Rule 3-9Rule 3-9 To convert a decimal to a ratio:

1. write the decimal as a fraction;

2. reduce the fraction to lowest terms;

3. restate the fraction as ratio.

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1. Write the decimal 0.25 as a fraction.

2. Reduce the fraction to lowest terms.

3. Restate the number as a ratio. 1:4

10025

41

10025

ExampleExample

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Rule 3-10 Rule 3-10 To convert a ratio to a percent:

1. convert the ratio to a decimal;

2. write the decimal as a percent by multiplying the decimal by 100 and adding the % symbol.

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Convert 2:3 to a percent.

1. 2:3 = 32

= 0.67

2. 0.67 X 100% = 67%

ExampleExample

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Rule 3-11Rule 3-11 To convert a percent to a ratio:

1. write the percent as a fraction;

2. reduce the fraction to lowest terms;

3. write the fraction as a ratio. Numerator = value A Denominator = value B A:B

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Convert 25% to a ratio.

1. 25% = 10025

25%4:141

10025

2.

ExampleExample

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Practice

Convert the following ratios to a fraction or mixed numbers:

5:33:4

Answer =43

Answer =32

135

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Practice

Convert the following decimals to ratios:

80.9

Answer =10:9

109

Answer =

1:818

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Writing Proportions A proportion is a mathematical statement that

two ratios or two fractions are equal.

2:3 is read “two to three”

2:3 = 4:6 is read “two is to three as four is to six”

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Writing Proportions (cont.)

Rule 3-12Rule 3-12 To change a proportion from ratios to fractions, convert both ratios to fractions.

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Write 5:10 = 50:100 as a proportion using fractions.

10050

105 5:10 = 50:100 same as

ExampleExample

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Rule 3-13Rule 3-13 To change a proportion from fractions to ratios, convert each fraction to a ratio.

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Write using ratios.1210

65

12:101210

6:565

and

5:6 = 10:12

ExampleExample

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Write the following as proportions using fractions:

Practice

50:25 = 10:5

4:5 = 8:10Answer

108

54

Answer 510

2550

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Using Proportions to Solve for an Unknown

Proportions are used to calculate dosages.

If three of four of the values of a proportion are known, the unknown quantity can be determined by using:

ratios; or

fractions.

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Using Proportions to Solve for an Unknown (cont.)

A : B = C : D

Means

Extremes

A proportion as the ratio – A:B = C:D.

3-40



Rule 3-14 Rule 3-14 To determine if a proportion is true:

1. multiply the means;

2. multiply the extremes;

3. compare the product of the means and the product of the extremes.

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Is 1:2=3:6 a true proportion?

1. Multiply the means: 2 X 3 = 6

2. Multiply the extremes: 1 X 6 = 6

3. Compare the products of the means and the extremes: 6=6

1:2=3:6 is a true proportion.

ExampleExample

3-42


Using Proportions to Solve for an Unknown (con’t)

Rule 3-15Rule 3-15 To find the unknown quantity in a proportion:

1. Write an equation:

product of the means = product of the extremes

2. Solve for the unknown quantity.

3-43


Using Proportions to Solve for an Unknown (con’t)

Rule 3-15Rule 3-15 (cont.)

3. Restate the proportion, inserting the unknown quantity.

4. Check your work. Determine if the ratio proportion is true.

3-44



Find the unknown quantity in 25:5=50:x

1. Write the equation:

5 x 50 = 25 X x becomes 250 = 25x

2. Solve the equation by dividing both sides by 25.

2525

25250 x

x = 10

ExampleExample

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3. Restate the proportion, inserting the unknown quantity. 25:5=50:10

4. Check your work.

5 X 50 = 25 X 10 250 = 250

The unknown quantity is 10.

Example (cont.) Example (cont.)

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ERROR ALERT!ERROR ALERT!

Do not forget the units of measurement.

Including units in the dosage strength will help you avoid errors.

3-47


Canceling Units in Proportions (cont.)

Rule 3-16Rule 3-16

1. If the units in the first part of the ratio in a proportion are the same, they can be canceled.

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Rule 3-16 Rule 3-16 (cont.)

2. If the units in the second part of the ratio in a proportion are the same, they can be canceled.

3-49



If 100 mL of solution contains 20 mg of drug, how many milligrams of the drug will be in 500 mL of the solution?

20 mg:100 mL=x:500 mL 20mg X 500 = 100 X x

x = 100mg100 X 100

100mg 10000 x

ExampleExample

3-50


Practice

Determine whether the following proportions are true:

3:8=9:32

Answer = True 6:12=12:24

Answer = Not true

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Practice

Use the means and extremes to find the unknown quantity.

3:12=x:36

Answer = 8 10:4=20:x

Answer = 9

3-52



Rule 3-17Rule 3-17 To determine if a proportion written as fractions is true:

1. Cross-multiply

2. Compare the products. The products must be equal.

DC

BA

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Determine if is a true proportion.2510

52

1. Cross-multiply. 2 X 25=5 X 10

2. Compare the products on both sides of the equal sign. 50 = 50

Therefore, is a true proportion.

2510

52

ExampleExample

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Rule 3-18Rule 3-18 To find the unknown quantity in a proportion written as fractions:

1. Cross-multiply. Write an equation setting the products equal to

each other.

2. Solve the equation to find the unknown quantity.

3-55



Rule 3-18Rule 3-18 (cont.)


4. Check your work.

3-56



Find the unknown quantity in x

653

1. Cross-multiply.

3 X x = 5 X 6

2. Solve the equation by dividing both sides by three.

x = 103

30

3

3

X x

ExampleExample

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4. Check your work by cross-multiplying.

3 X 10 = 5 X 6

30 = 30

The unknown quantity is 10.

106

53

Example (cont.)Example (cont.)

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Rule 3-19Rule 3-19If the units of the numerator of the two

fractions are the same, they can be dropped or canceled before setting up a proportion.

10mg X ?10

mL10mg75

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Rule 3-19 Rule 3-19 (cont.)

Likewise, if the units from the denominator of the two fractions are the same, they can be canceled.

5mL? 5

5mL25mg

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You have a solution containing 200mg drug in 5mL. How many milliliters of solution contain 500mg drug?

xmg

mL

mg 500

5

200

Cross-multiply to solve the equation.

Set up the fractions.

ExampleExample

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If 100 mL of solution contains 20mg of drug, how many milligrams of the drug will be in 500mL of solution?

mLmL

mg x500100

20Set up the fraction.

Cross-multiply to solve the equation.

ExampleExample

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Practice

Determine if the following proportions are true:

Answer = Not true

Answer = Not true

4828

167

300125

12550

3-63


Practice

Cross-multiply to find the unknown quantity.

Answer = 1

Answer = 2

5153 x

67525

x

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Practice

If 250 mL of solution contains 90 mg of drug, there would be 450 mg of drug in how many mL of solution?

Answer = 1,250 mL

xmg

mL

mg 450

250

90

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In Summary

In this chapter you learned to: convert values to and from a percent;

convert values to and from a ratio.

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In Summary (cont.)

In this chapter you learned to: write proportions;

use proportions to solve for an unknown quantity.

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Apply Your Knowledge

Convert to percent:

1.45 0.056

Convert to a decimal:

15.6 % 0.89%

ANSWER: 145% ANSWER: 5.6%

ANSWER: 0.156 ANSWER: 0.0089

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Convert to a ratio:

Convert to a fraction:

78:10

12575 ANSWER: 75:125

54

7ANSWER:

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Determine if the following are true proportions:

45:90 = 15:30

6/7 = 3/4

ANSWER: Not a true proportion

ANSWER: True proportion

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Solve for the unknown:

25mg:6mL=x:12mL

22/x=12/18

ANSWER: 50mg

ANSWER: 33

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End of Chapter 3

If what you're working for really matters, you'll give it all you've got.

~ Nido Qubein

Health & Medicine

Chapter 3 Dosages and Calculations