© 2010 The McGraw-Hill Companies, Inc. All rights reserved 2-1 McGraw-Hill Math and Dosage Calculations for Health Care Third Edition Booth & Whaley Chapter.

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McGraw-Hill

Math and Dosage Calculations for Health

Care Third EditionBooth & Whaley

Chapter 2: Percents, Ratios, and Proportions

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2.1 Describe the relationship among percents, ratios, decimals, and fractions.

2.2 Calculate equivalent measurements using percents, ratios, decimals, and fractions.

2.3 Indicate solution strengths by using percents and ratios.

Learning Outcomes

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2.4 Explain the concept of proportion.

2.5 Calculate missing values in proportions by using ratios (means and extremes) and fractions (cross-multiplying).

Learning Outcomes (cont.)

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Introduction

An understanding of percents, ratios, and proportion is needed to prepare solutions and solids in varying amounts.

Finding a missing value in a ratio or fraction proportion problem is necessary for dosage calculation.

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Percents

Provide a way to express relationship of parts to a whole.Indicated by the symbol %.

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

The whole is always 100 units.

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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

Percents (cont.)

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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

Rule 2-1Rule 2-1 To convert a percent to a decimal, remove the percent symbol. Then divide the remaining number by 100.

ExampleExample

Working with Percents

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Convert 0.02 to a percent:

Multiply by 100%

Move the decimal point two places to the right

0.02 x 100% =2.00% = 2%

Rule 2-2Rule 2-2To convert a decimal to a percent, multiply the decimal by 100. Then add the percent symbol.

Working with Percents (cont.)

ExampleExample

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Rule 2-3Rule 2-3To 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.

Working with Percents (cont.)

Convert 8% to an equivalent fraction.

8% = 252

1008

1008

2

25

ExampleExample

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Rule 2-4Rule 2-4To convert a fraction to a percent, first convert

the fraction to a decimal. Round the decimal to the nearest hundredth. Then follow the rule for converting a decimal to a percent.

Convert 2/3 to a percent.

Convert 2/3 to a decimal. Round to the nearest hundredth. 2/3 = 2 divided by 3 = 0.666 = 0.670.67 x 100% = 67%

Working with Percents (cont.)

ExampleExample

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Practice

Convert 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%

If you are still not sure, practice the rest of the exercises “Working with Percents.”

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Percent Strength of Mixtures

Percents commonly used to indicate concentration of ingredients in mixturesSolutions

Lotions

Creams

Ointments

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Percent Strength of Mixtures (cont.)

Two categories of mixtures:

Fluid • Mixtures that flow• Solute• Solvent or diluent• Solution

Solid or semisolid• Creams and ointments

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Percent Strength of Mixtures (cont.)

Rule 2-5Rule 2-5a. For fluid mixtures prepared with a drydry

medication, the percent strength represents the number of grams of the medication contained in 100 mLs of the mixture.

b. For fluid mixtures prepared with a liquidliquid medication, the percent strength represents the number of milliliters of the medication contained in 100 mLs of the mixture.

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Percent Strength of Mixtures (cont.)

Determine the amount of hydrocortisone per 100 mL of lotion.

A 2% hydrocortisone lotion will contain 2 grams of hydrocortisone powder in every 100 mL.

Therefore, 300 mL of the lotion will contain 3 times as much, or 6 grams of hydrocortisone powder.

ExampleExample

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Percent Strength of Mixtures (cont.)

Rule 2-6Rule 2-6a. For solid or semisolid mixtures prepared with

a drydry medication, the percent strength represents the number of grams of the medication contained in 100 grams of the mixture.

b. For solid or semisolid mixtures prepared with a liquidliquid medication, the percent strength represents the number of milliliters of the medication contained in 100 grams of the mixture.

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Percent Strength of Mixtures (cont.)

Determine the amount of hydrocortisone per 100 grams of ointment.

Each percent represents 1 gram of hydrocortisone per 100 grams of ointment.

A 1% hydrocortisone ointment will contain 1 gram of hydrocortisone powder in every 100 grams.

Therefore, 50 grams of the ointment will contain ½ as much or 0.5 grams of hydrocortisone powder.

ExampleExample

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Practice

How many grams of drug are in 100 mL of 10% solution?

How many grams of dextrose will a patient receive from a 20 mL bag of dextrose 5%?

Answer 10 grams

Answer 5 grams will be in 100 mL, so the patient will receive 1 gram of dextrose in 20 mL

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Ratios

Relationship of a part to the whole

Relate a quantity of liquid drug to a quantity of solution

Used to calculate dosages of dry medication such as tablets

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Ratios (cont.)

Rule 2-7Rule 2-7Reduce a ratio as you would a fraction. Find the

largest whole number that divides evenly into both values A and B.

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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Ratios (cont.)

Rule 2-8Rule 2-8To 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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Ratios (cont.)

Rule 2-9 Rule 2-9 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

Convert a mixed number to a ratio by first writing the mixed number as an improper fraction.

BA

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Ratios (cont.)

Convert the following into a ratio:

= 7:12127

1247 = 47:12

ExampleExample

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Ratios (cont.)

Convert the 1:10 to a decimal.

1. Write the ratio as a fraction.

1:10 = 10

1

Rule 2-10Rule 2-10 To convert a ratio to a decimal:

1. Write the ratio as a fraction

2. Convert the fraction to a decimal (Chapter 1)

ExampleExample

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Ratios (cont.)

2. Convert the fraction to a decimal.

= 1 divided by 10 = 0.1

Thus, 1:10 = 101

= 0.1

101

Example (cont.)Example (cont.)

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Ratios (cont.)

Rule 2-11Rule 2-11 To convert a decimal to a ratio:

1. Write the decimal as a fraction (see Chapter 1).

2. Reduce the fraction to lowest terms.

3. Restate the fraction as a ratio by writing the numerator as value A and the denominator as value B.

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Ratios (cont.)

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 2-12Rule 2-12 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.

Ratios (cont.)

Click for example

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Rule 2-13Rule 2-13 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 by writing the numerator as value A and the denominator as value B, in the form A:B.

Ratios (cont.)

Click for example

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Ratios (cont.)

Convert 2:3 to a percent.

Convert 25% to a ratio.

ExamplesExamples

1. 2:3 = 3

2= 0.67

2. 0.67 X 100% = 67%

1. 25% = 10025

25%4:141

10025

2.

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Practice

Convert the following ratio to 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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Ratio Strengths

Used to express the amount of drug In a solution In a solid dosage form

Equals the dosage strength of the medication

Ratio First number = the amount of drug Second number = amount of solution or number of

tablets or capsules 1 mg:5 mL = 1 mg of drug in every 5 mL of solution

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Ratio Strengths (cont.)

Write the ratio strength to describe 50 mL of solution containing 3 grams of drug.

First number represents amount of drug = 3 grams

Second number represents amount of solution = 50 mL

The ratio is 3 g:50 mL

ExampleExample

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ERROR ALERT!ERROR ALERT!

Do not forget the units of measurement.

Including units in the dosage strength will help avoid errors.

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Practice

Write a ratio to describe the following:

Two tablets contain 20 mg drug

Answer 5 g:100 mL

100 mL of solution contain 5 grams of drug

Answer 20 mg:2 tablets

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Writing Proportions

Mathematical statement that two ratios or two fractions are equal

2:3 is read “two to three”

Double colon in a proportion means “as”

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

Do not reduce the ratios to their lowest terms

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Writing Proportions (cont.)

Write proportion by replacing the double colon with an equal sign

2:3::4:6 is the same as 2:3 = 4:6

This format is a fraction proportion

64

32

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Writing Proportions (cont.)

Rule 2-14Rule 2-14 To write a ratio proportion as a fraction proportion:

1. Change the double colon to an equal sign.

2. Convert both ratios to fractions.

Rule 2-15Rule 2-15 To write a fraction proportion as a ratio proportion:

1. Convert each fraction to a ratio.

2. Change the equal sign to a double colon.Click to go to Example

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Writing Proportions (cont.)

1. Convert each fraction to a ratio so that 5:6 = 10:12

Write as a ratio proportion.1210

65

12:101210

6:565

and

2. 5:6 = 10:12 same as 5:6::10:12

Write 5:10::50:100 as a fraction proportion.

ExamplesExamples

1. 5:10::50:100 same as 5:10 = 50:100

10050

105

2. 5:10::50:100 same as

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Practice

Write the following ratio proportions as fraction proportions:

50:25::10:5

4:5::8:10Answer 4:5 = 8:10 or

108

54

Answer 50:25 = 10:5 or

510

2550

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Means and Extremes

Proportions are used to calculate dosages

If three of four of the values of a proportion are known, the missing value can be determined by:• Ratio proportion• Fraction proportion

The proportion must be set up correctly to determine the correct amount of medication.

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Means and Extremes (cont.)

A : B :: C : DA : B :: C : D

Extremes

Means

A ratio proportion in the form A:B::C:D.

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Means and Extremes (cont.)

Rule 2-16Rule 2-16 To determine if a ratio proportion is true:

1. Multiply the means.

2. Multiply the extremes.

3. Compare the product of the means and the product of the extremes. If the products are equal, the proportion is true.

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Means and Extremes (cont.)

Determine if 1:2::3:6 is 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

The statement 1:2::3:6 is a true proportion.

ExampleExample

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Means and Extremes (cont.)

Rule 2-17Rule 2-17 To find the missing value in a ratio proportion:

1. Write an equation setting the product of the means equal to the product of the extremes.

2. Solve the equation for the missing value.

3. Restate the proportion, inserting the missing value.

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

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Means and Extremes (cont.)

Find the missing value in 25:5::50:?

1. Write an equation setting the product of the means equal to the product of the extremes.

5 x 50 = 25 x ? and 250 = 25?

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

25? X 25

25250

? = 10

ExampleExample

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Means and Extremes (cont.)

3. Restate the proportion, inserting the missing value. 25:5::50:10

4. Check your work.

5 X 50 = 25 X 10 250 = 250

The missing value is 10.

Example (cont.)Example (cont.)

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Canceling Units in Proportions

Remember to include units when writing ratios.

Helps determine the correct units for the answer.

200 mg:5 mL::500 mg:?

Cancel units correctly.

200 mg:5 mL::500mg:?

Answer to ? will be in mL.

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Canceling Units in Proportions (cont.)

Rule 2-18Rule 2-18 If the units in the first part of the ratio in a

proportion are the same, they can be canceled.

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

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Canceling Units in Proportions (cont.)

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

ExampleExample

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

? = 100mg100

? X 100100

10000

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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 missing values.

3:12::?:36

Answer 8 10:4::20:?

Answer 9

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Cross-Multiplying

Determination if true or not

Proportion • Compare the products

of the extremes (A & D) with the products of the means (B & C)

Fractions• Cross-multiplying

to determine if it is true

A : B :: C : DA : B :: C : D

ExtremesExtremes

MeansMeans

DC

BA

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Cross-Multiplying (cont.)

Rule 2-19Rule 2-19 To determine if a fraction proportion is true:

1. Cross-multiply Multiply the numerator of the first fraction with

the denominator of the second fraction.

Then multiply the denominator of the first fraction with the numerator of the second fraction.

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

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Cross-Multiplying (cont.)

Determine if is a true proportion. 25

1052

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. 25

1052

ExampleExample

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Cross-Multiplying (cont.)

Rule 2-20Rule 2-20 To find the missing value in a fraction proportion:

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

2. Solve the equation to find the missing value.

3. Restate the proportion, inserting the missing value.

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

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Cross-Multiplying (cont.)

Find the missing value in ?6

53

1. Cross-multiply.

3 X ? = 5 X 6

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

? = 10330

3? X 3

ExampleExample

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Cross-Multiplying (cont.)

3. Restate the proportion, inserting the missing value.

4. Check your work by cross-multiplying.

3 X 10 = 5 X 6

30 = 30

The missing value is 10.

106

53

Example (cont.)Example (cont.)

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Canceling Units in Fraction Proportions

You can cancel units in fraction proportions.

Rule 2-21Rule 2-21If the units of the numerator of the two fractions

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

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

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

Canceling Units in Fraction Proportions (cont.)

?mg500

mL5mg200

The missing value can now be found by cross-multiplying and solving the equation as before.

ExampleExample

Set up the fraction

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500mL?

mL100mg20

Canceling Units in Fraction Proportions (cont.)

Set up the fraction.

Solve for ?, the missing value.

1. 100 X ? = 20 mg X 500

2. Divide each side by 100

3. ? = 100mg of drug in 500mL solution

100500X mg20

100? X 100

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

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Practice

Determine if the following proportions are true:

Answer Not true

Answer Not true

4828

167

300125

12550

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Practice

Cross-multiply to find the missing value.

Answer 1

Answer 1

5?

153

375

?25

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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

?mg450

mL250mg90

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End of Chapter 2

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