math for the pharmacy technician: concepts and calculations chapter 2: working with percents,...

Math for the Pharmacy Technician: Concepts and Calculations

Chapter 2: Working with Percents, Ratios, and Proportions

McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Egler • Booth

McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved

Percents, Ratios, and Proportions

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Learning Outcomes

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

Indicate solution strengths by using percents and ratios.

When you have successfully completed Chapter 2, you will have mastered skills to be able to:

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Learning Outcomes

Explain the concept of proportion. Calculate missing values in proportions

by using ratios (means and extremes) and fractions (cross-multiplying).

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Working with Percents

Provides a way to express the relationship of parts to a whole

Indicated by symbol % Percent literally means “per 100” or

“divided by 100” The whole is always 100 units/parts

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Working with Percents (con’t)

A number less than one is expressed as less than 100 percent

A number greater than one is expressed as greater than 100 percent

Any expression of one equals 100 percent

55

1.0 = = 100 percent

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Working with Percents

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

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

ExampleExample

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Working with Percents (con’t)

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%

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

ExampleExample

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Working with Percents (con’t)

To 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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Working with Percents (con’t)

Convert 8% to an equivalent fraction.

8% =

1008

1008

2

25252

ExampleExampleExampleExample

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Convert Fraction to 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 = 0.67

To 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.

ExampleExample

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Convert Fraction to Percent (con’t)

To 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.

Now convert to a percent.

2/3 = 0.67 = 0.67 X 100% = 67%

You can write this as 0.67% = 67%

ExampleExample

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Review and Practice

Convert the following percents to decimalsdecimals:

Answer = 0.14

300% Answer = 3.00

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

14%

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Review and Practice

Convert the following fractions to percentspercents:

Answer = 75%

4/5 Answer = 80%

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

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Working with 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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Working with Ratios (con’t)

Reduce 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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Working with Ratios (con’t)

To convert a ratio to a fraction, write value A

(the first number) as the numerator and value B

(the second number) as the denominator, so

that A:B =

Convert the following ratio to a fraction:

4:5 =

BA

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ExampleExample

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Working with Ratios (con’t)

To convert a fraction to a ratio, write the numerator as the first value A and the denominator as the second 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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Working with Ratios (con’t)

Convert the following into a ratio:

is 7:12127

1247 is 47:12

ExampleExample

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Working with Ratios (con’t)

Convert the 1:10 to a decimal.

1. Write the ratio as a fraction.

1:10 = 10

1

To convert a ratio to a decimal:1. Write the ratio as a fraction.2. Convert the fraction to a

decimal (see Chapter 1).

ExampleExample

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Working with Ratios (con’t)

2. Convert the fraction to a decimal.

= 1 divided by 10 = 0.1

Thus, 1:10 = 10

1= 0.1

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Example (con’t)Example (con’t)

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Working with Ratios (con’t)

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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Working with Ratios (con’t)

1. Write the decimal as a fraction.

2. Reduce the fraction to lowest terms.

3. Restate the number as a ratio.

0.25

10025

41

10025

1:4

ExampleExample

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Working with Ratios (con’t)

To convert a ratio to a percent:

Convert 2:3 to a percent.1. 2:3 = 3

2= 0.67

2. 0.67 X 100% = 67%

1. Convert the ratio to a decimal.

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

ExampleExample

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Working with Ratios (con’t)

To convert a percent to a ratio:

1. 25% =

25%4:141

10025

10025

2.

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.ExampleExample

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Review and Practice

Convert the following ratio to fraction or mixed numbers:

5:3

3:4 Answer =

43

Answer =

32

135

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Review and PracticeConvert the following decimals to ratios:

8

0.9Answer =

10:9109

Answer =1:8

18

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Ratio Strengths

Used to express the amount of drug In a solution In a solid dosage such as a tablet or capsule

This relationship = dosage strength of the medication

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 (con’t)

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

The first number represents amount of drug = 3 gramsThe second number represents amount of solution = 50 mL

The ratio is 3 g:50 mL

ExampleExample

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CAUTIONCAUTION

Do not forget the units of measurements.

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

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Review and Practice

Write a ratio to describe the following:

Two tablets contain 20 mg drug

Answer = 5 g:100 mL100 mL of solution contains 5 grams of drug

Answer = 20 mg:2 tablets

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Writing Proportions

Mathematical statement that two ratios are equal

Statement that 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 (con’t)

Write proportion by replacing the double colon with an equal sign

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

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32

This format is a fraction proportion

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Writing Proportions (con’t)

To write a ratio proportion as a fraction proportion:

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

10050

105

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

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

1. Change the double colon to an equal sign.

2. Convert both ratios to fractions.

Example Example

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Writing Proportions (con’t)

To write a fraction proportion as a ratio proportion:

ExampleExample Write

1. Convert each fraction to a ratio so

that 5:6 = 10:12

12

10

6

5 as a ratio

proportion.

12:101210

6:565

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

1. Convert each fraction to a ratio.

2. Change the equal sign to a double colon.

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Review and Practice

Write the following ratio proportions as fraction proportions:

50:25::10:5

4:5::8:10

Answer = 4:5 = 8:10 or

108

54

Answer = 50:25 = 10:5 or510

2550

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Cross-Multiplying

To find the missing value in a fraction proportions cross-multiply between numerators and denominator of the fractions. Write an equation setting the products equal to each other then solve the equation to find the missing value.

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Cross-Multiplying (con’t)

To find the missing value in a fractionproportion: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 (con’t)

ExampleExample

Find the missing value in ?6

53

1. Cross-multiply.

3 X ? = 5 X 62. Solve the equation by dividing both sides by three.

? = 10

330

3? X 3

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Cross-Multiplying (con’t)

Example (con’t) Example (con’t)

Find the missing value in

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.

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106

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Canceling Units in Fraction Proportions

You can cancel units in fraction proportions.Compare the units used in the top and

bottom of the two fractions in the proportion.

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

mg500mL5mg200

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

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Canceling Units in Fraction Proportions (con’t)

If 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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Canceling Units in Fraction Proportions (con’t)

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

500?

mL100mg20

Set up the fraction.

Solve for ?, the missing value.

1. 100 X ? = 20 mg X 500

2. Divide each side by 100 100500X mg20

100? X 100

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Canceling Units in Fraction Proportions (con’t)

Example (con’t)Example (con’t)

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

3. ? = 100

The second solution will contain 100 mg of drug in 500 mL of solution.

100500X mg 20

100? X 100

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Review and Practice

Cross-multiply to find the missing value.

Answer = 1

Answer = 1

5?

153

375

?25

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Review and 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

Simple isn’t Simple isn’t it !it !

mL?mg450

mL250mg90

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Means and ExtremesProportions are used to calculate dosages.When you know three of four of the values of a proportion, you will find the missing value. You can find the missing value

in a ratio proportion. in a fraction proportion.

Either method is correct. You must set the proportion up correctly to determine the correct amount of medication.

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Means and Extremes (con’t)

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

ExtremesExtremes

MeansMeans

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

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Means and Extremes (con’t)

To determine if a ratio proportion istrue:

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 (con’t)

Determine if 1:2::3:6 is a true proportion.

ExampleExample

1. Multiply the means: 2 X 3 = 6

2. Multiply the extremes: 1 X 6 = 63. Compare the products of the means and the extremes 6=6

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

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Means and Extremes (con’t)

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 (con’t)

ExampleExample1. Write an equation setting the product of the means equal to the product of the extremes. 5

X 50 = 25 X ? 250 = 25 X ?

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

25? X 25

25250

10 = ?

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

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Means and Extremes (con’t)

Example (con’t) Example (con’t)

Find the missing value in 25:5::50:?3. Restate the proportion, inserting the missing value. 25:5::50:104. Check your work.

5 X 50 = 25 X 10 250 = 250

The missing value is 10.

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Cross-Multiplying

To determine whether a proportion is

true, compare the products of the

extremes (A & D) with the products of

the means (B & C).

When written with fractions, use cross-

multiplying to determine if it is true.

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Cross-Multiplying

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

ExtremesExtremes

MeansMeans

To determine whether a proportion is true, compare the products of the extremes (A & D) with the products of the means (B & C).

Multiply the extremes

Multiply the means

Cross-multiplying

A C

B D

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Cross-Multiplying Means and Extremes (con’t)

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

ExampleExample Determine if is a true proportion.

2510

52

1. Cross-multiply.

2 X 25 = 5 X 102. Compare the products on both sides of the equal sign.

50 = 50

is a true proportion25

1052

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Canceling Units in Proportions

Remember to include units when writing ratios.Remember to include units when writing ratios.This will help you to determine the correct units for the

answer when solving problems using proportions. 200 mg:5 mL 500 mg:?

If the units of the first part of two ratios are the same, they can be dropped or canceled.

If the units of the second part of two ratios are the same, they can be canceled.

Units of the first part of each ratio are milligrams.

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Canceling Units in Proportions (con’t)

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.

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

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Review and Practice

Determine if the following proportions are true:

Answer = Not true

Answer = Not true

4828

167

300125

12550

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Review and PracticeDetermine whether the following

proportions are true:

3:8::9:32

Answer = True 6:12::12:24

Answer = Not true

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Review and 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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Percent Strength of Mixtures

Percents are commonly used to indicate concentration of ingredients in mixtures Solutions Lotions Creams Ointments

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Percent Strength of Mixtures (con’t)

Mixtures can be divided into two categories:

Fluid Mixtures that flow

Solvent or diluent

Solution

Solid or semisolidCreams and ointments

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Percent Strength of Mixtures (con’t)

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

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

Percent Strength of Mixtures (con’t)

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Percent Strength of Mixtures (con’t)

ExampleExample 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.

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Percent Strength of Mixtures (con’t)

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

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Percent Strength of Mixtures (con’t)

For solid or semisolid mixtures prepared with a liquid 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 (con’t)

ExampleExample 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.

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Review and 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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Solve the Mystery

Ready to compare the numbers?

THE END

Your turn to solve the mystery!