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 2-2

2-3 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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:

2-4 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Learning Outcomes  Explain the concept of proportion.  Calculate missing values in proportions by using ratios (means and extremes) and fractions (cross-multiplying).

2-5 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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

2-6 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 1.0 = = 100 percent

2-7 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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. Example

2-8 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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. Example

2-9 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-10 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Working with Percents (con’t) Convert 8% to an equivalent fraction. 8% = 2 25 Example

2-11 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 = = 0.67 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. Example

2-12 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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% Example

2-13 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice decimals Convert the following percents to decimals: Answer = % Answer = 3.00 If you are still not sure, practice the rest of the exercises “Working with Percents.” 14%

2-14 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice percents Convert the following fractions to percents: Answer = 75% 4/5Answer = 80% If you are still not sure, practice the rest of the exercises “Working with Percents.” 6/8

2-15 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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

2-16 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 Example

2-17 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 = Example

2-18 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-19 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Working with Ratios (con’t) Convert the following into a ratio: is 7:12 is 47:12 Example

2-20 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Working with Ratios (con’t) Convert the 1:10 to a decimal. 1.Write the ratio as a fraction. 1:10 = To convert a ratio to a decimal: 1.Write the ratio as a fraction. 2.Convert the fraction to a decimal (see Chapter 1). Example

2-21 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Working with Ratios (con’t) 2. Convert the fraction to a decimal. = 1 divided by 10 = 0.1 Thus, 1:10 == 0.1 Example (con’t)

2-22 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-23 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 :4 Example

2-24 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Working with Ratios (con’t) To convert a ratio to a percent: Convert 2:3 to a percent. 1. 2:3 == 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. Example

2-25 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Working with Ratios (con’t) To convert a percent to a ratio: 1. 25% = 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. Example

2-26 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice Convert the following ratio to fraction or mixed numbers: 5:3 3:4Answer =

2-27 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice Convert the following decimals to ratios: Answer =

2-28 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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

2-29 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 grams The second number represents amount of solution = 50 mL The ratio is 3 g:50 mL Example

2-30 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved CAUTION Do not forget the units of measurements. Including units in the dosage strength will help you avoid some common errors.

2-31 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice Write a ratio to describe the following: Two tablets contain 20 mg drug Answer = 5 g:100 mL 100 mL of solution contains 5 grams of drug Answer = 20 mg:2 tablets

2-32 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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

2-33 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 This format is a fraction proportion

2-34 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Writing Proportions (con’t) To write a ratio proportion as a fraction proportion: Write 5:10::50:100 as a fraction proportion. 1. 5:10::50:100 same as 5:10 = 50: :10::50:100 same as 1. Change the double colon to an equal sign. 2. Convert both ratios to fractions. Example

2-35 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Writing Proportions (con’t) To write a fraction proportion as a ratio proportion: Example Write 1.Convert each fraction to a ratio so that 5:6 = 10:12 as a ratio proportion. 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.

2-36 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 Answer = 50:25 = 10:5 or

2-37 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-38 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Cross-Multiplying (con’t) 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.

2-39 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Cross-Multiplying (con’t) Example Find the missing value in 1.Cross-multiply. 3 X ? = 5 X 6 2. Solve the equation by dividing both sides by three. ? = 10

2-40 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Cross-Multiplying (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.

2-41 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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. Example You have a solution containing 200 mg drug in 5 mL. How many milliliters of solution contain 500 mg drug? The missing value can now be found by cross- multiplying and solving the equation as before.

2-42 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-43 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Canceling Units in Fraction Proportions (con’t) Example If 100 mL of solution contains 20 mg of drug, how many milligrams of the drug will be in 500 mL of solution? Set up the fraction. Solve for ?, the missing value X ? = 20 mg X Divide each side by 100

2-44 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Canceling Units in Fraction Proportions (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.

2-45 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice Cross-multiply to find the missing value. Answer = 1

2-46 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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 it !

2-47 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Means and Extremes Proportions 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.

2-48 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Means and Extremes (con’t) A : B :: C : D Extremes Means A ratio proportion in the form A:B::C:D.

2-49 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Means and Extremes (con’t) 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.

2-50 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Means and Extremes (con’t) Determine if 1:2::3:6 is a true proportion. Example 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.

2-51 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-52 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Means and Extremes (con’t) Example 1. 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 = ? Find the missing value in 25:5::50:?

2-53 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Means and Extremes (con’t) Example (con’t) Find the missing value in 25:5::50:? 3. Restate the proportion, inserting the missing value. 25:5::50:10 4. Check your work. 5 X 50 = 25 X = 250 The missing value is 10.

2-54 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-55 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Cross-Multiplying A : B :: C : D Extremes Means 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

2-56 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-57 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Cross-Multiplying Example Determine if is a true proportion. 1. Cross-multiply. 2 X 25 = 5 X Compare the products on both sides of the equal sign. 50 = 50 is a true proportion

2-58 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Canceling Units in Proportions 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.

2-59 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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. Example If 100 mL of solution contains 20 mg of drug, how many milligrams of the drug will be in 500 mL of the solution?

2-60 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice Determine if the following proportions are true: Answer = Not true

2-61 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice Determine whether the following proportions are true: 3:8::9:32 Answer = True 6:12::12:24 Answer = Not true

2-62 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Review and Practice Use the means and extremes to find the missing values. 3:12::?:36 Answer = 8 10:4::20:? Answer = 9

2-63 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Percent Strength of Mixtures  Percents are commonly used to indicate concentration of ingredients in mixtures Solutions Lotions Creams Ointments

2-64 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Percent Strength of Mixtures (con’t)  Mixtures can be divided into two categories: Fluid  Mixtures that flow  Solvent or diluent  Solution Solid or semisolid  Creams and ointments

2-65 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Percent Strength of Mixtures (con’t) dry For fluid mixtures prepared with a dry medication, the percent strength represents the number of grams of the medication contained in 100 mLs of the mixture.

2-66 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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)

2-67 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Percent Strength of Mixtures (con’t) Example 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.

2-68 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-69 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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.

2-70 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Percent Strength of Mixtures (con’t) Example 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.

2-71 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved 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

2-72 McGraw-Hill ©2010 by the McGraw-Hill Companies, Inc All Rights Reserved Solve the Mystery Ready to compare the numbers? THE END Your turn to solve the mystery!