Math and Dosage Calculations for Health Care Third Edition Booth & Whaley

1. Math and Dosage Calculations for Health Care Third EditionBooth & Whaley Chapter 2: Percents, Ratios, and Proportions McGraw-Hill

2. Learning Outcomes 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.

3. Learning Outcomes (cont.) 2.4 Explain the concept of proportion. 2.5 Calculate missing values in proportions by using ratios (means and extremes) and fractions (cross-multiplying).

4. 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.

5. 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.

6. 1.0 = = 100 percent 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.

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

8. Working with Percents (cont.) Rule 2-2 To convert a decimal to a percent, multiply the decimal by 100. Then add the percent symbol. Convert 0.02 to a percent: • Multiply by 100% • Movethe decimal point two places to the right • 0.02x 100% =2.00% = 2% Example

9. Working with Percents (cont.) Rule 2-3 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. Convert 8% to an equivalent fraction. 8% = Example 2 25

10. Working with Percents (cont.) Rule 2-4 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. 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 • 0.67 x 100% = 67% Example

11. Convert the following percents to decimals: Convert the following fractions to percents: Practice Answer 0.14 Answer 75% 14% 6/8 300% Answer 3.00 4/5 Answer 80% If you are still not sure, practice the rest of the exercises “Working with Percents.”

12. Percent Strength of Mixtures • Percents commonly used to indicate concentration of ingredients in mixtures • Solutions • Lotions • Creams • Ointments

13. Percent Strength of Mixtures (cont.) • Two categories of mixtures: • Fluid • Mixtures that flow • Solute • Solventor diluent • Solution • Solid or semisolid • Creams and ointments

14. Percent Strength of Mixtures(cont.) Rule 2-5 • For fluid mixtures prepared with a drymedication, the percent strength represents the number of grams of the medication contained in 100 mLs of the mixture. • For fluid mixtures prepared with a liquid medication, the percent strength represents the number of milliliters of the medication contained in 100 mLs of the mixture.

15. Percent Strength of Mixtures (cont.) 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.

16. Percent Strength of Mixtures (cont.) Rule 2-6 • For solid or semisolid mixtures prepared with a drymedication, the percent strength represents the number of grams of the medication contained in 100 grams of the mixture. • 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.

17. Percent Strength of Mixtures (cont.) Determine the amount of hydrocortisone per 100 grams of ointment. Example 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.

18. Practice How many grams of drug are in 100 mL of 10% solution? Answer 10 grams How many grams of dextrose will a patient receive from a 20 mL bag of dextrose 5%? Answer 5 grams will be in 100 mL, so the patient will receive 1 gram of dextrose in 20 mL

19. Ratios A : B • 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

20. Ratios (cont.) Rule 2-7 Reduce a ratio as you would a fraction. Find the largest whole number that divides evenly into both values A and B. Example Reduce 2:12 to its lowest terms. Both values 2 and 12 are divisible by 2. 2:12 is written 1:6

21. Ratios (cont.) Rule 2-8 To 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 = Example Convert the following ratio to a fraction: 4:5 =

22. Ratios (cont.) 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.

23. Ratios (cont.) A : B Example Convert the following into a ratio: = 7:12 = 47:12

24. Ratios (cont.) Rule 2-10To convert a ratio to a decimal: 1. Write the ratio as a fraction 2. Convert the fraction to a decimal (Chapter 1) Example Convert the 1:10 to a decimal. • Write the ratio as a fraction. 1:10 =

25. Ratios (cont.) A : B Example (cont.) 2. Convert the fraction to a decimal. = 1 divided by 10 = 0.1 Thus, 1:10 = = 0.1

26. Ratios (cont.) Rule 2-11To 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.

27. Ratios (cont.) A : B Example 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

28. Click for example Ratios (cont.) Rule 2-12To 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.

29. Click for example Ratios (cont.) Rule 2-13To 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.

30. Convert 2:3 to a percent. Convert 25% to a ratio. 1.2:3 = = 0.67 Ratios (cont.) Examples 1. 25% = 2. 2. 0.67 X 100% = 67%

31. Answer Answer Practice Convert the following ratio to fraction or mixed numbers: 3:4 5:3

32. Answer Answer Practice Convert the following decimals to ratios: 0.9 8

33. 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

34. Ratio Strengths (cont.) Write the ratio strength to describe 50 mL of solution containing 3 grams of drug. Example First number represents amount of drug = 3 grams Second number represents amount of solution = 50 mL The ratio is 3 g:50 mL

35. ERROR ALERT! • Do not forget the units of measurement. • Including units in the dosage strength will help avoid errors.

36. Practice Write a ratio to describe the following: 100 mL of solution contain 5 grams of drug Answer 5 g:100 mL Two tablets contain 20 mg drug Answer 20 mg:2 tablets

37. 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

38. 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

39. Click to go to Example Writing Proportions (cont.) Rule 2-14To write a ratio proportion as a fraction proportion: • Change the double colon to an equal sign. • Convert both ratios to fractions. • Rule 2-15To write a fraction proportion as a ratio proportion: • Convert each fraction to a ratio. • Change the equal sign to a double colon.

40. 2. 5:10::50:100 same as Write as a ratio proportion. Writing Proportions (cont.) Write 5:10::50:100 as a fraction proportion. Examples 1. 5:10::50:100 same as 5:10 = 50:100 • Convert each fraction to a ratio so that 5:6 = 10:12 2. 5:6 = 10:12 same as 5:6::10:12

41. Answer 4:5 = 8:10 or Answer 50:25 = 10:5 or Practice Write the following ratio proportions as fraction proportions: 4:5::8:10 50:25::10:5

42. 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.

43. Means and Extremes (cont.) A ratio proportion in the form A:B::C:D. Extremes A : B :: C : D Means

44. Means and Extremes (cont.) Rule 2-16To 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.

45. Means and Extremes (cont.) 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.

46. Means and Extremes (cont.) Rule 2-17To 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.

47. 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. Means and Extremes (cont.) Example ? = 10

48. Means and Extremes (cont.) Example (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.

49. 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.

50. Canceling Units in Proportions (cont.) Rule 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.