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1. Math and Dosage Calculations for Health CareThird EditionBooth & Whaley Chapter 1: Fractions and Decimals McGraw-Hill

2. Learning Outcomes 1.1 Compare the values of fractions in various formats. 1.2 Accurately add, subtract, multiply, and divide fractions. 1.3 Convert fractions to mixed numbers and decimals. McGraw-Hill

3. Learning Outcomes (cont.) 1.4 Recognize the format of decimals and measure their relative value. 1.5 Accurately add, subtract, multiply, and divide decimals. 1.6 Round decimals to the nearest tenth, hundredth, or thousandth. McGraw-Hill

4. Introduction • Basic math skills are building blocks for accurate dosage calculations. • Need confidence in math skills. • A minor mistake can mean major errors in the patient’s medication. McGraw-Hill

5. Fractions and Mixed Numbers • Measure a portion or part of a whole amount • Written two ways: • Common fractions • Decimals McGraw-Hill

6. Common Fractions • Represent equal parts of a whole • Consist of two numbers and a fraction bar • Written in the form: Numerator (top part of the fraction) = part of whole Denominator (bottom part of the fraction) represents the whole one part of the whole is the whole McGraw-Hill

7. Common Fractions (cont.) • Scored (marked) tablet for 2 parts • You administer 1 part of that tablet each day • You would show this as 1 part of 2 wholes or ½ • Read it as “one half” McGraw-Hill

8. Fraction Rule Rule 1-1 When the denominator is 1, the fraction equals the number in the numerator. Examples Check these equations by treating each fraction as a division problem. McGraw-Hill

9. 2 (two and two-thirds) Mixed Numbers • Combine a whole number with a fraction. Example • Fractions with a value greater than 1 are written as mixed numbers. McGraw-Hill

10. Mixed Numbers(cont.) Rule 1-2 • If the numerator of the fraction is less than the denominator, the fraction has a value of < 1. • If the numerator of the fraction is equal to the denominator, the fraction has a value =1. • If the numerator of the fraction is greater than the denominator, the fraction has a value > 1. McGraw-Hill

11. Mixed Numbers (cont.) Rule 1-3To convert a fraction to a mixed number: • Divide the numerator by the denominator. The result will be a whole number plus a remainder. • Write the remainder as the number over the original denominator. • Combine the whole number and the fraction remainder. This mixed number equals the original fraction. Applied only if the numerator is greater than the denominator McGraw-Hill

12. Mixed Numbers(cont.) Convert to a mixed number: • Divide the numerator by the denominator • = 2 R3 (R3 means a remainder of 3) The result is the whole number 2 with a remainder of 3 • Write the remainder over the whole = ¾ • Combine the whole number and the fraction = 2¾ Example McGraw-Hill

13. Mixed Numbers(cont.) Rule 1-4To convert a mixed number ( ) to a fraction: • Multiply the whole number (5) by the denominator (3) of the fraction ( ) 5x3 = 15 • Add the product from Step 1 to the numerator of the fraction 15+1 = 16 McGraw-Hill

14. Mixed Numbers(cont.) Rule 1-4(cont.) To convert a mixed number to a fraction: • Write the sum from Step 2 over the original denominator • The result is a fraction equal to original mixed number. Thus McGraw-Hill

15. Practice What is the numerator in ? Answer 17 What is the denominator in ? Answer 100 Twelve patients are in the hospital ward. Four have type A blood.What fraction do not have type A blood? Answer McGraw-Hill

16. EquivalentFractions • Two fractions written differently that have the same value Rule 1-5 To find an equivalent fraction, multiply or divide both the numerator and denominator by the same number. Example same as same as McGraw-Hill

17. Equivalent Fractions(cont.) Find equivalent fractions for Example Exception: The numerator and denominator cannot be multiplied or divided by zero. McGraw-Hill

18. Rule 1-6To find missing numerator in an equivalent fraction: If the denominator of the equivalent fraction is larger than the original denominator: Divide the larger denominator by the smaller one. Multiply the original numerator by the quotient from Step a. Click to go to Example Equivalent Fractions(cont.) McGraw-Hill

19. Rule 1-6To find missing numerator in an equivalent fraction: (cont.) If the denominator of the equivalent fraction is smaller than the original denominator: Divide the larger denominator by the smaller one. Divide the original numerator by the quotient from Step a. Click to go to Example Equivalent Fractions(cont.) McGraw-Hill

20. Example 1 Example 2 Equivalent Fractions(cont.) Answer ?= 7 Answer ?= 8 McGraw-Hill

21. Answers Practice • Find 2 equivalent fractions for . • Find the missing numerator . Answer128 McGraw-Hill

22. Simplifying Fractions to Lowest Terms Rule 1-7 • To reduce a fraction to its lowest terms, find the largest whole number that divides evenly into both the numerator and denominator. Note:When 1 is the only number that divides evenly into the numerator and denominator, the fraction is reduced to its lowest terms. Prime numbers– whole numbers other than 1 that can be evenly divided only by themselves and 1 McGraw-Hill

23. Error Alert! • Reducing a fraction does not automatically mean it is simplified to lowest terms. McGraw-Hill

24. Simplifying Fractions to Lowest Terms(cont.) Both 10 and 15 are divisible by 5 Example Reduce McGraw-Hill

26. Finding Common Denominators • Any number that is a common multiple of all the denominators in a group of fractions Rule 1-8To find the least common denominator (LCD): • List the multiples of each denominator. • Compare the list for common denominators. • The smallest number on all lists is the LCD. McGraw-Hill

27. Finding Common Denominators (cont.) Rule 1-9To convert fractions with large denominators to equivalent fractions with a common denominator: • List the denominators of all the fractions. • Multiply the denominators. (The product is a common denominator.) • Convert each fraction to an equivalent with the common denominator. McGraw-Hill

28. Practice Find the least common denominator for: Answer 21 Answer 48 McGraw-Hill

29. Comparing Fractions Rule 1-10To compare fractions: • Write all fractions as equivalent fractions with a common denominator. • Write the fraction in order by size of the numerator. • Restate the comparisons with the original fractions. McGraw-Hill

30. Order from smallest to largest: Example Comparing Fractions Write as equivalent fractions with a common denominator. LCD = 10. Order fractions by size of numerator: McGraw-Hill

31. Click to go to Example Adding Fractions Rule 1-11To add fractions: • Rewrite any mixed numbers as fractions. • Write equivalent fractions with common denominators. The LCD will be the denominator of your answer. • Add the numerators. The sum will be the numerator of your answer. McGraw-Hill

32. Click to go to Example Subtracting Fractions Similar to adding fractions. Rule 1-12To subtract fractions: • Rewrite any mixed numbers as fractions. • Write equivalent fractions with common denominators. The LCD will be the denominator of your answer. • Subtract the numerators. The difference will be the numerator of your answer. McGraw-Hill

33. Example Addition Add LCD is 4 Adding and Subtracting Fractions Example Subtraction LCD is 12 Subtract McGraw-Hill

34. MultiplyingFractions Rule 1-13To multiply fractions: • Convert any mixed numbers or whole numbers to fractions. • Multiply the numerators and then the denominators. • Reduce the product to its lowest terms. McGraw-Hill

35. Multiplying Fractions (cont.) To multiply multiply the numerators and multiply the denominators Example McGraw-Hill

36. Answer will be Multiplying Fractions (cont.) Rule 1-14 To cancel terms when multiplying fractions, divide both the numerator and denominator by the same number, if they can be divided evenly. Cancel terms to solve 1 1 3 2 McGraw-Hill

37. Error Alert! • Avoid canceling too many terms. • Each time you cancel a term, you must cancel it from one numerator AND one denominator. McGraw-Hill

38. Answer Answer Practice Find the following products: A bottle of liquid medication contains 24 doses. The hospital has 9 ¾ bottles of medication. How many doses are available? Answer 234 McGraw-Hill

39. Dividing Fractions Rule 1-15 • Convert any mixed or whole number to fractions. • Invert (flip) the divisor to find its reciprocal. • Multiply the dividend by the reciprocal of the divisor and reduce. McGraw-Hill

40. 4 Multiply by the reciprocal of 1 Dividing Fractions (cont.) Example You have bottle of liquid medication available and you must give of this to your patient. How many doses are available in this bottle? McGraw-Hill

41. Error Alert! • Write division problems carefully to avoid mistakes. 1. Convert whole numbers to fractions, especially if you use complex fractions. 2. Be sure to use the reciprocal of the divisor when converting the problem from division to multiplication. McGraw-Hill

42. divided by Answer divided by Answer Practice Find the following quotients: A case has a total of 84 ounces of medication. Each vial in the case holds 1¾ounce. How many vials are in the case? Answer 48 vials McGraw-Hill

43. Decimals • Another way to represent whole numbers and their fractional parts • Used daily by health care practitioners • Metric system • Decimal based • Used in dosage calculations, calibrations, and charting McGraw-Hill

44. Working with Decimals • Location of a digit relative to the decimal point determines its value • The decimal point separates the whole number from the decimal fraction McGraw-Hill

45. Working with Decimals (cont.) McGraw-Hill

46. Decimal Place Values The number 1,542.567 is read: (1) - one thousand (5) - five hundred (42) - forty two and (0.5) - five hundred (0.067) – sixty-seven thousandths One thousand five hundred forty two and five hundred sixty-seven thousandths McGraw-Hill

47. Writing Decimals Rule 1-16When writing a decimal number: • Write the whole number part to the left of the decimal point • Write the decimal fraction part to the right of the decimal point. • Decimal fractions are equivalent to fractions that have denominators of 10, 100, 1000, and so forth. • Use zero as a placeholder to the right of the decimal point. • Example: 0.201 McGraw-Hill

48. Writing Decimals (cont.) Rule 1-17 • Always write a zero to the left of the decimal point when the decimal number has no whole number part. • Makes the decimal point more noticeable • Helps to prevent errors caused by illegible handwriting McGraw-Hill

49. Comparing Decimals Rule 1-18To compare values of a group of decimal numbers: • The decimal with the greatest whole number is the greatest decimal number. • If the whole numbers of two decimals are equal, compare the digits in the tenths place. • If the tenths place are equal, compare the hundredths place digits. • Continue moving to the right comparing digits until one is greater than the other. McGraw-Hill

50. 0.3isor three tenths 0.03 is or three hundredths 0.003 is or three thousandths Comparing Decimals (cont.) • The more places a number is to the right of the decimal point the smaller the value. Examples McGraw-Hill