Chapter 1: Fractions and Decimals Math and Dosage Calculations for Health Care Third Edition Booth & Whaley Chapter 1: Fractions and Decimals McGraw-Hill

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

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

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

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

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

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

Check these equations by treating each fraction as a division problem. 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

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

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

Applied only if the numerator is greater than the denominator Mixed Numbers (cont.) Rule 1-3 To 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

Mixed Numbers (cont.) Convert to a mixed number: Example 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

Mixed Numbers (cont.) Rule 1-4 To 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

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

Practice What is the numerator in ? What is the denominator 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

Equivalent Fractions same as same as 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

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

Equivalent Fractions (cont.) Rule 1-6 To 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 McGraw-Hill

Equivalent Fractions (cont.) Rule 1-6 To 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 McGraw-Hill

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

Practice Find 2 equivalent fractions for . Answers Find the missing numerator . Answers Answer 128 McGraw-Hill

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

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

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

Practice Answer Answer Reduce the following fractions: then, McGraw-Hill

Finding Common Denominators Any number that is a common multiple of all the denominators in a group of fractions Rule 1-8 To 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

Finding Common Denominators (cont.) Rule 1-9 To 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

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

Comparing Fractions Rule 1-10 To 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

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

Adding Fractions Rule 1-11 To 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. Click to go to Example McGraw-Hill

Subtracting Fractions Similar to adding fractions. Rule 1-12 To 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. Click to go to Example McGraw-Hill

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

Multiplying Fractions Rule 1-13 To 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

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

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 Answer will be McGraw-Hill

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

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

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

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? by the reciprocal of Multiply 1 4 McGraw-Hill

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

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

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

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

Working with Decimals (cont.) Table 1-3 Decimal Place Values The number 1,542.567 can be represented as follows: Whole Number Decimal Point Decimal Fraction Table 1-3 Decimal Place Values The number 1,542.567 can be represented as follows: Whole Number Decimal Point Decimal Fraction Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths 1, 5 4 2 6 7 Thousands Hundreds Tens Ones . Tenths Hundredths Thousandths 1, 5 4 2 6 7 McGraw-Hill

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

Writing Decimals Rule 1-16 When 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

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

Comparing Decimals Rule 1-18 To 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

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

Practice Write the following in decimal form: Answers = 0.2 = 0.17 = 0.023 McGraw-Hill

Rounding Decimals Rule 1-19 Decimals are usually rounded to the nearest tenth or hundredth. Rule 1-19 Underline the place value to which you want to round. Look at the digit to the right of this target. If 4 or less, do not change the digit If 5 or more, round up one unit Drop all digits to the right of the target place value. McGraw-Hill

Practice Answer 14.3 Answer 9.3 Answer 8.80 Answer 10.54 Round to the nearest tenth: 14.34 9.293 Round to the nearest hundredth: 8.799 10.542 Answer 14.3 Answer 9.3 Answer 8.80 Answer 10.54 McGraw-Hill

Converting Fractions into Decimals Rule 1-20 To convert a fraction to a decimal, divide the numerator by the denominator. Example McGraw-Hill

Converting Decimals into Fractions (cont.) Rule 1-21 Write the number to the left of the decimal point as the whole number. Write the number to the right of the decimal point as the numerator of the fraction. Use the place value of the digit farthest to the right of the decimal point as the denominator. Reduce the fraction part to its lowest term. McGraw-Hill

Practice 100.4 1.2 Convert decimals to fractions or mixed number: Answer or 1.2 100.4 Answer or McGraw-Hill

Adding and Subtracting Decimals Rule 1-22 Write the problem vertically. Align the decimal points. Add or subtract starting from the right. Include the decimal point in your answer. 2.47 + 0.39 2.86 McGraw-Hill

Adding and Subtracting Decimals (cont.) Examples Subtract 7.3 – 1.005 Answer 7.300 1.005 6.295 Add 13.561 + 0.099 Answer 13.561 + 0.099 13.660 McGraw-Hill

Practice Add or subtract the following pair of numbers: 48.669 + 0.081 Answer 48.75 16.250 – 1.625 Answer 14.625 McGraw-Hill

Multiplying Decimals Rule 1-23 First, multiply without considering the decimal points, as if the numbers were whole numbers. Count the total number of places to the right of the decimal point in both factors. To place the decimal point in the product, start at its right end and move the it to the left the same number of places. McGraw-Hill

Multiplying Decimals (cont.) Example Multiply 3.42 x 2.5 3.42 X 2.5 1710 684 8550 There are three decimal places so place the decimal point between 8 and 5. Answer 8.55 McGraw-Hill

Practice A patient is given 7.5 milliliters of liquid medication 5 times a day. How many milliliters does she receive per day? Answer 7.5 x 5 7.5 X 5 37.5 McGraw-Hill

Dividing Decimals Rule 1-24 Write the problem as a fraction. Move the decimal point to the right the same number of places in both the numerator and denominator until the denominator is a whole number. Insert zeros as needed. Complete the division as you would with whole numbers. Align the decimal point of the quotient with the decimal point of the numerator, if needed. McGraw-Hill

Dividing Decimals (cont.) Example Divide McGraw-Hill

Practice Answer 32 divided by 0.4 Take 0.4 into 32 A bottle contains 32 ounces of medication. If the average dose is 0.4 ounces, how many doses does the bottle contain? Answer 32 divided by 0.4 Take 0.4 into 32 Add a zero behind the 32 for each decimal place 320 divided by 4 = 80 or 80 doses McGraw-Hill

Apply Your Knowledge Convert the following mixed numbers to fractions: Answer Answer McGraw-Hill

Apply Your Knowledge Round to the nearest tenth: 7.091 Answer 7.1 McGraw-Hill

Apply Your Knowledge Add the following: 7.23 + 12.38 Answer 19.61 Multiply the following: 12.01 x 1.005 Answer 12.07005 McGraw-Hill

He who is ashamed of asking is ashamed of learning. End of Chapter 1 He who is ashamed of asking is ashamed of learning. ~ Danish Proverb McGraw-Hill