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Calculating with Fractions When something is divided into parts, each part is considered a fraction of the whole. For example, a pie might be divided into eight slices, each one of which is a fraction, or 1⁄8 , of the whole pie. The pie is still whole but has been divided into eight slices. Each slice is a selection—an eighth—of the whole pie, or 1 (the number of slices in the selection) over 8 (the number of slices in the whole pie). In the fraction 3⁄8 , the selection is for 3 of the 8 slices (see Figure 1). Just as we talked about dividing a pie into parts, we can speak of dividing a tablet into parts, a common procedure for both pharmacist and patient. Figure 2 shows how cutting a tablet into smaller parts relates to fractions. fraction a portion of a whole that is represented as a ratio Figure 1 Fractions of a Whole Pie 8 8 slices 5 1 whole pie 5 __ of the whole pie 8 3 3 slices 5 __ of the whole pie 8 1 1 slice 5 __ of the whole pie 8 Figure 2 Fractions of a Tablet 1 tablet 5 1000 mg numerator the number in the upper part of a fraction denominator the number in the bottom part of a fraction 1 __ tablet 5 500 mg 1 __ tablet 5 250 mg 2 4 Fractions are either common (3⁄2, 2⁄3, etc.) or decimal (0.5, 0.66, etc.). A common fraction is composed of a numerator (the number on the top) and a denominator (the number on the bottom). The numerator represents the portion (1 piece in the case of the pie), and the denominator represents the whole (8 pieces of pie). numerator 1 __ denominator 8 A fraction is simply a convenient way of representing an operation, the division of the numerator by the denominator. Thus the fraction 3⁄6 equals 6 divided by 3, which equals 2. The fraction 7⁄8 is 7 divided by 8, which equals 0.875. The number obtained upon dividing the numerator by the denominator is the value of the fraction. Therefore, a fraction with the same numerator and denominator has a value equivalent to 1. 8 5 __ 3 10 ___ 15 __ 5 __ 5 5 ___ 5 5 1 8 5 3 10 15 Fractions with the same value are said to be equivalent fractions. The following are equivalent fractions: 1 __ 5 1 4 2 5 0.5 2 2 __ 5 2 4 4 5 0.5 4 3 ___ 5 3 4 16 5 0.1875 16 © Paradigm Publishing, Inc. 4 __ 5 4 4 8 5 0.5 8 12 ___ 5 12 4 64 5 0.1875 64 1 proper fraction a fraction with a value of less than 1 (the value of the numerator is smaller than the value of the denominator) improper fraction a fraction with a value greater than 1 (the value of the numerator is larger than the value of the denominator) mixed number a whole number and a fraction A fraction with a value of less than 1 (the numerator smaller than the denominator) is called a proper fraction. Safety Note In pharmaceutical work, it is especially important not to misread a compound fraction as a simple one. For example, do not confuse the following fractions. They are not equal. Be careful when reading compound fractions. 1 __ 4 2 __ 3 7 __ 8 9 ___ 10 A fraction with a value greater than 1 (the numerator greater than the denominator) is called an improper fraction. 6 __ 5 7 __ 5 11 ___ 6 15 ___ 8 A mixed number, also called a compound fraction, is a whole number and a fraction. 1 5 __ 2 7 13 __ 8 23 99 ___ 24 99 111 ____ 100 3 33 3 __ ___ 8 8 Adding and Subtracting Fractions To add or subtract fractions, first convert any compound fractions to improper fractions containing no whole numbers. To do this, multiply the whole number part of the compound fraction by the denominator and add the result to the numerator. (whole number 3 denominator) 1 numerator compound fraction 5 _______________________________________ denominator common denominator a number into which each of the unlike denominators of two or more fractions can be divided evenly 3 (3 3 8) 1 3 ___ 27 3 __ 5 __________ 5 8 8 8 1 (4 3 3) 1 1 ___ 13 4 __ 5 __________ 5 3 3 3 The next step in adding or subtracting fractions is to check if the denominators are equal. If all of the fractions have the same denominator, addition or subtraction can proceed. If not, it is necessary to convert each fraction to an equivalent fraction such that all the fractions have the same denominator, or a common denominator. The least common denominator of a group of fractions is the smallest number that is evenly divisible by all of the denominators. To find the least common denominator, follow the steps shown in the following example. © Paradigm Publishing, Inc. Calculating with Fractions 2 Example Find the least common denominator of the following fractions: 9 ___ 28 1 __ 6 Step 1. Find the prime factors (numbers divisible only by 1 and themselves) of each denominator. Make a list of all the different prime factors that you find. Include in the list each different factor as many times as the factor occurs for any one of the denominators of the given fractions. The prime factors of 28 are 2, 2, and 7 (because 2 3 2 3 7 5 28). The prime factors of 6 are 2 and 3 (because 2 3 3 5 6). The number 2 occurs twice for one of the denominators, so it must occur twice in the list. The list will also include the unique factors 3 and 7; so the final list is 2, 2, 3, and 7. Step 2. Multiply all the prime factors on your list. The result of this multiplication is the least common denominator. 2 3 2 3 3 3 7 5 84 Step 3. To convert a fraction to an equivalent fraction with the common denominator, first divide the least common denominator by the denominator of the fraction, then multiply both the numerator and denominator by the result (the quotient). The least common denominator of 9⁄28 and 1⁄6 is 84. In the first fraction, 84 divided by 28 is 3, so multiply both the numerator and the denominator by 3. 9 933 27 ___ 5 ______ 5 ___ 28 28 3 3 84 In the second fraction, 84 divided by 6 is 14, so multiply both the numerator and the denominator by 14. 1 1 3 14 ___ 14 __ 5 ______ 5 6 6 3 14 84 The following are two equivalent fractions: 9 27 ___ 5 ___ 28 84 1 14 __ 5 ___ 6 84 Step 4. Once the fractions are converted to contain equal denominators, adding or subtracting them is straightforward. Simply add or subtract the numerators. 9 1 27 ___ 14 41 ___ 1 __ 5 ___ 1 5 ___ 28 6 84 84 84 9 1 27 ___ 14 13 ___ 2 __ 5 ___ 2 5 ___ 28 6 84 84 84 © Paradigm Publishing, Inc. Calculating with Fractions 3 Multiplying and Dividing Fractions To multiply fractions, multiply numerators by numerators and denominators by denominators. Table 1 shows some guidelines for multiplying fractions. 1 1 _____ 1 3 1 ___ 1 __ 3 __ 5 5 8 2 8 3 2 16 3 12 ______ 3 3 12 ___ 36 9 __ 3 ___ 5 5 5 ___ 4 17 4 3 17 68 17 1 1 __ 2 1 3 1 3 2 ___ 2 1 __ 3 __ 3 5 _________ 5 5 ___ 8 2 3 8 3 2 3 3 48 24 To divide by a fraction, invert the fraction and multiply. The inverted fraction is known as the reciprocal of the original fraction. Note that if the numerator of the original fraction is 1, the reciprocal will be a whole number. 3 1 __ 3 3 _____ 3 3 3 __ 9 1 __ 4 __ 5 3 __ 5 5 5 2 __ 4 3 4 1 431 4 4 10 1 ___ 10 __ 4 ___ 40 __ ___ 1 5 10 4 5 3 5 5 40 ⁄4 4 1 1 1 Table 1 Guidelines for Multiplying Fractions 1. Multiplying the numerator by a number increases the value of a fraction. 1 2 1 3 2 __ 2 1 __ 3 __ 5 ______ 5 5 __ 4 1 431 4 2 2. Multiplying the denominator by a number decreases the value of a fraction. 1 1 1 3 1 __ 1 __ 3 __ 5 ______ 5 4 2 432 8 3. The value of a fraction is not altered by multiplying or dividing both numerator and denominator by the same number. 1 4 1 3 4 ___ 4 1 __ 3 __ 5 ______ 5 5 __ 4 4 4 3 4 16 4 4. Dividing the denominator by a number is the same as multiplying the numerator by that number. 3 3 ___ 5 __ 20 ⁄5 4 3 3 5 ___ 15 3 ______ 5 5 __ 20 20 4 5. Dividing the numerator by a number is the same as multiplying the denominator by that number. 6 ⁄3 2 __ 1 __ 5 __ 5 4 4 2 © Paradigm Publishing, Inc. 6 6 1 ______ 5 ___ 5 __ 433 12 2 Calculating with Fractions 4