5.3 The Rational Numbers 243 Addition and Subtraction of Fractions Before we can add or subtract fractions, the fractions must have a common denominator. A common denominator is another name for a common multiple of the denominators. The lowest common denominator (LCD) is the least common multiple of the denominators. To add or subtract two fractions with a common denominator, we add or subtract their numerators and retain the common denominator. Did You Know? Mathematical Music The ancient Greeks believed that in nature, all harmony and everything of beauty could be explained with rational numbers. This belief was reinforced by the discovery that the sound of plucked strings, like those on a harp, could be quite pleasing if the strings plucked were in the ratio of 1 to 2 (an octave), 2 to 3 (a fifth), 3 to 4 (a fourth), and so on. Thus, the secret of harmony lies in the rational numbers such as , , and . 1 2 2 3 3 4 The theory of vibrating strings has applications today that go well beyond music. How materials vibrate, and hence the stress they can absorb, is a vital matter in the construction of rockets, buildings, and bridges. Addition and Subtraction of Fractions + = + ≠ − = − ≠ a c b c a b c c a c b c a b c c , 0; , 0 Example 13 Adding and Subtracting Fractions Evaluate. a) + 1 8 3 8 b) − 19 24 5 24 Solution a) + = + = = 1 8 3 8 1 3 8 4 8 1 2 b) − = − = = 19 24 5 24 19 5 24 14 24 7 12 7 Now try Exercise 67 Note that in Example 13, the denominators of the fractions being added or subtracted were the same; that is, they have a common denominator. When adding or subtracting two fractions with unlike denominators, first rewrite each fraction with a common denominator. Then add or subtract the fractions. Writing fractions with a common denominator is accomplished with the fundamental law of rational numbers. Fundamental Law of Rational Numbers If a b , , and c are integers, with ≠ b 0 and ≠ c 0, then = ⋅ = ⋅ ⋅ a b a b c c a c b c The terms a b and ⋅ ⋅ a c b c are called equivalent fractions . For example, since = ⋅ ⋅ = 7 12 7 5 12 5 35 60 , the fractions 7 12 and 35 60 are equivalent fractions. We will see the importance of equivalent fractions in the next two examples. Example 14 Adding and Subtracting Fractions with Unlike Denominators Evaluate. a) − 13 15 5 6 b) + 1 36 1 90 Solution a) Using prime factorization (Section 5.1), we determine that the LCM of 15 and 6 is 30. We will therefore express each fraction as an equivalent fraction with a Pavel L Photo and Video/Shutterstock
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