Section 1.2 Fractions Review 21 6d. 5 1 7 - 4 3 7 = 36 7 - 31 7 Write each mixed number as an improper fraction. = 36 - 31 7 = 5 7 Subtract the numerators and place over the common denominator. Simplify the numerator. Student Check 6 Add or subtract the fractions and simplify the result. a. 4 5 + 2 5 b. 3 8 + 1 8 c. 4 5 - 1 5 d. 9 2 9 - 7 4 9 Adding and Subtracting Fractions with Different Denominators If the fractions we need to add do not have the same denominator, we must first rewrite each fraction as an equivalent fraction with the same denominator, or a common denominator. Equivalent fractions are fractions that represent the same quantity. For example, 3 4 and 6 8 are equivalent fractions as shown. We can obtain equivalent fractions by multiplying a given fraction by 1. Recall multiplying by 1 does not change the value of the fraction. When we multiply a fraction by 1, we are applying the Fundamental Property of Fractions. This property not only enables us to simplify fractions, but also enables us to multiply the numerator and denominator by the same nonzero number—that is, by a form of 1. For example, to rewrite 3 4 as a fraction with a denominator of 8, we can multiply the numerator and denominator by 2 as shown. 3 4 = 3 4 · 1 = 3 4 · 2 2 = 3 · 2 4 · 2 = 6 8 If we want to add 4 9 and 3 2 , for example, we must first convert these fractions to equivalent fractions with the same, or common denominator. If we find the least common denominator, that will make the process somewhat easier. The least common denominator (LCD) is the smallest number that all denominators divide into evenly. A common denominator is a number that all the denominators divide evenly but may not be the smallest such number. So, if the denominators are 9 and 2, the LCD is the smallest number that both 9 and 2 divide into—that is, the number 18. If the LCD is not easily identified, we can use the prime factorizations of the denominators to determine it. For example, the least common denominator of 6 and 20 is 60. We know this by their prime factorizations as shown. 6 = 2 · 3 20 = 2 · 2 · 5 LCD = 2 · 2 · 3 · 5 = 60 Notice that 6 has a prime factor of 2 and 20 has two prime factors of 2. Since 2 appears twice in the prime factorization of 20, we use two factors of 2 in the LCD. Since 3 and 5 appear only once in the prime factorizations of 6 and 20, respectively, we include one factor of each of them in the product of the LCD. Objective 7 ▶ Add or subtract fractions with unlike denominators.
hendricks_beginning_algebra_1e_ch1_3
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