116 Chapter 2 Fractions and Mixed Numbers: Multiplication and Division Answers 6. 7. 13 8. 1 6 7 9 In Example 2, we show numerous steps to simplify fractions to lowest terms. However, the process is often made easier. For instance, we sometimes “divide out” common factors, and replace them with the new common factor of 1. 20 24 5 2 1 2 1 3 2 21 21 The largest number that divides evenly into two natural numbers is called the greatest common factor or (GCF). To find the greatest common factor of 20 and 24, factor each number into its prime factors. Then identify the common factors. The factors that are common to both lists are circled. The GCF is 20 2 2 5 By identifying the greatest common factor between the numerator and denominator of a fraction we can easily simplify the fraction. Notice that “dividing out” the common factor of 4 has the same effect as dividing the numerator and denominator by 4. This is often done mentally. 20 divided by 4 equals 5. 24 divided by 4 equals 6. 5 20 24 6 5 6 20 24 5 4 1 6 41 5 6 24 2 2 2 3 2 2 4 5 6 56 TIP: Simplifying a fraction is also called reducing a fraction to lowest terms. For example, the simplified (or reduced) form of 20 24 is . Simplifying Fractions to Lowest Terms Example 3 Simplify the fraction. Write the answer as a fraction or whole number. a. b. c. Solution: a. 1 Or alternatively: b. 1 Or alternatively: c. 1 Or alternatively: The greatest common factor in the numerator and denominator is 11. The greatest common factor in the numerator and denominator is 25. The greatest common factor in the numerator and denominator is 12. 12 divided by 12 equals 1. 60 divided by 12 equals 5. 3 10 1 12 60 5 1 5 12 60 1 12 5 12 1 1 5 3 75 divided by 25 equals 3. 25 divided by 25 equals 1. 75 25 1 3 1 75 25 3 25 1 25 1 3 1 3 110 divided by 11 equals 10. 99 divided by 11 equals 9. 110 99 9 10 9 110 99 10 11 9 11 1 10 9 12 60 75 25 110 99 TIP: Recall that any fraction of the form n. Therefore, 3. 31 n1 Skill Practice Simplify the fraction. Write the answer as a fraction or whole number. 6. 7. 8. 15 90 39 3 28 36 Avoiding Mistakes Do not forget to write the “1” in the numerator of the fraction . 15
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