. Multiplying Fractions. OBJECTIVES 1. Multiply two fractions. Multiply two mixed numbers. Simplify before multiplying fractions 4. Estimate products by rounding Multiplication is the easiest of the four operations with fractions. We can illustrate multiplication by picturing fractions as parts of a whole or unit. Using this idea, we show the 4 fractions and in Figure 1. 4 Figure 1 NOTE A fraction followed by the word of means that we want to multiply by that fraction. 4 Suppose now that we wish to find of. We can combine the diagrams as shown in Figure. The part of the whole representing the product is the purple region in 4 Figure. The unit has been divided into 1 parts and 8 of those parts are used, so must be 8 1. 4 4 Figure 8 1 The following rule is suggested by the diagrams. 001 McGraw-Hill Companies Step by Step: To Multiply Fractions Step 1 Multiply the numerators to find the numerator of the product. Step Multiply the denominators to find the denominator of the product. Step Simplify the resulting fraction if possible. Example 1 will require using steps 1 and. 169
10 CHAPTER MULTIPLYING AND DIVIDING FRACTIONS Example 1 Multiplying Two Fractions NOTE We multiply fractions in this way not because it is easy, but because it works! 4 4 8 1 8 9 8 9 CHECK YOURSELF 1 (a) (b) 8 10 4 Step indicates that the product of fractions should always be simplified to lowest terms. Consider the following. Example Multiplying Two Fractions Multiply and write the result in lowest terms. 4 9 4 9 6 6 1 6 CHECK YOURSELF Multiply and write the result in lowest terms. 6 Noting that is not in simplest form, 6 we divide numerator and denominator by 6 to write the product in lowest terms. 10 To find the product of a fraction and a whole number, write the whole number as a fraction (the whole number divided by 1) and apply the multiplication rule as before. Example illustrates this approach. NOTE We have written the resulting improper fraction as a mixed number. Example Multiplying a Whole Number and a Fraction Do the indicated multiplication. (a) Remember that 1. 4 1 4 1 4 1 4 4 001 McGraw-Hill Companies
MULTIPLYING FRACTIONS SECTION. 11 NOTE Write the product as a mixed number, then reduce the fractional portion to simplest form. (b) 1 6 1 6 1 6 1 1 0 1 6 1 1 CHECK YOURSELF (a) (b) 4 16 8 When mixed numbers are involved in multiplication, the problem requires an additional step. First, change any mixed numbers to improper fractions. Then apply our multiplication rule for fractions. Example 4 Multiplying a Mixed Number and a Fraction 1 1 4 4 Change the mixed number to an improper fraction. Here 1 1. 4 9 8 11 8 Multiply as before. The product is usually written in mixed-number form. 001 McGraw-Hill Companies CHECK YOURSELF 4 8 1 If two mixed numbers are involved, change both of the mixed numbers to improper fractions. Our next example illustrates.
1 CHAPTER MULTIPLYING AND DIVIDING FRACTIONS Example Multiplying Two Mixed Numbers 1 11 Change the mixed numbers to improper fractions. CAUTION 11 6 91 6 Be Careful! Students sometimes think of 1 as ( ) 1 This is not the correct multiplication pattern. You must first change the mixed numbers to improper fractions. CHECK YOURSELF 1 1 When multiplying fractions, it is usually easier to simplify, that is, remove any common factors in the numerator and denominator, before multiplying. Remember that to simplify means to divide by the same common factor. Example 6 Simplifying Before Multiplying Two Fractions NOTE Once again we are applying the fundamental principle to divide the numerator and denominator by. NOTE Because we divide by any common factors before we multiply, the resulting product is in simplest form. Simplify and then multiply. 4 1 9 4 9 1 4 4 1 CHECK YOURSELF 6 Simplify and then multiply. 8 1 To simplify, we divide the numerator and 1 denominator by the common factor. Remember that means 1, and 9 means 9 =. 001 McGraw-Hill Companies
MULTIPLYING FRACTIONS SECTION. 1 Our work in Example 6 leads to the following general rule about simplifying fractions in multiplication. Rules and Properties: Simplifying Fractions Before Multiplying In multiplying two or more fractions, we can divide any factor of the numerator and any factor of the denominator by the same nonzero number to simplify the product. When mixed numbers are involved, the process is similar. Consider Example. Example Simplifying Before Multiplying Two Mixed Numbers 1 4 8 9 4 8 9 4 1 1 1 1 First, convert the mixed numbers to improper fractions. To simplify, divide by the common factors of and 4. Multiply as before. 6 1 6 CHECK YOURSELF Simplify and then multiply. 1 The ideas of our previous examples will also allow us to find the product of more than two fractions. Example 8 Simplifying Before Multiplying Three Numbers 001 McGraw-Hill Companies NOTE Remember our earlier rule: We can divide any factor of the numerator and any factor of the denominator by the same nonzero number. Simplify and then multiply. 14 8 9 8 1 1 9 8 1 1 4 4 Write any mixed or whole numbers as improper fractions. To simplify, divide by the common factors in the numerator and denominator.
14 CHAPTER MULTIPLYING AND DIVIDING FRACTIONS CHECK YOURSELF 8 Simplify and then multiply. 8 44 1 6 We encountered estimation by rounding in our earlier work with whole numbers. Estimation can also be used to check the reasonableness of an answer when we are working with fractions or mixed numbers. Example 9 Estimating the Product of Two Mixed Numbers Estimate the product of 1 8 6 Round each mixed number to the nearest whole number. 1 8 6 6 Our estimate of the product is then 6 18 Note: The actual product in this case is 18 11 which certainly seems reasonable in view of 48, our estimate. CHECK YOURSELF 9 Estimate the product. 8 81 CHECK YOURSELF ANSWERS 1. (a) (b) 4 4 1 8 10 8 10 1 80 ; 8.. (a) 1 1 (b) 6 10 10 1 0 14 ; 1 8 1 4.. 8 1 6. 8 1 8 1 8 16 16 6 4 1. 1 8. 9. 4 10 1 4 10 1 8 1 8 1 1 001 McGraw-Hill Companies
Name. Exercises Section Date Be sure to write each answer in simplest form. 1... 4 11 9 4.. 6. 6. 8. 9. 9 6 1 4 9 11 4 11 6 11 8 6 11 9 ANSWERS 1... 4.. 6.. 8. 10. 11. 1. 10 9 9 1 14 9. 10. 11. 1. 8 1. 14. 1. 1 6 9 6 4 6 1. 14. 1. 16. 16. 1 1. 18. 9 11 19. 0. 1 1. 1 4 1 6 1 1 1 1. 18. 19. 0. 1.... 9 4. 14 6 1 6. 4.. 6. 1 10. 6.. 1 16 11 18 14 1 10 1. 8. 9. 0. 001 McGraw-Hill Companies 1 18 8. 9. 0. 8 0 4 1.. 1. 9 8 9 10 10 4. 1 1. 6. 11 4 1 1... 4.. 6. 1
ANSWERS. 8. 9. 40. 41. 4.. 4 1 8. 8 1 10 1 9 0 14 9. 40. 41. of 4 1 6 8 1 4. Estimate the following products. 4. 1 44. 1 4. 4 1 1 4 11 8 What is 11 4 1 4 6 of 9 10? 4. 44. 4. 46. 4. 48. 46. 4 4. 8 48. 6 9 11 1 9 10 Answers 1. 1 8... 9. 4 11 1 44 44 9 11. 14 1. 1. 4 6 6 4 18 8 9 10 9 1 90 1 6 1 14 1. 4 1 19. 1 1.. 1.. 10 6 9 9. 1. 9 1. 1 1. 9. 9. 41. 10 4. 1 4. 60 4. 64 1 1 44 001 McGraw-Hill Companies 16
Using Your Calculator to Multiply Fractions Scientific Calculator To multiply fractions on a scientific calculator, you enter the first fraction, using the a b/c key, then press the multiplication sign, next enter the second fraction, then press the equals sign. Example 1 Multiplying Two Fractions Find the product 1 1 The keystroke sequence is a b/c 1 a b/c 1 The result is 1 9. CHECK YOURSELF 1 Find the product 4 9 Graphing Calculator When using a graphing calculator, you must choose the fraction option 1: Frac the MATH menu before pressing Enter. For the fraction problem in Example 1, the keystroke sequence is 1 1, from 1 1 1: Frac Enter Again, the result will be 1 9. 001 McGraw-Hill Companies CHECK YOURSELF ANSWER 1. 16 9 1
Name Section ANSWERS 1.. Date Calculator Exercises Find the following products using your calculator. 1 1.. 0 8 1 6. 4. 4 8 4 1 8 4 1. 4.. 18. 6. 84 6. 8. 6 1 9 6 0 1 1 16 6.. 8. 9. 10. 9. 10. 1 6 6 1 11. 1. 4 1. 14. 4 4 8 4 64 18 1 6 6 81 16 84 1 11. 1. 1. Answers 1 1 1 1.... 9. 11. 1. or 11 10 1 14. 001 McGraw-Hill Companies 18