2.5 Multiplying Fractions - McGraw Hill Higher Education

1 2001 McGraw -Hill Companies169 Multiplying two two mixed before Multiplying products by roundingMultiplication is the easiest of the four operations with Fractions . We can illustrate multi-plication by picturing Fractions as parts of a whole or unit. Using this idea, we show thefractions and in Figure now that we wish to findof . We can combine the diagrams as shown inFigure 2. The part of the whole representing the productis the purple region inFigure 2. The unit has been divided into 15 parts and 8 of those parts are used, somust beThe following rule is suggested by the Figure 4523 4545235432 Figure 12345 NOTEA fraction followed bythe word of means that wewant to multiply by 1 will require using steps 1 and 1 Multiply the numerators to find the numerator of the 2 Multiply the denominators to find the denominator of the 3 Simplify the resulting fraction if by Step:To Multiply Fractions170 CHAPTER2 Multiplying ANDDIVIDINGFRACTIONSStep 3 indicates that the product of Fractions should always be simplified to lowestterms.

2 Consider the following. 2001 McGraw -Hill CompaniesNOTEWe multiply Fractions inthis way notbecause it is easy,but because it works!CHECK YOURSELF 1 Multiply.(a)(b)57 3478 310 Example 1 Multiplying Two 79 5 78 9 357223 45 2 43 5 815 Example 2 Multiplying Two FractionsMultiply and write the result in lowest 29 3 24 9 636 16 Noting that is not in simplest form,we divide numerator and denominator by 6 to writethe product in lowest YOURSELF 2 Multiply and write the result in lowest 310To 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 3illustrates this 3 Multiplying a Whole Number and a FractionDo the indicated that (a) 154 3345 34 51 34 5 31 45 have written theresulting improper fraction as amixed (b) 212 3012 2612 5 612 1 512 6 512 61 2001 McGraw -Hill CompaniesCHECK YOURSELF 3 Multiply.(a)(b)4 57316 8 When mixed numbers are involved in multiplication, the problem requires an additionalstep.

3 First, change any mixed numbers to improper Fractions . Then apply our multiplicationrule for the product as amixed number, then reduce thefractional portion to 4 Multiplying a Mixed Number and a FractionHere 98 118 3 32 4112 34 32 34 Change the mixed number to an improper as product is usually written in mixed-number YOURSELF 312If two mixed numbers are involved, change both of the mixed numbers to improperfractions. Our next example ANDDIVIDINGFRACTIONSM ultiplying Two Mixed Careful!Students sometimes think ofThis is notthe correct multiplication pattern. You must first change the mixed numbers toimproper 212 as (3 2) 23 12 11 53 2 556 916 Change the mixed numbersto improper 212 113 52 2001 McGraw -Hill CompaniesExample 5 CAUTIONCHECK YOURSELF 312 When Multiplying Fractions , it is usually easier to simplify, that is, remove any commonfactors in the numerator and denominator, before that to simplifymeans to divideby the same common 6 Simplifying Before Multiplying Two FractionsSimplify and then multiply.

4 415 1 45 335 49 3 45 913To simplify, we divide the numeratorand denominatorby the common factor 3. Remember that means 3 3 1, and means 9 3 = 3 13 NOTEOnce again we areapplying the fundamentalprinciple to divide thenumerator and denominatorby we divide byany common factors before wemultiply, the resulting productis in simplest YOURSELF 6 Simplify and then work in Example 6 leads to the following general rule about simplifying fractionsin multiplication. 2001 McGraw -Hill CompaniesIn Multiplying two or more Fractions , we can divide any factor of the numeratorand any factor of the denominator by the same nonzero number to simplify and Properties:Simplifying Fractions Before MultiplyingWhen mixed numbers are involved, the process is similar. Consider Example 7 Simplifying Before Multiplying Two Mixed NumbersMultiply. 61 6 2 31 1 8 93 4 223 214 83 94 First, convert the mixed numbers to improper simplify, divide by the common factors of 3 and as YOURSELF 7 Simplify and then 225 The ideas of our previous examples will also allow us to find the product of more thantwo 8 Simplifying Before Multiplying Three NumbersSimplify and then multiply.

5 34 2 9 53 5 8 23 145 58 23 95 582311131114 Write any mixed or whole numbers as improper simplify, divide by the common factorsin the numerator and our earlierrule: We can divide anyfactorof the numerator and anyfactor of the denominator bythe same nonzero ANDDIVIDINGFRACTIONSWe encountered estimation by rounding in our earlier work with whole numbers. Esti-mation can also be used to check the reasonableness of an answer when we are workingwith Fractions or mixed numbers. 2001 McGraw -Hill CompaniesCHECK YOURSELF 8 Simplify and then 445 16 Example 9 Estimating the Product of Two Mixed NumbersEstimate the product ofRound each mixed number to the nearest whole estimate of the product is then3 6 18 Note:The actual product in this case is which certainly seems reasonable in view ofour ,556 6318 3318 556 CHECK YOURSELF 9 Estimate the 813 CHECK YOURSELF ANSWERS1. (a)(b) (a)(b) 225 103 125 10 123 5 81 878 521 7 58 21 52481658 312 58 72 3516 2316267112;57 310 5 37 10 1570 31457 34 5 37 4 152878 310 7 38 10 2180;132411 2001 McGraw -Hill CompaniesExercisesMultiply.

6 Be sure to write each answer in simplest 213225 334113 1151027 335513 7849 335323 9101828 35222125 3071415 10211012 16251225 111815 569 5637 14213 212213 21625 314313 3723 225313 91134 67813 26579 65521 1425310 5979 35311 7959 611613 49611 8635 5725 3734 71127 5934 Section Date 2001 McGraw -Hill the following 67 3 64 7 1828 9141415310 59 1590 167338393734 711 21441544910 227829 71112345 5671134 514517 2213315 423 What is 56 of 910?Find 23 of 37412 556 815313 45 11878 513 514415 1021 2001 McGraw -Hill CompaniesUsing Your Calculator to Multiply FractionsScientific CalculatorTo multiply Fractions on a scientific calculator, you enter the first fraction, using the key, then press the multiplication sign, next enter the second fraction, then press the b/c177 Example 1 Multiplying Two FractionsFind the productThe keystroke sequence is7 155 21 The result is 19. a b/c a b/c715 521 CHECK YOURSELF 1 Find the product2433 2239 Graphing CalculatorWhen using a graphing calculator, you must choose the fraction option fromthe menu before pressing.

7 For the fraction problem in Example 1, the keystroke sequence is 715521 Again, the result will be : Frac 715 521,EnterMATH1: FracCHECK YOURSELF 2001 McGraw -Hill CompaniesCalculator ExercisesFind the following products using your or 11227251281136 841352772 244518132 36631245 2772827 4564712 36633235 15162726 139635 20121884 36272842 12353655 335478 4211520 812 Name Section Date