Section 6: Complex Fractions

Elementary and Intermediate Algebra: Graphs & Models (3rd Edition)

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Section 7.6 Complex Fractions 695 Version: Fall 2007 7.6 Complex Fractions In this section we learn how to simplify what are called complex fractions , an example of which follows. 1 2 + 1 3 1 4 + 2 3 (1) Note that both the numerator and denominator are fraction problems in their own right, lending credence to why we refer to such a structure as a “complex fraction.” There are two very different techniques we can use to simplify the complex fraction ( 1 ). The first technique is a “natural” choice. Simplifying Complex Fractions — First Technique. To simplify a complex fraction, proceed as follows: 1. Simplify the numerator. 2. Simplify the denominator. 3. Simplify the division problem that remains. Let’s follow this outline to simplify the complex fraction ( 1 ). First, add the fractions in the numerator as follows. 1 2 + 1 3 = 3 6 + 2 6 = 5 6 (2) Secondly, add the fractions in the denominator as follows. 1 4 + 2 3 = 3 12 + 8 12 = 11 12 (3) Substitute the results from ( 2 ) and ( 3 ) into the numerator and denominator of ( 1 ), respectively. 1 2 + 1 3 1 4 + 2 3 = 5 6 11 12 (4) The right-hand side of ( 4 ) is equivalent to 5 6 ÷ 11 12 . This is a division problem, so invert and multiply, factor, then cancel common factors. Copyrighted material. See: 1
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696 Chapter 7 Rational Functions Version: Fall 2007 1 2 + 1 3 1 4 + 2 3 = 5 6 · 12 11 = 5 2 · 3 · 2 · 2 · 3 11 = 5 2 · 3 · 2 · 2 · 3 11 = 10 11 Here is an arrangement of the work, from start to finish, presented without comment. This is a good template to emulate when doing your homework. 1 2 + 1 3 1 4 + 2 3 = 3 6 + 2 6 3 12 + 8 12 = 5 6 11 12 = 5 6 · 12 11 = 5 2 · 3 · 2 · 2 · 3 11 = 5 2 · 3 · 2 · 2 · 3 11 = 10 11 Now, let’s look at a second approach to the problem. We saw that simplifying the numerator in ( 2 ) required a common denominator of 6. Simplifying the denominator in ( 3 ) required a common denominator of 12. So, let’s choose another common denomina- tor, this one a common denominator for both numerator and denominator, namely, 12. Now, multiply top and bottom (numerator and denominator) of the complex fraction ( 1 ) by 12, as follows. 1 2 + 1 3 1 4 + 2 3 = 1 2 + 1 3 12 1 4 + 2 3 12 (5) Distribute the 12 in both numerator and denominator and simplify.
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Section 7.6 Complex Fractions 697 Version: Fall 2007 1 2 + 1 3 12 1 4 + 2 3 12 = 1 2 12 + 1 3 12 1 4 12 + 2 3 12 = 6 + 4 3 + 8 = 10 11 Let’s summarize this second technique. Simplifying Complex Fractions — Second Technique. To simplify a com- plex fraction, proceed as follows: 1. Find a common denominator for both numerator and denominator.
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