Fractions Review: Adding and Subtracting Fractions


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Fractions Review (page 4 of 5)

Sections: Reducing fractions, Mixed numbers and improper fractions, Multiplying and dividing fractions, Adding and subtracting fractions, Adding polynomial fractions

To add fractions, you have to have "common" (shared) denominators. As the proverb says, you can only add apples to apples, not apples to oranges. In the context of fractions, you can't combine ¹/₄ and ²/₅; you first have to convert to ⁵/₂₀ and ⁸/₂₀. Believe it or not, many civilizations (such as the ancient Egyptians) never figured out the common-denominator concept. So don't feel bad if you have some trouble with the computations!

The basic idea with common denominators is to multiply fractions by useful forms of 1. What does this mean? Take a look:

Simplify

Before I can add these fractions, I have to find their common denominator. The lowest (smallest) common denominator is just the Least Common Multiple (LCM) of the two denominators, 4 and 5. The prime factorizations and LCM are:

LCM: 2 * 2 * 5 = 20

In other words, I have to convert the fourths and fifths into twentieths. I'll do this by multiplying by a useful form of 1. In the case of the ¹/₄, the 4 needs to become a 20, so I'll multiply the 4 by 5. To keep the fraction equal to the same value, I'll multiply the top by 5, too. In other words, I'll multiply by ⁵/₅, which is just 1: Copyright © Elizabeth Stapel 2006-2008 All Rights Reserved

In the case of the ²/₅, the 5 needs to become a 20, so I'll multiply the 5 by 4. To keep the fraction equal to the same value, I'll multiply the top by 4, too. In other words, I'll multiply by ⁴/₄, which is just 1:

Only now can I actually add the fractions:

Note that your calculator may be able to do all of this for you; check your manual. But make sure you at least understand the basic idea, because you'll need this process later in algebra.