# Algebra: Encountering Complex Fractions

## Encountering Complex Fractions

If you hate fractions, then you'll be no fan of *complex fractions*. Just the name alone sounds scary, right? Fractions are hard enough, but *complex* fractions? Great! I imagine that brain surgery is pretty hard to do, but *complex* brain surgery sounds even worse. Actually, your gut fear is probably unjustified because the term "complex fraction" is false advertising for two reasons:

- The word "complex" might suggest that the fractions contain complex numbers, but they don't.
- You already know how to work with complex fractions; you just don't know that I know that you know how to. (But you know now.)

Enough mysterylet's cut to the chase. A *complex fraction* is a fraction that contains a fraction in its numerator or denominator (or both). Complex fractions are considered bad form, so your final answers shouldn't contain them; however, since a fraction translates into a division problem, they are extremely easy to simplifyjust divide the numerator by the denominator.

##### Talk the Talk

A **complex fraction** is a fraction that contains other fractions in its numerator, denominator, or both. It's sort of a double-decker fraction. (Complex fractions are also called *compound fractions*.)

**Example 4**: Simplify the complex fraction.

**Solution**: Rewrite the complex fraction as a quotientthe top fraction divided by the bottom fraction.

Suddenly, this has become a division problem very similar to Example 3(b). A little multiplication by the reciprocal should do the trick. Don't forget to factor.

Just simplify the fraction and you're done!

You're not quite done yetyou can still simplify the fraction further, since 21 and 9 are both divisible by 3.

^{7}_{3x}

##### You've Got Problems

Problem 4: Simplify the complex fraction

Excerpted from The Complete Idiot's Guide to Algebra © 2004 by W. Michael Kelley. All rights reserved including the right of reproduction in whole or in part in any form. Used by arrangement with **Alpha Books**, a member of Penguin Group (USA) Inc.

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