# 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 mysteryâ€”let'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 simplifyâ€”just 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 quotientâ€”the 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 yetâ€”you can still simplify the fraction further, since 21 and 9 are both divisible by 3.

^{7}â„_{3x}

##### You've Got Problems

Problem 4: Simplify the complex fraction

Excerpted from The Complete Idiot's Guide to Algebra Â© 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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