# How to Simplify Complex Fractions

Instructor: Michael Quist

Michael has taught college-level mathematics and sociology; high school math, history, science, and speech/drama; and has a doctorate in education.

When fractions are inside other fractions, it can get really confusing. In this lesson, we'll learn how to tackle complex fractions, using the tools math gives us to simplify and resolve the toughest ones.

## What is a Complex Fraction?

Fractions can be annoying enough when they show up in what could have been a fairly easy math problem, but they can get a lot scarier when they're made up of fractions over fractions! A complex fraction is any fraction where the numerator (top part of the fraction) contains its own fraction, the denominator (bottom part of the fraction) has its own fraction, or both. For example, if you have 1/2 divided by 3/4 you would have a complex fraction.

Notice that in every complex fraction there is a separate fraction stuck in either the numerator, the denominator, or both. These extra fractions make them a little harder to work with, so we will use some special math rules to simplify and get rid of any extra fractions.

## Math Rules That Will Help

There are many rules in math, and they can definitely get annoying, but these rules can be a lot of help when we're trying to simplify complex fractions:

1. Denominator-Numerator Same Quantity Rule: We can always multiply the denominator and the numerator of a fraction by the same value without changing the value of the fraction. For example, you could multiply 1/2 by 2/2 and you'd get 2/4, which is the same value as 1/2.
2. Multiplicative Property of Equality: You can multiply any number by 1 without changing its value. Since any number (except 0) divided by itself is equal to 1, we can multiply a number by anything we want, so long as we put that same number on both the numerator and the denominator. Multiplying 5 by 4/4 produces 20/4, which is still equal to 5.
3. Additive Property of Equality: We can add the same value to both sides of an equation without changing its value. If a = 1, then a + 1 = 1 + 1.

## Simplifying Complex Fractions

All right, we've got the tools, now let's solve some complex fractions! We'll start with an easy one. Let's say we have 1/2 divided by 3/4, like we mentioned earlier. Here are the steps:

1. Find a common denominator. You can multiply the two (or more) denominators together to get a common denominator, or compare prime factors, but often you can tell just by looking at them.
2. Multiply your number times a fraction that has the common denominator for both numerator and denominator.
3. Simplify and reduce the fraction.

So, here we go!

1. The lowest common denominator for 1/2 and 3/4 and would be 4, because it is the smallest number into which both denominators (2 and 4) can be evenly divided. We'll use 4/4 for our next step.
2. Multiplying both the top and the bottom of our original fraction by 4, we find that we now have 4/2 divided by 12/4. That looks worse than our original problem, but it will simplify nicely in the next step.
3. 4/2 = 2 and 12/4 = 3. This means that our original problem will now reduce to 2/3. That's much simpler!

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