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Multiplying and Dividing Fractions Study Guide

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Updated on Oct 5, 2011

Introduction to Multiplying and Dividing Fractions

If you ask your mother for one fried egg for breakfast and she gives you two fried eggs and you eat both of them, who is better in arithmetic, you or your mother?

—From "Arithmetic," by CARL SANDBURG, poet (1878–1967)

This fraction lesson focuses on multiplication and division with fractions and mixed numbers.

Fortunately, multiplying and dividing fractions is actually easier than adding and subtracting them. When you multiply, you can simply multiply both the top numbers and the bottom numbers. To divide fractions, you invert and multiply. Of course, there are extra steps when you get to multiplying and dividing mixed numbers. Read on.

Multiplying Fractions

Multiplication by a proper fraction is the same as finding a part of something. For instance, suppose a personal-size pizza is cut into 4 slices. Each slice represents of the pizza. If you eat of a slice, then you've eaten of of a pizza, or × of the pizza (of means multiply), which is the same as of the whole pizza.

Multiplying and Dividing Fractions

Multiplying Fractions by Fractions

To multiply fractions:

  1. Multiply their top numbers together to get the top number of the answer.
  2. Multiply their bottom numbers together to get the bottom number of the answer.

Example: ×

  1. Multiply the top numbers:
  2. Multiply the bottom numbers:

Example:

  1. Multiply the top numbers:
  2. Multiply the bottom numbers:
  3. Reduce:

Cancellation Shortcut

Sometimes you can cancel before multiplying. Cancelling is a shortcut that speeds up multiplication because you're working with smaller numbers. Cancelling is similar to reducing: If there is a number that divides evenly into a top number and a bottom number, do that division before multiplying. By the way, if you forget to cancel, don't worry. You'll still get the right answer, but you'll have to reduce it.

Example:

1. Cancel the 6 and the 9 by dividing 3 into both of them:
6 ÷ 3 = 2 and 9 ÷ 3 = 3. Cross out the 6 and the 9.
2. Cancel the 5 and the 20 by dividing 5 into both of them:
5 ÷ 5 = 1 and 20 ÷ 5 = 4. Cross out the 5 and the 20.
3. Multiply across the new top numbers and the new bottom numbers:

Tip

When multiplying three or more fractions, the cancelling shortcut can still be used between fractions that are not directly next to each other:

Nine and four cannot reduce and neither can five and 42. But you can reduce nine and 33 by dividing by three, and 42 and 21 can be reduced by dividing by 21:

Multiplying Fractions by Whole Numbers

To multiply a fraction by a whole number:

  1. Rewrite the whole number as a fraction with a bottom number of 1.
  2. Multiply as usual.

Example: 5 ×

1. Rewrite 5 as a fraction: 5 =
2. Multiply the fractions: × =
3. Optional: Change the product to a mixed number. =

Have you noticed that multiplying any number by a proper fraction produces an answer that's smaller than that number? It's the opposite of the result you get from multiplying whole numbers. That's because multiplying by a proper fraction is the same as finding a part of something.

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