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M15 • Lesson 29 of 105

Multiplying & Dividing Fractions

Multiplication and division algorithms for fractions

Middle School Bridge • 6-8

Prerequisites: M13

Key Concepts

  • multiplication
  • division
  • reciprocals

Multiplying and Dividing Fractions

Suppose a recipe calls for 3⁄4 cup of flour, but you want to make only half the recipe. How much flour do you need? You need 1⁄2 of 3⁄4 -- and that means multiplying fractions. Unlike addition and subtraction, multiplying fractions does not require a common denominator, which makes the process surprisingly direct.

Multiplying Fractions

To multiply two fractions, multiply the numerators together and multiply the denominators together.

a⁄b × c⁄d = a × c⁄b × d

Example 1 -- Multiplying Two Fractions

Compute 2⁄3 × 4⁄5.

  1. Multiply numerators: 2 × 4 = 8
  2. Multiply denominators: 3 × 5 = 15
  3. Result: 8⁄15 (already simplified)

Cross-Canceling Saves Time

Before multiplying, you can simplify diagonally. If a numerator and the opposite denominator share a common factor, divide both by that factor first. This keeps numbers small and avoids simplifying at the end.

Example 2 -- Using Cross-Canceling

Compute 3⁄8 × 4⁄9.

  1. Notice that 3 and 9 share a factor of 3: divide 3 by 3 to get 1, and 9 by 3 to get 3.
  2. Notice that 4 and 8 share a factor of 4: divide 4 by 4 to get 1, and 8 by 4 to get 2.
  3. Now multiply: 1⁄2 × 1⁄3 = 1⁄6.

Dividing Fractions: Multiply by the Reciprocal

The reciprocal of a fraction a⁄b is b⁄a -- you flip the numerator and denominator. To divide by a fraction, multiply by its reciprocal.

a⁄b ÷ c⁄d = a⁄b × d⁄c

Why does this work? Division asks "how many groups of this size fit into that amount?" Multiplying by the reciprocal answers exactly that question.

Example 3 -- Dividing Fractions

Compute 3⁄4 ÷ 2⁄5.

  1. Find the reciprocal of 2⁄5: it is 5⁄2.
  2. Multiply: 3⁄4 × 5⁄2 = 15⁄8.
  3. Convert to a mixed number if desired: 1 7⁄8.

Common Mistake

When dividing, only flip the second fraction (the divisor). A frequent error is flipping both fractions or flipping the first one. Remember the phrase: "Keep, Change, Flip" -- keep the first fraction, change division to multiplication, flip the second fraction.

Practice Problems

1. 5⁄6 × 3⁄10

Show Solution

Cross-cancel: 5 and 10 share factor 5 (giving 1 and 2); 3 and 6 share factor 3 (giving 1 and 2). Multiply: 1⁄2 × 1⁄2 = 1⁄4.

2. 7⁄8 × 2⁄3

Show Solution

Cross-cancel 2 and 8 (factor 2): gives 1 and 4. Multiply: 7⁄4 × 1⁄3 = 7⁄12.

3. 4⁄5 ÷ 2⁄3

Show Solution

Keep, Change, Flip: 4⁄5 × 3⁄2 = 12⁄10 = 6⁄5 = 1 1⁄5.

4. 1⁄2 ÷ 3⁄4

Show Solution

1⁄2 × 4⁄3 = 4⁄6 = 2⁄3.

5. A ribbon is 3⁄4 meter long. You cut it into pieces that are each 1⁄8 meter long. How many pieces do you get?

Show Solution

3⁄4 ÷ 1⁄8 = 3⁄4 × 8⁄1 = 24⁄4 = 6 pieces.

Lesson Summary

Multiply fractions straight across: numerator times numerator, denominator times denominator. Use cross-canceling to simplify before multiplying. To divide fractions, Keep, Change, Flip -- keep the first fraction, change the operation to multiplication, and flip the second fraction. Always simplify your final answer.

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