Dividing fractions

Dividing by a fraction asks “how many of these fit?”. Dividing a fraction by a whole number shares it into equal parts. To divide by a fraction, multiply by its reciprocal: keep, change, flip.

Key facts

  • Dividing a fraction by a whole number shares it: 1/2 ÷ 3 = 1/6, the same as 1/2 × 1/3.
  • The reciprocal of a/b is b/a; the reciprocal of 5 is 1/5.
  • To divide by a fraction, multiply by its reciprocal: a/b ÷ c/d = a/b × d/c (keep, change, flip).
  • Change mixed numbers into improper fractions before multiplying or dividing.
  • Multiplying by a fraction less than 1 makes a number smaller; dividing by it makes it bigger.

Worked examples

Example 1

Work out 1/2 ÷ 3.

  1. Split a half into 3 equal parts.
  2. Each part is a sixth. 1/2 ÷ 3 = 1/6

Answer: 1/6

Example 2

Work out 2/3 ÷ 4.

  1. Split each third into 4 equal parts: twelfths. 2/3 = 8/12
  2. Share the 8 twelfths between 4. 8/12 ÷ 4 = 2/12 = 1/6

Answer: 1/6

Example 3

Half a pizza is shared equally between 4 friends. What fraction of the whole pizza does each friend get?

  1. Split the half into 4 equal parts.
  2. Each part is an eighth of the whole pizza. 1/2 ÷ 4 = 1/8

Answer: 1/8

Example 4

How many 1/3s are there in 4?

  1. Each whole has 3 thirds. 4 × 3 = 12
  2. So: 4 ÷ 1/3 = 12

Answer: 12

Example 5

Work out 3/4 ÷ 2/5.

  1. Multiply by the reciprocal of 2/5. 3/4 × 5/2 = 15/8
  2. As a mixed number: 15/8 = 1 7/8

Answer: 1 7/8

Example 6

How many 3/8-litre cups can be filled from a 4 1/2-litre jug?

  1. Write the jug as an improper fraction. 4 1/2 = 9/2
  2. Divide. 9/2 ÷ 3/8 = 9/2 × 8/3
  3. Multiply and simplify. 72/6 = 12

Answer: 12 cups

Example 7

Work out 3 3/4 ÷ 1 1/2.

  1. Change both to improper fractions. 15/4 ÷ 3/2
  2. Keep, change, flip. 15/4 × 2/3 = 30/12
  3. Simplify. 30/12 = 5/2 = 2 1/2

Answer: 2 1/2