Reciprocals first
The reciprocal of a fraction is the fraction flipped upside down.
| Number | Reciprocal |
|---|---|
| 2/3 | 3/2 |
| 5/8 | 8/5 |
| 4 (= 4/1) | 1/4 |
| 1/7 | 7 |
A number times its reciprocal is always 1: 2/3 × 3/2 = 6/6 = 1.
Zero has no reciprocal.
The rule: multiply by the reciprocal
To divide by a fraction, multiply by its reciprocal.
a/b ÷ c/d = a/b × d/c
Memory aid: keep, change, flip.
- Keep the first fraction.
- Change ÷ to ×.
- Flip the second fraction (the divisor).
Example: 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 = 1 7/8
Why does it work?
Division asks "how many of these fit into that?"
- 3 ÷ 1/2 asks "how many halves fit into 3?" There are 2 halves in each whole, so 3 × 2 = 6.
Dividing by 1/2 is the same as multiplying by 2, its reciprocal.
Dividing with whole numbers
- 6 ÷ 2/3 = 6/1 × 3/2 = 18/2 = 9 (nine two-thirds fit into 6)
- 4/5 ÷ 2 = 4/5 × 1/2 = 4/10 = 2/5 (half of 4/5)
Mixed numbers
Convert to improper fractions first.
Example: 2 1/4 ÷ 1 1/2
- 9/4 ÷ 3/2 = 9/4 × 2/3 = 18/12 = 3/2 = 1 1/2
Bigger or smaller?
| Divide by | Quotient is |
|---|---|
| a number greater than 1 | smaller than the dividend |
| a number between 0 and 1 | bigger than the dividend |
Dividing by a fraction less than 1 makes things bigger, because many small pieces fit.
Worked examples
Example 1: 5/6 ÷ 1/3 = 5/6 × 3/1 = 15/6 = 5/2 = 2 1/2
Example 2: How many 3/4-cup servings are in 6 cups of soup?
6 ÷ 3/4 = 6 × 4/3 = 8 servings.
Example 3: 7/8 ÷ 7/8 = 1 (any non-zero number divided by itself).
Common mistakes
- Flipping the first fraction instead of the second.
- Flipping both fractions.
- Forgetting to change ÷ to ×.
- Not converting mixed numbers first.
Try it yourself
- 1/2 ÷ 1/4
- 2/3 ÷ 4/9
- 10 ÷ 2/5
- 3/8 ÷ 3
- 3 1/3 ÷ 1 2/3
Answers: 1) 2 2) 3/2 = 1 1/2 3) 25 4) 1/8 5) 2
Quick recap
- The reciprocal flips a fraction; a number times its reciprocal is 1.
- To divide, keep, change, flip.
- Dividing by a fraction less than 1 gives a bigger quotient.