Division as a multiplication question

See division as a multiplication question, a way that also works for fractions.

12 ÷ 4 = 3. But there's another way to ask the same question, and it works for fractions too: what times 4 makes 12? 4 × ? = 12. The answer is 3. So a division is a multiplication with a missing number.

1 ÷ 2/3 asks: 2/3 × ? = 1. What cancels the 2 and the 3? 3/2: 2/3 × 3/2 = 6/6 = 1. So 1 ÷ 2/3 = 3/2.

3 ÷ 2/3 asks: 2/3 × ? = 3. We know 2/3 × 3/2 = 1, so to get 3 we need three times as much: 3/2 × 3 = 9/2.

1/5 ÷ 1/2 asks: 1/2 × ? = 1/5. 1/2 × 2 = 1, then × 1/5 gives 1/5. So the missing number is 2 × 1/5 = 2/5.

2/3 ÷ 3/5 asks: 3/5 × ? = 2/3. 3/5 × 5/3 = 1, then × 2/3 gives 2/3. So the answer is 5/3 × 2/3 = 10/9. Do you see a pattern in all of these?

Division asks what the divisor must be multiplied by to give the dividend — which works for fractions just as for whole numbers.

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