How to divide fractions and why the reciprocal works

You have 3/4 litre of drink. Each serving uses 3/8 litre. The picture shows two servings inside your drink. Division asks for that count: (3/4) ÷ (3/8) = 2. The answer counts servings, not litres.

Two coloured servings fill the drink

The first bar fills six eighths, which is three quarters of a litre. Each group of three eighths is one serving.

3/4 litre = two servings

One serving = 3/8 litre

Count in servings instead of litres

How many 3/8-litre servings fit in 3/4 litre?

Find what turns one serving into one unit

$\frac38\times\frac83=1$

The reciprocal of 3/8 is 8/3: the fraction turned over. Multiplying by it makes one serving count as one.
Count the full amount in that same unit

$\frac34\times\frac83=\frac{24}{12}=2$

Use the same change for the full drink. It counts as two servings. You changed the unit of the count, not the amount of drink.
Check by rebuilding the drink

$2\times\frac38=\frac68=\frac34$

Two servings use exactly the drink you started with. Multiplying back checks the division.

Keep, turn over, multiply

This gives the general method: keep the first fraction, turn the second one over, then multiply. The second fraction gives the group size, which is why you turn that one over.

The count need not be a whole number. With half a litre and a serving size of three quarters, you cannot fill one serving. The calculation (1/2) × (4/3) = 2/3 means two thirds of a serving.

A small serving can make the count larger than the starting number. That is not extra drink. It is more small groups. You cannot use zero as a serving size or divide by zero.

Try it yourself

Work out an answer and explain your reasoning. Then check the explanation.

Divide by a different fraction

$\frac23\div\frac45$

How many groups of size 4/5 fit in 2/3? Find the count and check that its size makes sense.

Show the answerHide the answer

$\frac56$

Keep 2/3 and multiply by 5/4. The result 10/12 reduces to 5/6.

Turn over the correct fraction

$\frac23\div\frac45=\frac32\cdot\frac45$

Fix the method. Which fraction gives the group size, and which one must turn over?

Show the answerHide the answer

$\frac23\cdot\frac54$

You are dividing by the second fraction, so that is the one to turn over.

Test the portion-counting idea

Try dividing new fractions. Keep the first amount, turn over the group size and multiply. An answer can be a whole number or part of a group.

2-minute practice · keep going after mistakes · opens in a new tab

Try fraction division