Maths

How do you divide fractions?

Quick answer

To divide by a fraction, multiply by its reciprocal. Keep the first fraction, change ÷ to ×, flip the second fraction, then simplify. 3/4 ÷ 1/2 = 3/4 × 2/1 = 3/2.

Ages 10–13Primary

In one sentence

To divide by a fraction, flip that fraction upside down and multiply instead.

Explained

Dividing asks a question: how many of these fit into that? 6 ÷ 2 asks how many 2s fit into 6 (three). In the same way, 3 ÷ 1/2 asks how many halves fit into 3. Each whole holds two halves, so three wholes hold 6 halves. Notice that 3 ÷ 1/2 gave the same answer as 3 × 2. That is the secret of dividing fractions: dividing by a fraction is the same as multiplying by its reciprocal.

The reciprocal of a fraction is the fraction flipped upside down. The reciprocal of 2/3 is 3/2; the reciprocal of 1/4 is 4; the reciprocal of 5 (which is 5/1) is 1/5. A number times its reciprocal always makes 1.

pizza
juice
cake

3 pizzas ÷ 1/4 pizza each = 12 slices. Dividing by a fraction can make the answer bigger.

Many students learn the method as Keep, Change, Flip (KCF):

Keep the first fraction as it is.
Change the ÷ sign to ×.
Flip the second fraction.

So 2/3 ÷ 3/4 becomes 2/3 × 4/3 = 8/9. Then you just multiply the fractions: tops times tops, bottoms times bottoms. You never flip the first fraction, and you never need a common denominator.

Unit fractions and whole numbers. A unit fraction has 1 on top, like 1/3. Dividing a unit fraction by a whole number makes smaller pieces: 1/3 ÷ 2 = 1/6 (half of a third is a sixth). Dividing a whole number by a unit fraction counts pieces: 4 ÷ 1/3 = 12 (four wholes contain twelve thirds). Drawing a bar model makes both easy to see.

Same denominators shortcut. When the denominators match, just divide the numerators: 6/7 ÷ 2/7 = 6 ÷ 2 = 3, because you are asking how many 2-sevenths fit into 6-sevenths.

Mixed numbers. Change them to improper fractions first (see improper fractions and mixed numbers): 2 1/4 ÷ 1 1/2 = 9/4 ÷ 3/2 = 9/4 × 2/3 = 18/12 = 1 1/2.

Examples

Calculation	Keep, change, flip	Answer
2/3 ÷ 1/3	2/3 × 3/1 = 6/3	2
3/5 ÷ 2/7	3/5 × 7/2 = 21/10	2 1/10
4 ÷ 2/3	4/1 × 3/2 = 12/2	6
5/6 ÷ 5	5/6 × 1/5 = 5/30	1/6
1/4 ÷ 3	1/4 × 1/3	1/12
2 1/4 ÷ 1 1/2	9/4 × 2/3 = 18/12	1 1/2

Worked example: 5/6 ÷ 2/3. Keep 5/6, change ÷ to ×, flip 2/3 to 3/2: 5/6 × 3/2 = 15/12. Simplify by dividing top and bottom by 3: 5/4 = 1 1/4. Check by multiplying back: 1 1/4 × 2/3 = 5/4 × 2/3 = 10/12 = 5/6. Correct.

Common mistakes

Flipping the wrong fraction. Only the second fraction (the one you divide by) is flipped. Fix: say "keep, change, flip" in that order.
Flipping but forgetting to change ÷ to ×. For 3/4 ÷ 1/2, writing 3/4 ÷ 2/1 gives 3/8, which is wrong; 3/4 × 2/1 = 1 1/2 is right. Flip and change go together.
Dividing the parts of mixed numbers separately. 3 1/3 ÷ 1 2/3 is not (3 ÷ 1) and (1/3 ÷ 2/3), which gives 3 1/2. Change to improper fractions first: 10/3 × 3/5 = 2.
Thinking division always makes things smaller. Dividing by a fraction less than 1 gives a bigger answer: 6 ÷ 1/2 = 12.

Step by step

Write everything as fractionsWhole numbers go over 1; mixed numbers become improper fractions.
KeepLeave the first fraction exactly as it is.
ChangeSwap the ÷ sign for ×.
FlipTurn the second fraction upside down (its reciprocal).
Multiply and simplifyMultiply tops and bottoms, cancel common factors, and turn top-heavy answers into mixed numbers.

To check an answer, multiply it by the fraction you divided by. You should get back the number you started with.

Practice questions

Divide the fractions

Try each one, then check your answer.

1/2 ÷ 1/4
Show answer
1/2 × 4/1 = 4/2 = 2
3 ÷ 1/2
Show answer
3 × 2 = 6
5 ÷ 1/3
Show answer
5 × 3 = 15
1/3 ÷ 2
Show answer
1/3 × 1/2 = 1/6
1/5 ÷ 4
Show answer
1/5 × 1/4 = 1/20
4/5 ÷ 2/5
Show answer
Same denominators: 4 ÷ 2 = 2
2/3 ÷ 3/4
Show answer
2/3 × 4/3 = 8/9
3/4 ÷ 3/8
Show answer
3/4 × 8/3 = 24/12 = 2
5/6 ÷ 2/3
Show answer
5/6 × 3/2 = 15/12 = 5/4 = 1 1/4
7/10 ÷ 1/5
Show answer
7/10 × 5/1 = 35/10 = 3 1/2
6 ÷ 3/4
Show answer
6 × 4/3 = 24/3 = 8
9/10 ÷ 3
Show answer
9/10 × 1/3 = 9/30 = 3/10
1 1/2 ÷ 3/4
Show answer
3/2 × 4/3 = 12/6 = 2
3 1/3 ÷ 1 2/3
Show answer
10/3 × 3/5 = 30/15 = 2
2 1/2 ÷ 1/4
Show answer
5/2 × 4/1 = 20/2 = 10
−2/5 ÷ 4/5
Show answer
−2/5 × 5/4 = −10/20 = −1/2

Word problems

Dividing fractions in real life

Try each one, then check your answer.

You have 3 pizzas and each person gets 1/4 of a pizza. How many people can you feed?
Show answer
3 ÷ 1/4 = 3 × 4 = 12 people
3/4 litre of juice is shared equally between 3 glasses. How much goes in each glass?
Show answer
3/4 ÷ 3 = 3/4 × 1/3 = 3/12 = 1/4 litre
A ribbon is 4 1/2 m long. How many 3/4 m pieces can you cut from it?
Show answer
9/2 ÷ 3/4 = 9/2 × 4/3 = 36/6 = 6 pieces
One batch of cookies needs 2/3 cup of flour. You have 4 cups. How many batches can you make?
Show answer
4 ÷ 2/3 = 4 × 3/2 = 12/2 = 6 batches
A walker covers 2 1/2 km in 3/4 of an hour. What is her speed in km per hour?
Show answer
5/2 ÷ 3/4 = 5/2 × 4/3 = 20/6 = 3 1/3 km/h

Questions people ask

What is the rule for dividing fractions?

Multiply by the reciprocal of the second fraction: a/b ÷ c/d = a/b × d/c. Then simplify.

What does KCF mean in dividing fractions?

KCF stands for Keep, Change, Flip: keep the first fraction, change ÷ to ×, and flip the second fraction. It is a memory aid for the reciprocal rule.

Why do you flip the second fraction?

Dividing by a number is the same as multiplying by its reciprocal. Dividing by 2 is the same as taking half (× 1/2), and dividing by 1/2 is the same as doubling (× 2).

Do you need a common denominator to divide fractions?

No. Keep, change, flip works for any denominators. If the denominators already match, you can simply divide the numerators: 6/7 ÷ 2/7 = 3.

How do you divide a fraction by a whole number?

Write the whole number over 1, flip it and multiply. 3/5 ÷ 2 = 3/5 × 1/2 = 3/10. In effect, you multiply the denominator by the whole number.