How to divide fractions

Dividing one fraction by another asks a simple question: how many times does the second one fit into the first? 3/4 ÷ 1/4 asks how many quarters fit into three quarters, and the answer is 3. Most fractions are harder to picture than that, so there is a method that turns the division into a multiplication.

The rule

Keep the first fraction as it is, change ÷ into ×, and flip the second fraction upside down. Then multiply top times top and bottom times bottom.

How to do it

Keep: write the first fraction down as it is.
Change: turn the ÷ sign into ×.
Flip: turn the second fraction upside down, so its top goes to the bottom. 3/4 becomes 4/3, and a whole number such as 2, which is 2/1, becomes 1/2.
Multiply the top numbers, then multiply the bottom numbers.
Then simplify if you can. If the top is bigger than the bottom, the answer is more than 1, and you can write it as a whole number or a mixed number.

Worked examples

What is 3/4 ÷ 1/4?

3/4 ÷ 1/4 = 3/4 × 4/1 = 12/4 = 3

The picture asks how many quarters fit into 3/4, and you can count them: 3. Keep, change, flip agrees: 3/4 × 4/1 = 12/4, and 12 ÷ 4 = 3.

What is 2 ÷ 1/2?

2 ÷ 1/2 = 2/1 × 2/1 = 4/1 = 4

Each whole holds two halves, so two wholes hold four. Dividing by 1/2 is the same as doubling.

What is 1/2 ÷ 3/4?

1/2 ÷ 3/4 = 1/2 × 4/3 = 4/6 = 2/3

3/4 is bigger than 1/2, so it does not fit in even once, and the answer is less than 1: only 2/3 of a 3/4 fits into a half. Check by multiplying back: 2/3 × 3/4 = 6/12, which is 1/2.

What is 5/6 ÷ 1/3?

5/6 ÷ 1/3 = 5/6 × 3/1 = 15/6 = 5/2 = 2 1/2

A third is 2 sixths. 5 sixths hold two lots of 2 sixths with 1 sixth left over, and that sixth is half of a third. So 1/3 fits into 5/6 two and a half times.

Multiplying across, the way the app does it

Keep, change, flip and multiply comes down to two multiplications across the ÷ sign. The top of the answer is the first top times the second bottom. The bottom of the answer is the first bottom times the second top. For 1/2 ÷ 3/4: 1 × 4 = 4 on top, and 2 × 3 = 6 underneath, so 4/6.

It is the same sum without writing the flipped fraction out. Some teachers draw it as an X across the two fractions.

Making the bottoms the same

Another way is to give both fractions the same bottom number and then divide the tops. 5/6 ÷ 1/3 becomes 5/6 ÷ 2/6, and 5 sixths ÷ 2 sixths is 5 ÷ 2, which is 5/2 or 2 1/2.

It is closer to what division means, since it counts how many pieces of one size fit into the other, and it gives the same answer as keep, change, flip.

Other names for the method

Keep, change, flip, or KCF, is what many classrooms call it. Books also say invert and multiply, or multiply by the reciprocal. The reciprocal of a fraction is the fraction flipped, so the reciprocal of 3/4 is 4/3.

Why flipping works

Division asks how many times one number fits into another. 6 ÷ 2 = 3 because 2 fits into 6 three times. With fractions, 3/4 ÷ 1/4 asks how many quarters fit into 3/4, and the answer is 3.

Dividing by 1/2 asks how many halves there are. Every whole holds 2 halves, so there are twice as many halves as wholes, and dividing by 1/2 is the same as multiplying by 2. In the same way, dividing by 1/4 is multiplying by 4, because every whole holds 4 quarters. The flipped fraction, 4/1, is exactly that 4.

Dividing by 3/4 asks how many lots of three quarters there are. Count the quarters, which is × 4, then put them into groups of 3, which is ÷ 3. Multiplying by 4 and dividing by 3 is the same as multiplying by 4/3, the flipped fraction.

To check an answer, multiply it by the fraction you divided by. You should get back to the first fraction: 5/2 × 1/3 = 5/6.

Where mistakes happen

Flipping the first fraction instead of the second. 3/4 ÷ 1/4 done that way is 4/3 × 1/4 = 4/12 = 1/3, not 3. Only the fraction you divide by gets flipped.
Flipping the second fraction but leaving the ÷ sign. Keep, change and flip all happen together, and after them the sum is a multiplication.
Expecting the answer to be smaller. Dividing by a fraction less than 1 makes the answer bigger, so 2 ÷ 1/2 is 4, not 1.
Flipping a mixed number as it is. Turn 1 1/2 into 3/2 first, then flip it to 2/3: so 3 ÷ 1 1/2 = 3/1 × 2/3 = 6/3 = 2.

Questions people ask

Why does dividing by a fraction give a bigger answer?

Because a piece smaller than a whole fits into a number more times than a whole does. 1 fits into 2 twice, and 1/2 fits into 2 four times.

How do I divide a fraction by a whole number?

Write the whole number over 1 and flip it. 3/4 ÷ 3 = 3/4 × 1/3 = 3/12, which is 1/4. That makes sense, because three quarters shared between 3 is one quarter each.

Do I have to simplify the answer?

Simplified is the usual way to finish at school. The Math Basics practice asks for the answer exactly as the multiplying gives it, 4/6 for 1/2 ÷ 3/4, so that the two multiplications are what get practised.

Do the bottom numbers have to be the same?

No. Keep, change, flip works on any two fractions as they are. Making the bottoms the same is a different way to divide, shown above, and it gives the same answer.