To divide one fraction by another, turn the second fraction upside down and multiply. So 6/7 divided by 2/3 becomes 6/7 multiplied by 3/2, which is 18/14, or 9/7 once simplified. The upturned fraction is called the reciprocal, and the rule is often shortened to keep, change, flip: keep the first fraction, change the divide sign to a multiply sign, flip the second fraction.
Divisions of this kind turn up throughout numerical reasoning assessments, usually buried inside a word problem rather than written out as a sum, so recognising one is half the work.
A few points of vocabulary that the working below relies on:
- The top number in a fraction is the numerator and the bottom number is the denominator
- The line between them means divide, so 3/4 is another way of writing 3 divided by 4
- A fraction can be smaller than 1, equal to 1 or larger than 1: 3/4, 4/4 and 9/4 are all fractions
- The reciprocal of a fraction is that fraction with its two numbers swapped, so the reciprocal of 2/3 is 3/2, and the reciprocal of the whole number 5 is 1/5
- Simplifying as you go keeps the numbers small, which matters when you are working against a clock
Step 1: Keep the first fraction, change the sign, flip the second
Here’s an example:
3/8 divided by 2/4
The first fraction is left exactly as it is.
Change the division sign to a multiplication sign, then turn the second fraction upside down, so 2/4 becomes 4/2.
That makes the problem look like this:
3/8 x 4/2
Step 2: Multiply across and simplify
Multiply the two numerators together, then the two denominators.
3 x 4 = 12
8 x 2 = 16
That gives 12/16, which simplifies to 3/4.
You can also cancel before you multiply. Here 4/2 is just 2, and 3/8 multiplied by 2 is 6/8, which is 3/4 again. Same answer, smaller numbers.
Why turning the second fraction over works
Dividing by a number asks how many of that number fit into the first one. Dividing by 1/2 asks how many halves fit in, and since every whole contains two halves, dividing by 1/2 doubles the number. Doubling is multiplying by 2, and 2 is exactly 1/2 turned over.
The general reason is that any fraction multiplied by its own reciprocal gives 1. Multiply 2/3 by 3/2 and you get 6/6, which is 1. So multiplying by 3/2 undoes multiplying by 2/3, and undoing a multiplication is precisely what dividing does.
That is worth knowing because it means the rule is not a trick to be memorised and hoped for. It is the reciprocal doing the work, which is why the same two lines handle proper fractions, improper fractions, whole numbers and mixed numbers without any special cases.
Worked examples
Question one: a fraction divided by a fraction
What is 6/8 divided by 4/12?
Keep the first fraction, change the sign, flip the second:
6/8 x 12/4
6 x 12 = 72
8 x 4 = 32
That gives 72/32, which simplifies to 9/4, or 2 and 1/4 as a mixed number.
Question two: getting the order right
Divide 6/7 by 2/3.
Order matters in division, and “divide 6/7 by 2/3” means 6/7 is the number being divided. Written as a sum it is 6/7 divided by 2/3, with 6/7 first.
Keep, change, flip:
6/7 x 3/2
6 x 3 = 18
7 x 2 = 14
That gives 18/14, which simplifies to 9/7, or 1 and 2/7.
Check the size of the answer against the question. Dividing by 2/3, a number smaller than 1, has to make the result bigger than 6/7, and 9/7 is bigger. Reverse the order and you get a different number altogether: 2/3 divided by 6/7 is 2/3 x 7/6, which is 14/18, or 7/9.
Question three: dividing by a whole number
What is 1/4 divided by 2?
Write the whole number as a fraction over 1, so 2 becomes 2/1, then keep, change, flip:
1/4 x 1/2 = 1/8
The shortcut worth remembering is that dividing by a whole number multiplies the denominator. 1/4 divided by 2 is 1/8, and 1/4 divided by 3 would be 1/12. Half of a quarter is an eighth, which is the sanity check.
Take care not to confuse this with doubling. Twice 1/4 is 1/2, while 1/4 divided by 2 is 1/8, and the two are easy to transpose under time pressure.
Question four: mixed numbers on both sides
What is 2 and 3/4 divided by 1 and 2/3?
Convert both mixed numbers into improper fractions before anything else. 2 and 3/4 is 11/4, and 1 and 2/3 is 5/3.
Keep, change, flip:
11/4 x 3/5
11 x 3 = 33
4 x 5 = 20
That gives 33/20, or 1 and 13/20. Because the question was written with mixed numbers, give the answer as one too.
Question five: a mixed number as the divisor
What is 1/4 divided by 2 and 2/3?
2 and 2/3 is 8/3, so the sum is 1/4 divided by 8/3.
1/4 x 3/8
1 x 3 = 3
4 x 8 = 32
The answer is 3/32. It is smaller than 1/4, which is what dividing by a number bigger than 1 should always do.
Mistakes that cost marks
- Flipping the wrong fraction. The first fraction never changes. Only the divisor turns over.
- Half flipping. The reciprocal of 6/7 is 7/6. Both numbers swap, so 7/3 and 6/1 are neither of them right.
- Reading the order backwards. “Divide A by B” and “A divided into B” are opposites, and so are their answers. Decide which number is being divided before you write anything down.
- Leaving mixed numbers in place. Convert to improper fractions first, then convert back at the end so the answer matches the form of the question.
- Stopping one step early. An answer left as 72/32 is not wrong, but in a multiple choice test the options are usually in lowest terms, so simplify before you look at them.
Practice dividing fractions
Dividing fractions is quick once the reciprocal step is automatic, and it is one of the few numerical topics where a single method covers every case. Work through it under timed conditions with our numerical reasoning tests, build the arithmetic underneath it with our basic numeracy tests, or turn to the step before this one with our guide to multiplying fractions.
