How to Divide Fractions
Keep, change, flip — and why the answer always comes out bigger than the fraction you started with.
Keep the first fraction, change the division sign to multiplication, flip the second fraction, then multiply straight across and simplify. So 3⁄5 ÷ 2⁄3 becomes 3⁄5 × 3⁄2 = 9⁄10. You never need a common denominator.
Key takeaways
- Flip the second fraction, never the first. Flipping the wrong one gives a different answer 97.8% of the time — the two agree only when both fractions are identical.
- Take all 2,025 ways of dividing one proper fraction by another (the 45 fractions in lowest terms with denominators up to 12). The answer is bigger than the fraction you started with in 100% of them. Dividing by something smaller than 1 always grows a number.
- 51.1% of those answers come out at 1 or more, and 45.2% have to be written as a mixed number.
- 43.5% of the raw answers still share a common factor and need simplifying. Nearly half of the answers marked wrong are arithmetically right but unreduced.
- You never need a common denominator to divide. That rule belongs to addition and subtraction, and carrying it over is the most common wrong turn.
- Multiplying straight across without flipping gives a different answer 100% of the time. There is no case where skipping the flip happens to work.
On this page
How do you divide fractions?
Three steps, in this order. The name most classrooms use is keep, change, flip.
- Keep the first fraction exactly as written. 3⁄5 stays 3⁄5.
- Change the ÷ to a ×.
- Flip the second fraction — swap its top and bottom. 2⁄3 becomes 3⁄2.
- Multiply across: 3 × 3 = 9 on top, 5 × 2 = 10 underneath. The answer is 9⁄10.
- Simplify if the top and bottom share a factor. Here they do not, so 9⁄10 is final.
Why do you flip the second fraction?
Because dividing by a number is exactly the same operation as multiplying by its reciprocal — the fraction turned upside down. The reciprocal is defined by one property: a number times its reciprocal is 1.
2⁄3 × 3⁄2 = 6⁄6 = 1. So multiplying by 3⁄2 undoes a division by 2⁄3 perfectly, and “flip and multiply” is not a trick to memorise but a restatement of what division means.
The same logic explains why it has to be the second fraction. The reciprocal you need is the reciprocal of the thing you are dividing by. Flipping the first fraction instead inverts the wrong quantity, and it produces a different answer in 97.8% of all fraction pairs — the only cases where it accidentally works are the 45 where both fractions are the same to begin with.
Why does the answer get bigger?
This is the part that feels wrong, so it is worth making concrete. Division asks: how many of these fit inside that? When the divisor is smaller than 1, lots of them fit, so the answer grows.
Run that across every case and the pattern is total. Taking the 45 proper fractions in lowest terms with denominators up to 12 and dividing each by each — 2,025 problems — the answer is larger than the starting fraction every single time. There is no exception, because a proper fraction is always less than 1 and dividing by anything less than 1 always increases a positive number.
| Property of the answer | Share of cases | What it means for you |
|---|---|---|
| Bigger than the fraction you started with | 100% | If your answer shrank, you skipped the flip |
| Equal to 1 or more | 51.1% | Just over half the time you cross into improper territory |
| Has to be written as a mixed number | 45.2% | Check whether the question asked for one |
| Raw answer still needs simplifying | 43.5% | Almost half. Never hand in the unreduced form |
| Comes out as a whole number | 5.9% | Uncommon, and usually a sign the numbers were chosen for you |
The first row is the most useful thing on this page as a check. If your answer is smaller than the fraction you started with, you have made one of the two flip errors — either you did not flip at all, or you flipped the wrong fraction. You can catch it without redoing the arithmetic.
Four worked examples
| Problem | After the flip | Multiply across | Simplify | Final |
|---|---|---|---|---|
| 3⁄5 ÷ 2⁄3 | 3⁄5 × 3⁄2 | 9⁄10 | already lowest | 9⁄10 |
| 1⁄2 ÷ 1⁄4 | 1⁄2 × 4⁄1 | 4⁄2 | ÷ 2 | 2 |
| 3⁄4 ÷ 9⁄10 | 3⁄4 × 10⁄9 | 30⁄36 | ÷ 6 | 5⁄6 |
| 7⁄8 ÷ 1⁄6 | 7⁄8 × 6⁄1 | 42⁄8 | ÷ 2 | 21⁄4 = 51⁄4 |
Row three is the one worth studying. 30⁄36 is a correct answer, and it is also the answer that loses marks. Both numbers are divisible by 6, and the reduced form 5⁄6 is what the question wants. This shape of near-miss accounts for the 43.5% figure above.
Row four shows the other half of the finishing work: 21⁄4 is fully simplified but improper, and if the question asks for a mixed number it becomes 51⁄4.
Mixed numbers and whole numbers
The rule does not change. What changes is the preparation: everything has to be a single fraction before you flip anything.
Dividing by a whole number
Write the whole number over 1. 3⁄4 ÷ 2 becomes 3⁄4 ÷ 2⁄1, which becomes 3⁄4 × 1⁄2 = 3⁄8. In shortcut form: dividing a fraction by a whole number just multiplies the denominator by it.
Note that this is the one family where the answer does get smaller — because the divisor, 2, is bigger than 1. The 100% figure earlier applies to dividing by a proper fraction.
Dividing a whole number by a fraction
Same move, other way round. 6 ÷ 2⁄3 becomes 6⁄1 × 3⁄2 = 18⁄2 = 9. Nine two-thirds fit inside 6, which you can check by counting.
Dividing mixed numbers
Convert both to improper fractions first. 21⁄2 ÷ 11⁄4 becomes 5⁄2 ÷ 5⁄4, then 5⁄2 × 4⁄5 = 20⁄10 = 2.
The four mistakes, by how much they cost
All four are worked on the same problem, 3⁄4 ÷ 2⁄3, whose correct answer is 9⁄8.
| Mistake | What you get | How often it differs from the right answer | How to catch it |
|---|---|---|---|
| Multiplied straight across, no flip | 6⁄12 = 1⁄2 | 100% | Answer shrank instead of growing |
| Flipped the first fraction | 8⁄9 | 97.8% | Answer is below 1 when it should be above |
| Found a common denominator first, then divided tops and bottoms | varies | — | You did work the method never asked for |
| Correct arithmetic, never simplified | 9⁄8 is already lowest — but 43.5% of problems are not | 43.5% of cases | Check the greatest common factor before writing the answer |
The first two are catchable without redoing anything, using the rule from the section above: dividing by a proper fraction must make the number bigger. 1⁄2 and 8⁄9 are both smaller than the 9⁄8 that a correct answer has to exceed, and both are smaller than 1 when the right answer is above it.
Simplifying the answer
Divide the numerator and the denominator by their greatest common factor. For 30⁄36, the GCF of 30 and 36 is 6, so the fraction reduces to 5⁄6 in one step.
There is a labour-saving version of this worth knowing: cancel before you multiply, not after. In 3⁄4 × 10⁄9, the 3 and the 9 share a factor of 3, and the 4 and the 10 share a factor of 2. Cancelling first turns the problem into 1⁄2 × 5⁄3 = 5⁄6, and you never write 30 or 36 at all. Across the 2,025 problems, raw numerators or denominators climb above 100 in 3.6% of cases — the largest is 132 — and cancelling first keeps every one of them small.
Terms worth knowing
- Numerator
- The top number. How many parts you have.
- Denominator
- The bottom number. How many parts make one whole.
- Dividend and divisor
- In a ÷ b, a is the dividend and b is the divisor. The divisor is the one you flip.
- Reciprocal
- A fraction with its numerator and denominator swapped. A number times its reciprocal is always 1, which is the entire reason the flip works.
- Proper fraction
- Numerator smaller than denominator, so the value is below 1. All 45 fractions used for the counts on this page are proper.
- Improper fraction
- Numerator at least as large as the denominator, so the value is 1 or more. Not an error — often the tidiest way to write an answer.
- Mixed number
- A whole number beside a proper fraction, such as 51⁄4. Convert to improper before doing any arithmetic with it.
- Greatest common factor (GCF)
- The largest number dividing both the numerator and the denominator. Dividing by it reduces a fraction to lowest terms in a single step.
- Lowest terms
- A fraction whose numerator and denominator share no factor above 1. The form nearly every answer is expected in.
All 13 fraction tools in this guide
Authoritative sources
- Fractions — Khan Academy — worked lessons and practice on every operation covered here.
- Multiplicative inverse — Wikipedia — the reciprocal, and why multiplying by it is division.
- Fraction — Wikipedia — proper, improper and mixed forms, and lowest terms.
- Maths — Oak National Academy — curriculum-aligned lessons on fraction division for school use.
- Greatest common divisor — Wikipedia — the Euclidean algorithm behind every simplification step.