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Dividing Fractions Calculator
Enter two fractions and this fraction division calculator shows the answer plus the full keep-change-flip working behind it.
Fraction division has a reputation it does not deserve. It looks harder than the other three operations because the method seems arbitrary: you flip a fraction over for no obvious reason and suddenly a division problem becomes a multiplication problem. Most people are taught the mnemonic, pass the test, and forget it by the following term, because nobody explained what the flip is doing. The short version is that dividing by a number and multiplying by its reciprocal are the same operation, so once you flip the second fraction you are allowed to switch the sign. That is the whole trick, and it works for whole numbers too.
This calculator does the arithmetic and shows the reasoning at the same time. Type in the fraction you are dividing and the fraction you are dividing by, and you get the reciprocal it used, the multiplication it turned the problem into, the unsimplified product, the reduced answer, and the mixed number form when the result is larger than 1. It also runs a check in reverse: multiply the answer by the divisor and you should land back on the fraction you started with. Whole numbers work the same way, just put the whole number on top and 1 underneath. Below the calculator you will find the keep-change-flip method spelled out, three worked examples, and the four mistakes that account for nearly every wrong answer.
Worked example: 2/3 ÷ 4/5
- Keep. The first fraction stays exactly as it is: 2/3.
- Change. The division sign becomes a multiplication sign.
- Flip. The second fraction turns upside down: 4/5 becomes 5/4.
- Multiply across. 2/3 × 5/4 = (2×5)/(3×4) = 10/12.
- Simplify. 10 and 12 share a factor of 2, so 10/12 = 5/6.
- Check. Multiply back: 5/6 × 4/5 = 20/30 = 2/3, the fraction you started with.
The keep, change, flip method
Keep-change-flip, sometimes shortened to KCF, is a four-step routine for turning any fraction division into a fraction multiplication:
- Keep the first fraction exactly as it is.
- Change the division sign to a multiplication sign.
- Flip the second fraction, swapping its numerator and denominator.
- Multiply straight across and simplify the result.
Why flipping is allowed
Dividing by a number is always the same as multiplying by its reciprocal. You already use this with whole numbers without thinking about it: dividing by 2 gives the same answer as multiplying by 1/2, and dividing by 5 gives the same answer as multiplying by 1/5. Fractions follow the identical rule. The reciprocal of 4/5 is 5/4, so dividing by 4/5 is multiplying by 5/4.
Written out as algebra, the example above looks like this:
(2/3) ÷ (4/5)
= (2/3) × 1 ÷ (4/5)
= (2/3) × (5/4)
= 10/12
= 5/6The expression 1 divided by 4/5 simplifies to 5/4, which is just the flip. Nothing magic happens: keep-change-flip converts a division into the multiplication it was equal to all along. That is also why the reverse check works, and why multiplying any fraction by its own reciprocal gives 1.
Three more worked examples
Dividing by a whole number: 3/4 ÷ 3
Rewrite the whole number as a fraction over 1, then apply the method.
- Rewrite: 3 becomes 3/1.
- Keep 3/4, change ÷ to ×, flip 3/1 to 1/3.
- Multiply: 3/4 × 1/3 = 3/12.
- Simplify: 3/12 = 1/4.
Splitting three quarters into three equal shares gives one quarter each, which is exactly what the answer says.
Dividing by a smaller fraction: 1/2 ÷ 1/8
- Keep 1/2, change ÷ to ×, flip 1/8 to 8/1.
- Multiply: 1/2 × 8/1 = 8/2.
- Simplify: 8/2 = 4.
The answer is bigger than what you started with, and it should be: the question is really "how many eighths fit into a half?" Four of them do.
Dividing mixed numbers: 1 3/4 ÷ 1/2
- Convert first: 1 3/4 becomes 7/4.
- Keep 7/4, change ÷ to ×, flip 1/2 to 2/1.
- Multiply: 7/4 × 2/1 = 14/4.
- Simplify: 14/4 = 7/2, which is 3 1/2 as a mixed number.
The four mistakes that cause most wrong answers
- Flipping the first fraction. Only the divisor, the fraction you are dividing by, gets flipped. 2/3 ÷ 4/5 becomes 2/3 × 5/4, never 3/2 × 4/5.
- Flipping both fractions. That sets up a completely different problem and the answer will not be close.
- Starting with a mixed number. Convert to an improper fraction first. 2 1/2 has to become 5/2 before you flip anything.
- Leaving the answer unsimplified. 10/12 is arithmetically right but not finished. Reduce it to 5/6.
A fast sanity check: dividing by a fraction smaller than 1 makes the answer bigger, and dividing by a fraction bigger than 1 makes it smaller. If your answer moved the wrong way, you probably flipped the wrong fraction.
Practice until you can drop the mnemonic
Repetition is what makes this stick. The fraction division practice page generates unlimited problems at three difficulty levels with hints when you get stuck. After a dozen problems most students stop reciting keep-change-flip and just flip the second fraction automatically.
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Frequently Asked Questions
Common questions about dividing fractions