How do you simplify fractions?
Short answer. Find the greatest common factor (GCF) of the numerator and denominator, then divide both by it. For 18/24 the GCF is 6, so 18/24 = 3/4. When the only factor they still share is 1, the fraction is in simplest form. Dividing by small common factors over and over gets you to the same place.
Simplifying is the last line of almost every fraction problem. You add fractions, or subtract them, and end up with 14/24 when the answer key says 7/12. Same number, fewer digits. Here is how to get there without guessing.
Why dividing both sides is allowed
Dividing the top and bottom by the same number is the same as dividing the whole fraction by 6/6, which is 1. The value does not move. Only the size of the pieces does: 18 small pieces out of 24 is the same amount of pizza as 3 big pieces out of 4.
18/24 = 0.75 3/4 = 0.75 same number, smaller digits
The greatest common factor method
List what divides each number. The biggest one that appears on both lists is the greatest common factor:
Factors of 18 1, 2, 3, 6, 9, 18 Factors of 24 1, 2, 3, 4, 6, 8, 12, 24 GCF = 6 18 ÷ 6 = 3 24 ÷ 6 = 4 18/24 = 3/4
One division, done. The catch is that listing factors gets tedious once the numbers pass about 100.
Can't spot the GCF? Divide in small steps
You do not need the greatest factor, just any common one. Divide by it, look again, repeat. Both numbers even? Halve them. It takes more lines and never gives a wrong answer:
48/60 both even ÷ 2 = 24/30 24/30 both even ÷ 2 = 12/15 12/15 both divide by 3 ÷ 3 = 4/5 2 × 2 × 3 = 12, which was the GCF all along.
The only way to go wrong here is stopping early. 12/15 looks finished if you only check for 2.
Divisibility tests that do the spotting for you
| Divides by | If the number | Example |
|---|---|---|
2 | ends in 0, 2, 4, 6 or 8 | 38 |
3 | has a digit sum divisible by 3 | 123: 1 + 2 + 3 = 6 |
5 | ends in 0 or 5 | 85 |
9 | has a digit sum divisible by 9 | 405: 4 + 0 + 5 = 9 |
10 | ends in 0 | 70 |
So 135/405 gives itself away: both digit sums are 9, so both divide by 9.
135/405 ÷ 9 = 15/45 15/45 ÷ 15 = 1/3 GCF = 9 × 15 = 135. The top divides the bottom exactly.
Big numbers: Euclid's algorithm
Some fractions look finished and are not. 391/667 is odd on both sides, fails the tests for 3 and 5, and still simplifies. The tool for this is over 2,000 years old. Divide the bigger number by the smaller, keep the remainder, and repeat with the last two numbers until the remainder is 0:
667 = 1 × 391 + 276 391 = 1 × 276 + 115 276 = 2 × 115 + 46 115 = 2 × 46 + 23 46 = 2 × 23 + 0 GCF = 23 (the last remainder that was not 0) 391 ÷ 23 = 17 667 ÷ 23 = 29 391/667 = 17/29
A smaller one for practice: 252/378. 378 = 1 × 252 + 126, then 252 = 2 × 126 + 0, so the GCF is 126 and the fraction is 2/3. This is exactly what a fraction simplifier does under the hood. There is no cleverness in it, which is why it never fails.
Prime factors: the method that shows its working
Break both numbers into primes and cross out the pairs. Whatever survives is the answer:
84 = 2 × 2 × 3 × 7 126 = 2 × 3 × 3 × 7 cancel one 2, one 3, one 7 (2 × 3 × 7 = 42, the GCF) 84/126 = 2/3
It is the same cancelling you do before multiplying fractions, and it makes the last check obvious: if no prime appears on both sides, you are done.
How to know you are finished
A fraction is in simplest form when the top and bottom share no factor except 1. Neither number needs to be prime. 8/9 is fully simplified: 8 is 2 × 2 × 2, 9 is 3 × 3, nothing in common. And a prime on one side does not guarantee anything either: 7/21 is 1/3.
Improper fractions, mixed numbers, negatives
Same rule everywhere. With a mixed number, only the fraction part needs simplifying. With a negative, simplify the digits and keep the sign:
28/12 GCF 4 = 7/3 = 2 1/3 3 10/15 GCF 5 = 3 2/3 −6/8 GCF 2 = −3/4
Where this actually comes up
Tape measures and clocks. A mark at 12/16 of an inch is 3/4 inch. A 6/8 inch drill bit is sold as 3/4. Forty minutes is 40/60 of an hour, which is 2/3, and 45 minutes is 45/60, which is 3/4. Recipes that were scaled up tend to come out as 6/8 cup and deserve the same treatment.
The decimal trap
You can only simplify what is exact. Round 1/3 to 0.333 and write it as a fraction, and you get 333/1000. The GCF of 333 and 1000 is 1, so that is already in simplest form, and it is not 1/3. The rounding is baked in and no amount of simplifying removes it. (Converting decimals to fractions covers the repeating-decimal rule that does get you back to 1/3.)
Computers have a stranger version of this. Most programming languages and spreadsheets store 0.1 in binary floating point, and the number actually stored is 3602879701896397/36028797018963968, fully simplified and not quite 1/10. That is the root of 0.1 + 0.2 = 0.30000000000000004. Keep numbers as fractions and simplify at the end, and there is nothing to drift.
Common mistakes, quickly
| Mistake | What happens | Fix |
|---|---|---|
| Stopping too early18/24 ÷ 2 = 9/12 | 9 and 12 still share a 3 | Check again after every division: 3/4 |
| Subtracting instead of dividing18/24 as (18 − 6)/(24 − 6) | 12/18 = 2/3, a different number | Divide top and bottom: 3/4 |
| Dividing only the top18/24 as (18 ÷ 6)/24 | 3/24 = 1/8 | Whatever the top gets, the bottom gets |
| Crossing out digits12/24, strike the 2s | 1/4 instead of 1/2 | Cancel factors, never digits |
| Cancelling inside a sum(2 + 3)/(2 + 5), strike the 2s | 3/5 instead of 5/7 | Add first, then simplify |
Every number on this page was computed exactly and checked before publishing.
Or type 391/667 and get 17/29 back
Prism has a fraction mode that computes exactly and reduces every answer to lowest terms on its own, so 18/24 comes back as 3/4, not a decimal. Fractions are free forever, no subscription. See the app.