Simplify 12/16
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12/1612 is 2² · 3, 16 is 2⁴. They share 2², so the greatest common factor is 4: divide both parts to get 3/4. Dividing by 2 instead would have given 6/8, which is correct but not finished - 6 and 8 still share a 2.
Simplify Fractions Calculator
Reduce any fraction to lowest terms, with the common factor found rather than assumed.
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Want to solve these without the calculator?
The Coddy math course teaches the method itself - you work each step on an interactive board and get told exactly where a move went wrong.
Simplifying a fraction is one division, done to both parts
A fraction does not change value when both of its parts are multiplied or divided by the same number - 12/16, 6/8 and 3/4 are one quantity written three ways. Simplifying means dividing top and bottom by everything they have in common, until they have nothing left in common. The number you divide by is their greatest common factor, and it is the only thing you need to find.
Most calculators print the reduced fraction and stop, which hides the interesting half. Here both parts are broken into primes first: 12 is 2² · 3 and 16 is 2⁴, they share 2², so the factor is 4. That is why 4 and not 2 and not 8 - and once you have seen it two or three times you stop needing the calculator, which is the point.
Everything is exact. 1/3 is reported as 0.333… with the repeating part marked rather than as 0.3333333333, one sixth as a percentage is 16⅔% rather than 16.67%, and a mixed number keeps its sign on the whole quantity, so -2 3/4 is -11/4 and never -5/4.
What to notice
- Lowest terms means the two parts share no factor above 1. That is the finishing condition - not "the numbers look small".
- The factor to divide by is the greatest common factor. Dividing by a smaller shared factor works too, it just takes more than one round: 12/16 by 2 gives 6/8, which still needs another 2.
- Reducing a fraction and finding a GCF are the same problem. If you can do one, you can do the other - the two calculators here walk the same route on purpose.
- A fraction whose parts are coprime is already done. 7/9 cannot be simplified, and being told so is an answer rather than a failure.
- Simplifying never changes the decimal or the percentage, because it never changes the value. That is a free check on your own work: if the decimal moved, you divided only one part.
How to simplify a fraction
Deal with a mixed number first
If there is a whole number in front, fold it in: multiply the whole by the denominator and add the numerator. 2 6/8 becomes (2 · 8 + 6)/8 = 22/8. The sign, if any, belongs to the whole quantity.
Break both parts into prime factors
22 is 2 · 11 and 8 is 2³. Writing them out is what makes the shared factors visible instead of guessed at.
Multiply the shared primes together
Take each prime both numbers have, as many times as they both have it. Here that is a single 2, so the greatest common factor is 2.
Divide both parts by it
22 ÷ 2 = 11 and 8 ÷ 2 = 4, so the fraction is 11/4. Check the finish: 11 and 4 share nothing, so this is lowest terms.
Write it the way the question wants
11/4 is more than one whole, so it can also be written as the mixed number 2 3/4 - the same value, and usually the expected form when the question started with one.
Fractions in lowest terms
The same eight quantities in every costume: the fraction as given, the factor that comes out, the reduced form, the exact decimal, and the exact percentage.
| Fraction | GCF | Lowest terms | Decimal | Percent |
|---|---|---|---|---|
| 12/16 | 4 | 3/4 | 0.75 | 75% |
| 8/12 | 4 | 2/3 | 0.666… | 66⅔% |
| 84/126 | 42 | 2/3 | 0.666… | 66⅔% |
| 15/25 | 5 | 3/5 | 0.6 | 60% |
| 45/60 | 15 | 3/4 | 0.75 | 75% |
| 18/6 | 6 | 3 | 3 | 300% |
| 7/9 | 1 | 7/9 | 0.777… | 77⁷⁄₉% |
| 2 6/8 | 2 | 2 3/4 | 2.75 | 275% |
Worked examples
A fraction that is already done: 7/9
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7/97 is prime and 9 is 3², so they share nothing above 1 and the fraction cannot be simplified. The page says so rather than pretending to work: "already in lowest terms" is the answer to the question that was asked.
A big one: 84/126
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84/12684 is 2² · 3 · 7 and 126 is 2 · 3² · 7. Shared: one 2, one 3, one 7 - so the factor is 42, and the fraction collapses straight to 2/3. Spotting 42 by eye is hard; reading it off the two factorisations is not.
A mixed number: 2 6/8
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2 6/8Fold it first: 2 · 8 + 6 = 22, so 22/8. Then reduce by 2 to reach 11/4, and write it back as 2 3/4. The whole number did not need touching - only the fraction part was ever reducible.
One that becomes a whole number: 18/6
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18/6The greatest common factor is 6, and dividing both parts gives 3/1 - which is just 3. A bar over nothing reads as a mistake, so the answer is written as a plain 3.
Common mistakes
- Dividing only one part. 12/16 is not 3/16. Whatever you do to the top, you must do to the bottom, or you have changed the value.
- Stopping too early. 6/8 is a correct simplification of 12/16 and still not the answer - most marking wants lowest terms.
- Subtracting instead of dividing. 12/16 is not 8/12 "because I took 4 off both". Fractions are ratios; only multiplying and dividing leave them alone.
- Cancelling across an addition. In (4 + 6)/2 you cannot cancel the 4 with the 2 - the 2 divides the whole numerator, so the sum comes first.
- Putting a mixed number's sign on the fraction part. -2 3/4 means -(2 + 3/4) = -11/4. Reading it as -2 + 3/4 gives -5/4, which is a different number entirely.
- Rounding on the way. If a decimal or percentage is wanted, get the exact fraction first: 1/3 written as 0.33 is no longer one third.
Simplifying fractions FAQ
How do I simplify a fraction?
Divide the numerator and the denominator by their greatest common factor. For 12/16 that factor is 4, giving 3/4. If you cannot see the factor, break both numbers into primes and multiply the ones they share - that product IS the greatest common factor.
What does "lowest terms" or "simplest form" mean?
It means the numerator and denominator have no common factor above 1. 3/4 is in lowest terms; 6/8 is not, because both parts are still even. The two phrases mean exactly the same thing, and so does "fully reduced".
How do I know when a fraction cannot be simplified?
When the greatest common factor of the two parts is 1 - the numbers are then called coprime. A quick check: if the numerator is prime and does not divide the denominator, you are already done. 7/9, 5/12 and 13/20 are all finished as they stand.
Can this simplify mixed numbers and improper fractions?
Yes. A mixed number is folded into a single fraction first (2 6/8 becomes 22/8), reduced, and then written back as a mixed number if it is bigger than one whole. Improper fractions are handled the same way, and you are given both forms of the answer.
Why does the answer show a repeating decimal instead of a rounded one?
Because they are different numbers. One third is 0.333… forever, and 0.333 is not one third - it is smaller by one three-thousandth. Marking the repeating part is the only way to write the value truthfully, which is why the page does that instead of rounding.
Is simplifying a fraction the same as finding the GCF?
Practically, yes - the whole job is finding the greatest common factor and then doing one division with it. That is why this page shows both factorisations, and why the GCF calculator here walks the same route: they are one skill with two names.
Do I have to simplify my answers?
Almost always, yes. An unsimplified fraction is not wrong in value, but most exam schemes and textbooks award the mark for lowest terms, and comparing fractions is far easier once both are reduced. The exception is a question that asks for a specific denominator.
