Simplify Fractions Calculator
This calculator will simplify fractions to lowest terms. Type a fraction such as 42/56, a mixed number such as 3 6/8, or one whole number over another. It finds the greatest common factor, divides the top and the bottom by it, and shows the result as a decimal, a percent and, when the fraction is improper, a mixed number, along with a row of equivalent fractions.
42/56, 3 6/8 and 45/12 all work.How to use the simplify fractions calculator
- Type a fraction into the box. It opens with
42/56. - The simplified fraction appears straight away:
42/56 = 3/4. - Below it you see the greatest common factor the calculator used (14), and the division written out:
42 ÷ 14 = 3and56 ÷ 14 = 4. - Read the same value as a decimal (0.75) and a percent (75%).
- Check the row of equivalent fractions: 3/4, 6/8, 9/12, 12/16, 15/20. Every one of them, and 42/56, names the same number.
You can also type a mixed number such as 3 6/8, or an improper fraction such as 45/12. When the simplified fraction is bigger than 1, the calculator also gives it as a mixed number, so 45/12 comes back as 15/4 and as 3 3/4.
For the method behind the result, with three ways of finding the common factor by hand, read how to simplify fractions.
How to simplify fractions with the GCF
Simplifying a fraction means rewriting it with the smallest whole numbers possible while keeping its value. Dividing the numerator and the denominator by the same number never changes the value, because it is the same as dividing by n/n, which is 1. Dividing by the greatest common factor gets you to lowest terms in one move instead of several.
Finding the GCF
There are three ways to find it. Any of them works; they differ in speed.
- List the factors. Factors of 42: 1, 2, 3, 6, 7, 14, 21, 42. Factors of 56: 1, 2, 4, 7, 8, 14, 28, 56. The largest one on both lists is 14. Fine for small numbers.
- Prime factors.
42 = 2 × 3 × 7and56 = 2³ × 7. Take the primes they share, each at the lower power:2 × 7 = 14. The prime factorization calculator breaks down any number this way. - Euclid’s method. Divide the larger number by the smaller and keep the remainder, then repeat with the smaller number and the remainder until the remainder is 0. The last non-zero remainder is the GCF.
Euclid’s method is the one to use for large numbers, because it never needs you to factor anything. The GCF calculator shows both the prime and the Euclid working, and how to find the GCF and LCM covers the ladder method as well.
Worked examples
Example: 18/24
Both numbers are even and both have digit sums divisible by 3, so 6 divides both. Nothing larger does.
You could also halve first (9/12) and then divide by 3 (3/4). Same destination, one more step.
Example: 84/126 with Euclid’s method
Large enough that listing factors gets tedious.
Dividing by small numbers one at a time would take three steps here (by 2, then 3, then 7). Euclid needs two lines.
Example: an improper fraction, 45/12
As a decimal that is 3.75, or 375%. The improper fraction calculator goes into the mixed-number conversion in more detail.
Example: a hidden factor, 91/119
Neither number is even, neither digit sum is divisible by 3, and neither ends in 0 or 5. It looks finished. It is not.
Sevens, elevens and thirteens have no quick divisibility test, which is why fractions built on them slip past a glance. Euclid’s method catches them every time.
Reducing fractions by repeated division
Reducing fractions and simplifying fractions are the same job under two names. If you do not want to find the GCF first, divide by any common factor you can see, then look again, and keep going until nothing but 1 divides both numbers.
This route is slower but hard to get wrong, as long as you remember to check again after each division. The quick tests below tell you which small primes to try.
| Divides by | Test | Example |
|---|---|---|
| 2 | Last digit is even | 56 ends in 6 |
| 3 | Digit sum divides by 3 | 84 → 8 + 4 = 12 |
| 4 | Last two digits divide by 4 | 116 → 16 |
| 5 | Last digit is 0 or 5 | 45 |
| 9 | Digit sum divides by 9 | 126 → 1 + 2 + 6 = 9 |
| 10 | Last digit is 0 | 250 |
How to tell a fraction is already in lowest terms
A fraction is in lowest terms when the GCF of its numerator and denominator is 1. The numbers are then called coprime. A few signs tell you this without any working:
- The numerator is 1, as in 1/8.
- The numerator and denominator are consecutive whole numbers, as in 7/8 or 15/16. Two numbers that differ by 1 can share no factor larger than 1.
- The numerator is prime and does not divide the denominator, as in 11/24. The guide what is a prime number lists the primes up to 100.
Other cases need a check. 35/48 is in lowest terms because 35 = 5 × 7 and 48 = 2⁴ × 3 share no prime. But one odd number and one even number is not enough on its own: 91/119 is odd over odd and still reduces, and 15/42 is odd over even and still reduces by 3 to 5/14.
Simplified fractions reference table
| Fraction | GCF | Simplest form | Decimal |
|---|---|---|---|
| 2/4 | 2 | 1/2 | 0.5 |
| 4/6 | 2 | 2/3 | ≈ 0.6667 |
| 6/8 | 2 | 3/4 | 0.75 |
| 8/12 | 4 | 2/3 | ≈ 0.6667 |
| 9/12 | 3 | 3/4 | 0.75 |
| 10/15 | 5 | 2/3 | ≈ 0.6667 |
| 12/16 | 4 | 3/4 | 0.75 |
| 14/49 | 7 | 2/7 | ≈ 0.2857 |
| 15/25 | 5 | 3/5 | 0.6 |
| 16/24 | 8 | 2/3 | ≈ 0.6667 |
| 18/27 | 9 | 2/3 | ≈ 0.6667 |
| 20/32 | 4 | 5/8 | 0.625 |
| 24/36 | 12 | 2/3 | ≈ 0.6667 |
| 25/100 | 25 | 1/4 | 0.25 |
| 36/48 | 12 | 3/4 | 0.75 |
| 45/60 | 15 | 3/4 | 0.75 |
| 48/64 | 16 | 3/4 | 0.75 |
| 75/100 | 25 | 3/4 | 0.75 |
Many different-looking fractions land on 2/3 and 3/4. That is the point of simplifying: it gives every value one standard name, which makes answers easy to compare. The guide to equivalent fractions works the same idea in reverse, building up instead of reducing. For more decimal values, use the fraction to decimal calculator.
Ratios simplify by exactly the same rule. 42:56 reduces to 3:4 by dividing both parts by 14, and the ratio calculator does that for ratios with more than two parts.
Common mistakes
- Dividing only the top or only the bottom. That changes the value. Both must be divided by the same number.
- Subtracting instead of dividing. Taking 2 off the top and bottom of 6/8 gives 4/6, which is 2/3, not 3/4. Subtraction does not preserve the value.
- Stopping after one common factor. Dividing 24/36 by 2 gives 12/18, which is still reducible. Check again after every division, or use the GCF.
- Missing factors of 7, 11 and 13. They have no quick test. Use Euclid’s method when nothing obvious divides.
- Simplifying the whole part of a mixed number. In
3 6/8, only 6/8 reduces. The 3 stays, giving 3 3/4. - Cancelling digits. Striking the 6s in 16/64 happens to give 1/4, which is correct by coincidence. Striking the 2s in 12/24 gives 1/4, which is wrong; the answer is 1/2.
Simplifying answers is also the last step of every fraction sum and product, which the fraction calculator does for you automatically.
Frequently asked questions
What is 42/56 simplified?
3/4. The greatest common factor of 42 and 56 is 14, and 42 ÷ 14 = 3, 56 ÷ 14 = 4. You can confirm it with prime factors: 42 = 2 × 3 × 7 and 56 = 2³ × 7 share a 2 and a 7. As a decimal 3/4 is 0.75, or 75%.
What is 8/12 in simplest form?
2/3. Both 8 and 12 divide by 4, their greatest common factor, giving 8 ÷ 4 = 2 and 12 ÷ 4 = 3. If you halve first you get 4/6, which still reduces, so halve again to reach 2/3. As a decimal it is about 0.6667.
What is 6/8 simplified?
3/4. The greatest common factor of 6 and 8 is 2, so divide both by 2: 6 ÷ 2 = 3 and 8 ÷ 2 = 4. Since 3 and 4 share no factor other than 1, the fraction is now in lowest terms. Its decimal value is 0.75.
How do you simplify an improper fraction?
Exactly as you would a proper one: divide the top and bottom by their greatest common factor. 45/12 has a GCF of 3, giving 15/4. Then, if a mixed number is wanted, divide 15 by 4 to get 3 remainder 3, so the answer is 3 3/4. Simplify first; the division is easier with smaller numbers.
Is reducing fractions the same as simplifying fractions?
Yes. Reducing fractions, simplifying fractions and writing a fraction in lowest terms or simplest form all mean the same thing: dividing the numerator and denominator by their common factors until only 1 divides both. The value never changes, only how the fraction is written. 42/56 and 3/4 are the same number.
How do you simplify fractions with big numbers?
Use Euclid’s method to find the greatest common factor. Divide the larger number by the smaller, keep the remainder, and repeat with the smaller number and the remainder until the remainder is 0. For 84/126 the steps are 126 ÷ 84 remainder 42, then 84 ÷ 42 remainder 0, so the GCF is 42 and the answer is 2/3.
Can every fraction be simplified?
No. A fraction whose numerator and denominator share no factor other than 1 is already in lowest terms, such as 7/8, 11/24 or 35/48. Those cannot be reduced further. Every fraction does have exactly one simplest form, though, and many fractions that look finished, like 91/119, still hide a common factor.
Related calculators
Guides that use this calculator
Longer explanations of the method behind it, with worked examples.
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