How to use the fraction calculator
- Enter the top (numerator) and bottom (denominator) of each fraction. For a mixed number such as 2 ½, also fill in the small whole box.
- Choose +, −, × or ÷ between them. The answer is shown in lowest terms, as a mixed number when it is bigger than one, as a decimal and as a percentage.
- Read the numbered steps underneath to see how the answer was reached, which is useful for checking homework or learning the method.
The other two tabs simplify a single fraction and turn a decimal into a fraction. All arithmetic uses exact whole numbers, so results never suffer from rounding, even for very large fractions.
The four operations
| Operation | Rule | Example |
|---|---|---|
| Add | Rewrite both fractions over a common denominator, then add the tops | 2/3 + 3/4 = 8/12 + 9/12 = 17/12 |
| Subtract | Same common denominator, then subtract the tops | 5/6 − 1/4 = 10/12 − 3/12 = 7/12 |
| Multiply | Multiply the tops together and the bottoms together | 3/4 × 2/5 = 6/20 = 3/10 |
| Divide | Multiply by the second fraction turned upside down | 3/4 ÷ 2/5 = 3/4 × 5/2 = 15/8 |
For addition and subtraction the tool uses the lowest common denominator, which is the lowest common multiple of the two bottoms. Using the product of the bottoms also works, but gives bigger numbers and needs more simplifying afterwards.
Simplifying fractions
A fraction is in lowest terms when the top and bottom share no common factor other than 1. To get there, divide both by their greatest common divisor (GCD), which the tool finds with Euclid's algorithm: repeatedly replace the larger number with the remainder after dividing by the smaller one, until the remainder is zero.
GCD(18, 24): 24 = 1 × 18 + 6, then 18 = 3 × 6 + 0, so the GCD is 6 18/24 = (18 ÷ 6) / (24 ÷ 6) = 3/4
Simplifying never changes the value. 18/24, 9/12 and 3/4 are the same number written in different sizes.
Worked examples
- 2/3 + 3/4: the lowest common denominator of 3 and 4 is 12. 2/3 = 8/12 and 3/4 = 9/12, so the sum is 17/12, which is the mixed number 1 5/12, or 1.41667 as a decimal.
- 2 1/2 × 1 1/3: as improper fractions this is 5/2 × 4/3 = 20/6 = 10/3, which is 3 1/3.
- 3/4 ÷ 2/5: 3/4 × 5/2 = 15/8 = 1 7/8 = 1.875.
- Decimal to fraction: 0.375 is 375/1000. The GCD of 375 and 1000 is 125, so it simplifies to 3/8. In the same way 0.125 is 1/8 and 2.5 is 5/2.
- Fraction to decimal: 1/7 is 0.(142857), with the digits in brackets repeating forever. 1/6 is 0.1(6).
Common fraction and decimal equivalents
| Fraction | Decimal | Percent |
|---|---|---|
| 1/2 | 0.5 | 50% |
| 1/3 | 0.(3) | 33.33% |
| 1/4 | 0.25 | 25% |
| 1/5 | 0.2 | 20% |
| 1/6 | 0.1(6) | 16.67% |
| 1/8 | 0.125 | 12.5% |
| 1/10 | 0.1 | 10% |
| 3/4 | 0.75 | 75% |
| 2/3 | 0.(6) | 66.67% |
| 5/8 | 0.625 | 62.5% |
A fraction in lowest terms ends in a plain decimal exactly when its denominator has no prime factors other than 2 and 5. That is why 1/8 (2³) and 1/20 (2² × 5) terminate, while 1/3, 1/6 and 1/7 repeat.
Fractions in everyday measurements
| Inch fraction | Decimal inches | Millimetres |
|---|---|---|
| 1/16 | 0.0625 | 1.59 |
| 1/8 | 0.125 | 3.18 |
| 3/16 | 0.1875 | 4.76 |
| 1/4 | 0.25 | 6.35 |
| 3/8 | 0.375 | 9.53 |
| 1/2 | 0.5 | 12.7 |
| 5/8 | 0.625 | 15.88 |
| 3/4 | 0.75 | 19.05 |
| 7/8 | 0.875 | 22.23 |
Cooking and carpentry still depend on fractions. Doubling a recipe that needs 3/4 cup of flour needs 1 1/2 cups, and a board cut to 5 3/8 inches plus another 2 3/4 inches adds up to 8 1/8 inches. Use the Calculate tab for sums like these, entering whole numbers in the small boxes. A tape measure divides each inch into sixteenths, so the sixteenth, eighth, quarter and half are the fractions you meet most often.
Accuracy and limits
The calculator works with exact integer arithmetic (up to 40 digits per number), so results are exact, not rounded. Decimals that repeat are shown with the repeating block in brackets, up to 60 digits. Negative mixed numbers take their sign from the whole part, so enter −2 with 1/2 for −2½. Square roots, exponents and equations with unknowns are not supported; use the scientific calculator for those.
Frequently asked questions
How do I add fractions with different denominators?
Find a common denominator, rewrite each fraction over it, then add the tops. For 2/3 + 3/4, use 12: 8/12 + 9/12 = 17/12.
How do I divide fractions?
Flip the second fraction upside down and multiply. 3/4 ÷ 2/5 becomes 3/4 × 5/2 = 15/8.
How do I turn an improper fraction into a mixed number?
Divide the top by the bottom. The quotient is the whole part and the remainder goes over the original bottom. For 17/12: 17 ÷ 12 = 1 remainder 5, so 1 5/12.
Can the calculator handle mixed numbers?
Yes. Put the whole part in the small box, with the fraction part beside it. A negative mixed number takes its minus sign on the whole part.
Why does a decimal show brackets such as 0.(3)?
The digits inside the brackets repeat forever. 1/3 is 0.333…, written as 0.(3). It is the exact value, since a plain rounded decimal would be only approximate.