Every decimal that terminates or repeats is a fraction in disguise. This calculator finds that fraction, reduces it to lowest terms and, when it is larger than 1, writes it as a mixed number too. It handles ordinary decimals such as 0.625 and repeating decimals such as 0.1666…, which you can type as 0.1(6).
How to use the converter
- Type the decimal. Negative numbers and numbers without a leading zero (.5) are fine.
- For a repeating decimal, put the repeating block in parentheses — 0.(3), 0.1(6), 2.(142857) — or end with three dots to repeat the final digit (0.333…).
- The tape shows the simplified fraction, the mixed number, the numerator and denominator, and a decimal check.
Terminating decimals: the place-value method
- Count the decimal places, k.
- Write the digits (without the point) over 10k.
- Simplify by the greatest common factor.
Example: 0.625. Three decimal places, so 0.625 = 625/1000. The GCF of 625 and 1000 is 125, and 625/1000 = 5/8.
Repeating decimals: the algebra method
- Call the decimal x.
- Multiply by a power of 10 that moves one full repeating block to the left of the point.
- Multiply by a power of 10 that moves just the non-repeating digits to the left of the point.
- Subtract the two equations — the endless tails cancel — and solve for x.
Example: 0.1(6) = 0.1666….
- x = 0.1666…
- 100x = 16.666…
- 10x = 1.666…
- Subtract: 90x = 15, so x = 15/90 = 1/6.
When the repeating block starts right after the point, step 3 is just x itself: for 0.(27), 100x = 27.2727…, and 100x − x = 27 gives x = 27/99 = 3/11.
Common decimals as fractions
| Decimal | Fraction | Decimal | Fraction |
|---|---|---|---|
| 0.5 | 1/2 | 0.(3) | 1/3 |
| 0.25 | 1/4 | 0.(6) | 2/3 |
| 0.75 | 3/4 | 0.1(6) | 1/6 |
| 0.2 | 1/5 | 0.8(3) | 5/6 |
| 0.125 | 1/8 | 0.(142857) | 1/7 |
| 0.0625 | 1/16 | 0.(1) | 1/9 |
Rounded decimals versus exact ones
A decimal that has already been rounded, such as 0.33, is treated exactly as typed: 0.33 = 33/100, not 1/3. If you mean one-third, enter 0.(3). Likewise 3.14 converts to 157/50, not to π, which is not a fraction at all. For a decimal you measured, consider whether the nearest simple fraction (like 1/3 or 5/16) is what you really want — the fraction to decimal chart helps you spot it.
To go the other way, use the fraction to decimal calculator; for percentages, the percent to fraction calculator applies the same idea with a denominator of 100.
Frequently asked questions
How do you convert a decimal to a fraction?
Count the digits after the decimal point and write the number without its point over 10 raised to that count. Then simplify. 0.625 has three decimal places, so it is 625/1000, which reduces to 5/8.
How do I enter a repeating decimal?
Put the repeating digits in parentheses: 0.(3) means 0.333…, 0.1(6) means 0.1666… and 2.(142857) means 2.142857142857…. You can also type 0.333... with three dots, and the last digit is treated as repeating.
How do you turn a repeating decimal into a fraction?
Use algebra. Let x = 0.1666…. Multiply by 100 to get 100x = 16.666… and by 10 to get 10x = 1.666…. Subtracting removes the repeating tail: 90x = 15, so x = 15/90 = 1/6.
Is 0.999… really equal to 1?
Yes. With x = 0.999…, 10x = 9.999…, and subtracting gives 9x = 9, so x = 1. The calculator returns 1 for 0.(9). There is no number between 0.999… and 1, so they are the same number.