Decimal to Fraction Calculator
Turn terminating or repeating decimals into simplified fractions, with steps.
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How to Turn a Decimal into a Fraction
Converting a terminating decimal is a three-step job:
- Write the decimal over 1. For 0.75, write 0.75 / 1.
- Multiply top and bottom by 10 for each digit after the decimal point. 0.75 has two digits, so multiply by 100: 75 / 100.
- Simplify. Divide the top and bottom by their greatest common factor. 75 / 100 simplifies to 3 / 4.
More examples: 0.5 is 5 / 10, which simplifies to 1 / 2. 0.125 is 125 / 1000, which simplifies to 1 / 8. 0.2 is 2 / 10, which simplifies to 1 / 5. Once you know the pattern, you can do most of them in your head.
Repeating Decimals
A repeating decimal never ends: 0.333..., 0.1666..., 0.142857142857... You cannot write it over a power of 10, because there is no last digit. Instead, use a small algebra trick.
Let x = 0.333... Then 10x = 3.333... Subtract the first equation from the second: 10x - x = 3.333... - 0.333..., so 9x = 3, and x = 3 / 9 = 1 / 3. The repeating part cancels out, and you are left with a normal fraction.
For a longer repeat, use a bigger power of 10. For 0.142857 repeating, the repeating block has 6 digits, so multiply by 1,000,000. The same subtraction trick then gives 142857 / 999999, which simplifies to 1 / 7. The scientific notation calculator is the companion tool when your decimals get very small or very large.
Terminating or Repeating? How to Tell
Whether a fraction gives a terminating or a repeating decimal depends on its denominator. Write the fraction in lowest terms and look at the prime factors of the denominator:
- Only 2s and 5s: the decimal terminates. 3 / 8 = 0.375, 7 / 20 = 0.35.
- Any other prime factor: the decimal repeats. 1 / 3 = 0.333..., 5 / 6 = 0.8333...
This is useful in reverse. If a decimal terminates, you know its fraction has a denominator made only of 2s and 5s. That is why 0.75 is 3 / 4 (denominator 4 = 2²) and 0.2 is 1 / 5.
Mixed Numbers and Negative Decimals
For a decimal larger than 1, split it into a whole part and a decimal part. For 2.75, the whole part is 2 and the decimal part is 0.75, which is 3 / 4. So 2.75 is 2 and 3 / 4, or 11 / 4 as an improper fraction.
For a negative decimal, convert the positive version first and then put the minus sign back. -0.6 becomes -3 / 5. The sign never changes the method — it only changes the answer at the end.
Fractions are exact, while decimals are often rounded. That is why maths classes prefer fractions: 1 / 3 is exact, but 0.333 is only an approximation. When an answer must be exact, convert to a fraction.
Practice With Our Free Tool
Use the decimal to fraction calculator above to check your homework. Enter any decimal — terminating or repeating — and see the simplified fraction at once. For graphing work, the function graphing calculator plots the functions behind these numbers. Everything is free, with no download and no sign-up.
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Frequently Asked Questions
Write the decimal over 1, then multiply the top and bottom by 10 for each digit after the decimal point. Then simplify the fraction. For 0.75, write 0.75 over 1, multiply by 100 to get 75 over 100, and simplify to 3 over 4.
Let x equal the repeating decimal. Multiply x by a power of 10 that moves one full repeating block past the decimal point. Then subtract the original x. The repeating part cancels, leaving a normal equation. For 0.333..., let x = 0.333..., then 10x = 3.333..., and 10x - x = 3, so 9x = 3 and x = 1 over 3.
0.125 is 1 over 8. Write 0.125 over 1, multiply the top and bottom by 1000 to get 125 over 1000, then divide the top and bottom by 125 to simplify. The result is 1 over 8.
A terminating decimal is a decimal that ends, like 0.25 or 0.8. It comes from a fraction whose denominator, in lowest terms, has only the prime factors 2 and 5. A repeating decimal never ends, like 0.333... or 0.1666..., and it comes from a fraction with other prime factors in the denominator.
Yes. The decimal to fraction calculator is free to use online. There is no download and no sign-up. You can convert as many decimals as you want, right in your browser.