Decimal to fraction calculator
This decimal to fraction calculator converts any decimal into an exact fraction in lowest terms — including a repeating decimal, solved algebraically rather than rounded. Typing 0.75 returns 3/4. Typing 0.(3) — the notation for 0.333... repeating forever — returns exactly1/3, not a rounded guess.
- Multiply to align the repeating block10x − 1x
- Subtract to cancel the repeating digits9x = 3
- Solve for xx = 3/9
- Reduce to lowest terms1/3
How to convert a decimal to a fraction
A terminating decimal is built from place value: each digit after the point is a tenth, a hundredth, a thousandth, and so on. Writing the decimal over the matching power of 10, then reducing, is the whole method.
- Write the decimal over the power of 10 that matches its last place.
0.75has two digits after the point — hundredths — so it becomes75/100. - Reduce to lowest terms:
3/4.
A repeating decimal has no last place, so that method never finishes. The algebraic method instead setsx equal to the decimal, multiplies by two powers of 10 far enough apart to line the repeating block up with itself, then subtracts — the repeating part cancels, leaving a plain equation to solve for x.
Worked examples
A terminating decimal
0.375 has three digits after the point — thousandths — so it's written over 1000: 375/1000. Reducing that gives 3/8.
A repeating decimal with a non-repeating start
0.1(6) means 0.1666... — the 1 appears once, then the 6 repeats forever. Let x = 0.1666.... Multiplying by100 and by 10, then subtracting one from the other, cancels the repeating part and leaves a plain equation to solve — the calculator above shows every step for this exact value. Solving it reduces to 1/6.
A six-digit repeating block
The same algebra works whatever the block's length — it just needs a bigger power of 10 to line the block up with itself. 0.(142857) reduces all the way down to 1/7, the same fraction this site's fraction-to-decimal calculator expands the other way.
Common values
Click a decimal to load it into the calculator above.
| Decimal | Fraction |
|---|---|
| 1/2 | |
| 1/4 | |
| 3/4 | |
| 1/5 | |
| 4/5 | |
| 1/8 |
Where people get it wrong
Rounding a repeating decimal before converting it bakes the rounding error into the fraction. 1/3 is exactly 1/3, but truncating it to 0.333 and converting that gives 333/1000 — a real fraction, just not the one 0.333... actually equals. Typing the repeating notation directly, rather than a rounded-off approximation, is what keeps the answer exact.
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Frequently asked questions
How do you convert a decimal to a fraction?
Write it over the matching power of 10 — tenths over 10, hundredths over 100, thousandths over 1000 — then reduce to lowest terms. A repeating decimal needs a different method, since it never reaches a last digit to write over a power of 10.
What does the notation 0.1(6) mean?
The digits in parentheses repeat forever. 0.1(6) means 0.1666..., where the 6 never stops. It is the same notation this site's fraction-to-decimal calculator displays a repeating result in, so a result copied from there can be pasted straight back in here.
Why can't you just round a repeating decimal and convert that?
Rounding first bakes an error into the fraction before the conversion even starts. 0.333, truncated from 1/3, converts to 333/1000 — a real fraction, but not the one 0.333... actually equals. The algebraic method used here solves for the exact value instead.
Does this calculator accept repeating decimals as input?
Yes — type the repeating notation directly, like 0.(3) or 0.1(6), and it solves the exact fraction algebraically rather than truncating.
Is every terminating decimal a fraction with a denominator that is a power of 10?
Before reducing, yes — that is exactly what writing it over the matching power of 10 means. After reducing to lowest terms, the denominator can end up being anything, once the common factors with the numerator cancel.