Guide · Recurring decimals
Repeating Decimal to Fraction
Updated: June 2026
A repeating decimal looks like it should be impossible to pin down — the digits run on forever. Yet every recurring decimal is a perfectly ordinary fraction. There is a short formula that converts any of them exactly, and the logic behind it is worth seeing once because it makes the formula impossible to forget.
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The algebra trick
Suppose x = 0.3333… The repeating block is one digit long, so multiply by 10 to line the decimals up, then subtract the original:
10x = 3.333…
10x − x = 3
9x = 3 → x = 3/9 = 1/3
The subtraction wipes out the infinite tail entirely, leaving a clean whole number. That single idea powers every repeating-to-fraction conversion.
The shortcut formula
You do not have to redo the algebra each time. For a decimal that is purely repeating from the first digit, put the repeating block over as many nines as it has digits:
0.(27) = 27/99 = 3/11
0.(142857) = 142857/999999 = 1/7
One digit, one nine; two digits, two nines; six digits, six nines. Then simplify. That last example is a lovely surprise: the ungainly-looking 0.(142857) is simply one seventh.
When digits come before the repeat
Many decimals have a non-repeating part first, like 0.16666… Here you also need zeros. The rule: the numerator is the whole string up to the end of the first repeat, minus the non-repeating part; the denominator is one nine per repeating digit followed by one zero per non-repeating digit.
0.58(3): (583 − 58) / 900 = 525/900 = 7/12
Count carefully: in 0.1(6) there is one non-repeating digit (the 1) and one repeating digit (the 6), giving a denominator of 90 — one nine, one zero.
Why nines, exactly?
Multiplying by ten for each repeating digit and subtracting is what produces the nines. Two repeating digits means multiplying by 100; 100x − x leaves 99x, hence the denominator 99. The zeros appear when you first have to shift past the non-repeating digits before the subtraction lines up. Seeing it this way means you can always rebuild the formula from scratch if you forget it.
A reference table
| Repeating decimal | Fraction |
|---|---|
| 0.(1) | 1/9 |
| 0.(3) | 1/3 |
| 0.(6) | 2/3 |
| 0.(9) | 1 |
| 0.0(3) | 1/30 |
| 0.1(6) | 1/6 |
| 0.(09) | 1/11 |
Yes, 0.(9) really does equal 1 — the formula gives 9/9, and there is no number squeezed between 0.999… and 1. It is one of the cleanest demonstrations that repeating decimals are genuine, exact numbers rather than approximations.
Frequently asked questions
What is 0.333… as a fraction?
It is 1/3. The single repeating digit 3 goes over one nine, giving 3/9, which reduces to 1/3.
How do I convert 0.1666… to a fraction?
Use (16 − 1)/90 = 15/90 = 1/6, because there is one non-repeating digit and one repeating digit.
Does 0.999… equal 1?
Yes. The formula gives 9/9 = 1, and there is no number between 0.999… and 1, so they are equal.