Repeating Decimal to Fraction Converter
Convert a pure or mixed repeating decimal to an exact simplified fraction. Parentheses identify the recurring digits, so the tool never guesses which part repeats.
Written and checked by TheToolNet Editorial Team · Last reviewed: September 20, 2026 · Editorial method
Convert a repeating decimal
Why the subtraction method works
Multiplying by a suitable power of ten moves one complete repeating block to the left of the decimal. A second multiplication aligns the repeating tails. Subtracting the equations removes those identical infinite tails and leaves a finite equation.
For 0.1(6), let x = 0.1666…. Then 100x = 16.666… and 10x = 1.666…. Subtract: 90x = 15, so x = 15/90 = 1/6.
Pure and mixed recurring decimals
| Type | Example | Fraction |
|---|---|---|
| Pure recurring | 0.(3) | 1/3 |
| Mixed recurring | 0.1(6) | 1/6 |
| Whole plus recurring | 2.(47) | 245/99 |
| Recurring nines | 0.(9) | 1 |
Frequently asked questions
Is 0.333 a repeating decimal?
Not by itself. It is a finite decimal unless a bar, dots, parentheses, or trailing ellipsis says the 3 continues forever.
Can the converter handle negative values?
Yes. Put the minus sign before the number, as in −0.(8).
Where can I convert ordinary fractions?
Use the fraction and decimal converter.