A repeating decimal never ends, but its digits fall into a pattern that repeats. The repeating block is often marked with a bar or with an ellipsis. Fractions such as 1/3 (0.333…) and 5/11 (0.4545…) produce repeating decimals, whereas 3/40 produces the terminating decimal 0.075.
A fraction in lowest terms terminates only when its denominator has no prime factors other than 2 and 5; otherwise it repeats.
Converting to a fraction
Count the digits n in the repeating block, multiply the number by 10ⁿ, and subtract the original so the repeating part cancels.
x = 0.4545…
100x = 45.4545…
100x − x = 45 → 99x = 45 → x = 45/99 = 5/11
Example
For 0.777…, one digit repeats: 10x − x = 7, so 9x = 7 and x = 7/9. The denominator is one nine per repeating digit, and the result is then reduced with the greatest common factor. The how to convert a decimal to a fraction guide covers terminating and repeating cases, and the decimal to fraction calculator does it automatically.
Common mistakes
- Treating a rounded display (0.3333) as exact. The finite decimal 0.3333 is 3333/10000, not 1/3.
- Forgetting to simplify, for example leaving 45/99 instead of 5/11.