# Decimal to Fraction Calculator - Repeating Decimals & Steps

> Convert terminating and repeating decimals (0.1(6), 0.[142857], 0.1666…) to fractions with steps, and fractions or percents to decimals with the repetend shown.

- URL: https://qilingtools.com/en/math/decimal-fraction
- Category: Math Tools (https://qilingtools.com/en/math)
- Tags: [Math](https://qilingtools.com/en/tags/math), [Education](https://qilingtools.com/en/tags/education), [Converters](https://qilingtools.com/en/tags/convert)
- Last updated: 2026-10-26
- Price: Free, no sign-up
- Privacy: Runs locally in the browser; files are never uploaded

Type any number and the converter works out whether it is a decimal, a fraction or a percent, then shows every other form. Decimal to fraction: a terminating decimal is written over a power of ten and reduced (0.375 = 375/1000 = 3/8). A repeating decimal can be typed as 0.1(6), 0.[142857], with an overline like 0.16̅, or simply as 0.1666… — the repetend is detected from the repeated digits at the end. The steps use the classic trick of multiplying by powers of ten and subtracting so the endless tail cancels (100x − 10x), then reduce to lowest terms. Fraction to decimal: long division tracks the remainders; the first repeated remainder marks the start of the cycle, and the result is shown with an overline, in bracket notation and with an ellipsis. It also explains why a denominator with only 2s and 5s gives a terminating decimal, and how many digits come before the cycle. Mixed numbers (1 3/8) and percents (12.5%, 33.(3)%) work too, and repetends thousands of digits long (like 1/99991) are handled exactly.

## How to use

1. Type a decimal (0.375, 0.1(6), 0.1666…), a fraction (22/7, 1 3/8) or a percent (12.5%), or pick an example
2. Read the fraction in lowest terms, the mixed number, the overlined decimal, the percent and the type of decimal
3. Follow the worked steps and the long-division remainders, and open the long division tool if you want the full layout

## Features

- Four ways to enter a repeating decimal: brackets, square brackets, overline or dots, and a trailing ellipsis
- Full derivation for repeating decimals: let x, multiply by powers of ten, subtract, reduce
- Fraction to decimal with a table of long-division remainders and the cycle start marked
- Overline, bracket and ellipsis notation, each ready to copy
- Classifies terminating, purely repeating and eventually repeating decimals with repetend length
- Negative numbers, mixed numbers and percents (including repeating percents)
- Exact BigInt arithmetic with repetends up to 20,000 digits
- Also shows the value rounded to 10 decimal places

## FAQ

### How do you convert a repeating decimal to a fraction?

Let x be the decimal. Multiply by a power of ten to move the non-repeating digits in front of the point, then by another power so one full repetend is in front too, and subtract — the infinite tails cancel. For x = 0.1(6): 10x = 1.(6) and 100x = 16.(6), so 90x = 15 and x = 15/90 = 1/6. A purely repeating decimal is quicker: 0.(142857) = 142857/999999 = 1/7, with as many 9s as the repetend has digits.

### Which fractions give terminating decimals?

Reduce the fraction and factor the denominator. If its only prime factors are 2 and 5 (8 = 2³, 20 = 2² × 5), the decimal terminates, and the number of decimal places is the larger of the two exponents. Any other prime factor (3, 7, 12 = 2² × 3) makes it repeat forever. With no 2s or 5s the decimal is purely repeating (1/7); with some it is eventually repeating (1/6 = 0.1(6)), and the non-repeating part again has as many digits as the larger exponent.

### Is 0.999… really equal to 1?

Yes. Using the same method, x = 0.(9) gives 10x = 9.(9), so 9x = 9 and x = 1. Another way to see it: 1/3 = 0.333…, and multiplying both sides by 3 gives 1 = 0.999…. Type 0.(9) or 0.999… here and you get exactly 1. In the same way 0.4(9) = 0.5 — any decimal ending in endlessly repeating 9s equals a terminating decimal.

### How does the tool guess the repetend of 0.1666…?

It looks at the digits before the ellipsis for the shortest block that repeats at least twice in full (a single digit must appear three times), then extends the repeating part as far left as it can: 0.1666… gives 6 and 0.142857142857… gives 142857. Because this is a guess, the page tells you what it assumed. If there are too few digits, or you mean a different repetend, mark it with brackets, e.g. 0.1(6) or 0.(142857).

## Related tools

- [Graphing Calculator](https://qilingtools.com/en/math/graph): Free online function graphing calculator: type sin(x), x^2 or 1/x and see the curve at once, with several functions, parametric and polar curves, PNG export.
- [Scientific Calculator](https://qilingtools.com/en/math/calculator): Free online scientific calculator: type 2*(3+4)^2, sin(pi/6) or log(8,2) and see the answer as you type, with degrees or radians and a history list.
- [Equation Solver](https://qilingtools.com/en/math/equation): Free online equation solver: enter coefficients for linear, quadratic (with discriminant, vertex and complex roots) and cubic equations, plus 2×2 systems.
- [Fraction Calculator](https://qilingtools.com/en/math/fraction): Free online fraction calculator: add, subtract, multiply and divide fractions with automatic reduction, plus fraction ↔ decimal with repeating digits.
- [Percentage Calculator](https://qilingtools.com/en/math/percentage): Free online percentage calculator: X% of Y, X as a percent of Y, increase and decrease, a value after a percentage change and the original behind it.
- [Prime Factorization](https://qilingtools.com/en/math/prime): Free online prime factorization: factor an integer with exponents, list every divisor and their sum, test primality with neighbouring primes, or list a range.
- [GCD and LCM](https://qilingtools.com/en/math/gcd-lcm): Free online GCD and LCM calculator: works with two or more integers, says whether they are coprime, shows each prime factorization and every Euclidean step.
- [Statistics Calculator](https://qilingtools.com/en/math/statistics): Free online statistics calculator: paste a data set for the mean, median, mode, range, quartiles, variance, standard deviation, standard error and outliers.