➗

Repeating Decimal to Fraction Calculator

Convert a repeating (recurring) decimal such as 2.1363636… to a fraction in lowest terms and a mixed number, with the multiply-and-subtract steps shown.

Decimal before the repeating part

Repeating digits

Loading calculator…

How it works

x = 0.a(b) with n non-repeating and k repeating digits: (10ⁿ⁺ᵏ − 10ⁿ)·x = (all digits as an integer) − (non-repeating digits as an integer) · simplify by the GCF

Every repeating decimal is a rational number. Multiplying x by 10ⁿ⁺ᵏ and by 10ⁿ produces two numbers with exactly the same infinite tail after the decimal point, so subtracting them removes the tail and leaves a whole number. The result is a fraction whose denominator is a string of 9s (one per repeating digit) followed by 0s (one per non-repeating decimal digit) — for example 0.(36) = 36/99 and 0.1(6) = 15/90. The fraction is then reduced to lowest terms with the greatest common factor.

Worked example

Convert 2.1(36) = 2.1363636… to a fraction. Let x = 2.1363636…. Then 10x = 21.363636… and 1000x = 2136.363636…. Subtracting, 990x = 2136 − 21 = 2115, so x = 2115/990. Dividing by the GCF 45 gives 47/22, or the mixed number 2 3/22.

Sources: Repeating decimal — Wikipedia · Converting repeating decimals to fractions — Math is Fun

Frequently asked questions

How do I convert 0.333… to a fraction?

Enter 0 as the non-repeating part and 3 as the repeating digits. x = 0.333…, 10x = 3.333…, so 9x = 3 and x = 1/3.

Is 0.999… really equal to 1?

Yes. With x = 0.999…, 10x = 9.999…, so 9x = 9 and x = 1. Two different decimal expansions can name the same number.

What is the quick rule for pure repeating decimals?

Put the repeating block over as many 9s as it has digits: 0.(142857) = 142857/999999 = 1/7, and 0.(45) = 45/99 = 5/11.

Which fractions give repeating decimals?

A fraction in lowest terms terminates only if its denominator has no prime factors other than 2 and 5. Any other prime factor (3, 7, 11 …) makes the decimal repeat forever.

Related calculators

Decimal to Fraction Calculator

Turn any decimal into a simplified fraction (and mixed number), with denominators up to 10,000.

Fraction Simplifier Calculator

Reduce any fraction or mixed number to lowest terms with the steps shown three ways — Euclid’s GCF, prime factorization or dividing by common factors one at a time — plus the mixed number, decimal (repeating digits marked), percent and equivalent fractions.

Fraction to Decimal Calculator

Convert a fraction or mixed number to a decimal with the repeating digits marked — 1/3 = 0.(3), 5/6 = 0.8(3) — rounded to any number of places, as a percent, with the long division worked out step by step.

Continued Fraction Calculator

Expand a decimal into its continued fraction and read off the best rational approximations (convergents) to the value.

Percent to Fraction Calculator

Convert a percentage to a fraction in simplest form, a mixed number and a decimal, with the steps. Works with decimal percents like 35.5%, repeating percents like 33.3…% (1/3) and percents over 100%, or finds the nearest simple fraction.

This calculator is for educational purposes. Double-check important results before acting on them.