Convert any repeating (recurring) decimal into an exact fraction. Put the repeating digits in brackets, like 0.1(6) for 0.1666…, or type the decimal with dots and let the tool find the pattern. You get the simplified fraction, mixed number and the algebra behind it.
Last updated:
Input style
Put the repeating digits in brackets: 0.(3), 2.1(45), −0.(142857). You can also type 0.1666... and the pattern is detected.
Digits after the point before the repeat (may be empty)
Enter your values to see the result.
Advertisement
Advertisement
How to use the Repeating Decimal to Fraction
Type the decimal with the repeating digits in brackets, for example 0.1(6) or 3.(142857).
Or switch to separate parts and enter the whole number, non-repeating and repeating digits.
Press Convert; results also update as you type.
Read the simplified fraction and mixed number.
Follow the steps and the back-conversion check.
The algebra method
For a decimal x with a non-repeating digits and a repeating block of b digits:
x = (all digits up to the end of the first block − the digits before the block) ÷ (10ᵃ × (10ᵇ − 1))
The denominator is b nines followed by a zeros. Example: 2.1(45) → (2145 − 21) ÷ 990 = 2124/990 = 118/55.
Common repeating decimals
Decimal
Fraction
Decimal
Fraction
0.(1)
1/9
0.1(6)
1/6
0.(3)
1/3
0.8(3)
5/6
0.(6)
2/3
0.08(3)
1/12
0.(09)
1/11
0.(142857)
1/7
0.(18)
2/11
0.(285714)
2/7
0.(27)
3/11
0.(037)
1/27
0.(9)
1
0.0(6)
1/15
Frequently asked questions
How do you convert a repeating decimal to a fraction?
Let x be the decimal. Multiply x by a power of 10 that shifts one full repeating block past the decimal point, and by another power of 10 that shifts only the non-repeating digits. Subtract the two equations so the repeating tails cancel, then solve for x and simplify. For 0.1(6): 100x = 16.66…, 10x = 1.66…, so 90x = 15 and x = 1/6.
What is 0.333… as a fraction?
0.(3) = 1/3. With x = 0.333…, 10x = 3.333…, and subtracting gives 9x = 3, so x = 3/9 = 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, and 0.(9) is another way to write 1.
Is there a quick rule for purely repeating decimals?
Yes. If all digits after the point repeat, the fraction is the repeating block over as many 9s as it has digits: 0.(27) = 27/99 = 3/11 and 0.(142857) = 142857/999999 = 1/7.
What if some digits do not repeat?
Write the digits up to the end of the first block minus the non-repeating digits, over 9s followed by 0s. For 0.1(6): (16 − 1)/90 = 15/90 = 1/6. For 2.1(45): (2145 − 21)/990 = 2124/990 = 118/55.
Which fractions give repeating decimals?
A fraction in lowest terms has a terminating decimal only when its denominator has no prime factors other than 2 and 5. Any other prime factor, such as 3, 7 or 11, makes the decimal repeat forever.
How should I type a repeating decimal?
Put the repeating part in round or square brackets: 0.(3), 0.12(34), 5.[6]. You can also type the decimal with three dots, like 0.272727..., and the tool will detect the shortest repeating pattern.
Are all the tools on FreeToolCabin tested for quality and accuracy?
Yes! Every tool on our site undergoes automated Q/A testing in real browser environments to ensure high precision, correct output, and zero errors. If you happen to spot a bug or have a suggestion, please contact us at [email protected] and our team will review it right away.