Repeating Decimal to Fraction Calculator

How it works

A repeating decimal equals (all the digits − the digits before the repeat) divided by a number made of one 9 for each repeating digit followed by one 0 for each non-repeating decimal digit. For 0.1666…: (16 − 1) ÷ 90 = 15/90 = 1/6. The result is then simplified by the greatest common divisor.

The algebra method, step by step

Every repeating decimal is a fraction. To find it, call the number x and multiply it so the repeating part lines up, then subtract.

1. Let x equal the decimal. x = 0.1666…
2. Multiply to move the non-repeating digits in front of the point: 10x = 1.666…
3. Multiply again by 10 for each repeating digit: 100x = 16.666…
4. Subtract the two lines so the endless 6s cancel: 100x − 10x = 16.666… − 1.666…, so 90x = 15.
5. Solve and simplify: x = 15/90 = 1/6.

The calculator above uses the same idea in a single formula: (all the digits − the digits before the repeat) ÷ (a 9 for each repeating digit followed by a 0 for each non-repeating decimal digit). For 0.1666…: (16 − 1) ÷ 90 = 15/90 = 1/6.

Repeating decimals and their fractions

Notice the pattern: one repeating digit goes over 9, two over 99, six over 999,999. That is why 0.142857… is 142857/999999, which simplifies to 1/7.

Does 0.999… really equal 1?

Yes. Using the method: x = 0.999…, 10x = 9.999…, subtract to get 9x = 9, so x = 1. Another way to see it: 1/3 = 0.333…, and three times that is 0.999…, which must equal 3/3 = 1. There is no number between 0.999… and 1, so they are the same number written two ways.

How to tell if a decimal repeats

A calculator display cuts numbers off, so 0.3333333 might be a rounded 1/3 or exactly 3,333,333/10,000,000. If you know where the number came from — a division, a fraction, a ratio — it almost certainly repeats. A fraction gives a repeating decimal whenever its simplified denominator has a prime factor other than 2 or 5, such as 3, 7, 11 or 13.

For terminating decimals such as 0.375, use the regular decimal to fraction calculator. To go the other way, the fraction to decimal converter shows the repeating block of any fraction.