How to Convert a Repeating Decimal to a Fraction

Decimals like 0.444… never end, but every repeating decimal is an exact fraction in disguise. A little algebra removes the endless tail.

Case 1: everything after the point repeats

Take x = 0.444…

1. One digit repeats, so multiply by 10: 10x = 4.444…
2. Subtract the original: 10x − x = 4.444… − 0.444… = 4.
3. So 9x = 4 and x = 4/9.

For two repeating digits, multiply by 100. With x = 0.2727…, 100x = 27.2727…, so 99x = 27 and x = 27/99 = 3/11.

Case 2: the repeat starts later

Take x = 0.8333…, where only the 3 repeats.

1. Move the non-repeating part past the point: 10x = 8.333…
2. Move one full cycle further: 100x = 83.333…
3. Subtract: 100x − 10x = 75, so 90x = 75.
4. x = 75/90 = 5/6.

The 9s and 0s shortcut

Write the repeating block over as many 9s as it has digits. For a delay before the repeat, put the digits up to the end of the first cycle, minus the non-repeating part, over 9s followed by one 0 per delayed digit.

• 0.(4) = 4/9
• 0.(27) = 27/99 = 3/11
• 0.(123) = 123/999 = 41/333
• 0.8(3) = (83 − 8)/90 = 75/90 = 5/6

Numbers larger than 1

Handle the whole-number part separately. 2.(4) = 2 + 4/9 = 22/9.

A surprising result

Apply the method to 0.999…: 10x − x = 9, so x = 1. The decimal 0.999… is exactly equal to 1, not just very close to it.

Tip Always simplify at the end. 27/99 looks unfamiliar until you divide by 9 and see 3/11.

½Try it yourself: Decimal to Fraction CalculatorChange any decimal, including repeating ones like 0.(6), into a fraction in lowest termsOpen →

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