Convert a recurring decimal to an exact fraction

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Convert a recurring decimal to an exact fraction

A recurring decimal repeats a digit block indefinitely. A finite calculator display is not the full recurring value. To find its fraction, multiply by a power of ten that shifts one full repeat block, then subtract matching tails.

If a non-repeating prefix comes first, create two shifted expressions with identical recurring tails before subtracting. The method is algebraic cancellation of the same infinite tail, not a claim that a long finite decimal is exact. Reduce the resulting fraction.

Worked example: Convert 0.272727… to a fraction.

Let x = 0.272727….

Two digits repeat, so 100 x = 27.272727….

Subtract x: 99 x = 27, then divide and reduce 27/99.

Answer: 3/11.

Check your understanding: Convert 0.166666… to a fraction.

Answer: 1/6.

Why: Let x equal the decimal; 100 x − 10 x = 15, so 90 x = 15.