Turn a repeating decimal into a fraction

Question

Solve in your head, no long division. Write 0.181818\ldots = 0.\overline{18} as a fraction in lowest terms. Aim: under 5 seconds.

Accepted Answer

A repeating block goes over as many 9s as it has digits. The block 18 has two digits, so it goes over 99: 0.\overline{18} = \frac{18}{99}. Now reduce: 18 and 99 share a factor of 9, giving \frac{18}{99} = \frac{2}{11}. Cross-check with the elevenths rule: 	frac{2}{11} has repeating block 9 	imes 2 = 18, so 	frac{2}{11} = 0.\overline{18}, which closes the loop. The general recipe: a one-digit repeat goes over 9, a two-digit repeat over 99, a three-digit repeat over 999, then reduce. Takeaway. To turn a purely repeating decimal into a fraction, put the block over a string of 9s the same leng…

Turn a repeating decimal into a fraction

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