Turn 0.2727... into a fraction

Question

A decimal repeats a two-digit block forever: 0.272727\ldots Solve in your head. Write 0.\overline{27} as a fraction in lowest terms. Aim: under 20 seconds.

Accepted Answer

Let x = 0.\overline{27}. The block is two digits long, so multiply by 100 to line up the repeats: 100x = 27.2727\ldots, \qquad x = 0.2727\ldots Subtract: 99x = 27 \;\Rightarrow\; x = \frac{27}{99} = \frac{3}{11}. Check: 	frac{3}{11} = 0.2727\ldots Takeaway. A repeating block of n digits over \underbrace{99\ldots9}{n} is the whole trick: 0.\overline{27} = 	frac{27}{99}, 0.\overline{6} = 	frac{6}{9} = 	frac23, 0.\overline{142857} = 	frac{142857}{999999} = 	frac17. Just remember to reduce.

Turn 0.2727... into a fraction

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