Elevenths as a two-digit repeat

Question

Solve in your head, no long division. What is 	frac{7}{11} as a decimal? And what is the rule for any 	frac{k}{11}? Aim: under 5 seconds.

Accepted Answer

For 	frac{k}{11} the decimal is a repeating two-digit block, and that block is 9k (nine times k, written as two digits): 	frac{7}{11} = 0.636363\ldots = 0.\overline{63}, \qquad 	ext{since } 9 	imes 7 = 63. The reason is that 	frac{1}{11} = 0.090909\ldots = 0.\overline{09}, so multiplying by k gives the block 09 	imes k = 9k. A couple more: 	frac{2}{11} = 0.\overline{18} (since 9 	imes 2 = 18) and 	frac{5}{11} = 0.\overline{45} (since 9 	imes 5 = 45). Takeaway. For 	frac{k}{11}, the repeating block is 9k as a two-digit number: 	frac{7}{11} = 0.\overline{63}. One multiplication and …

Elevenths as a two-digit repeat

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