13/17 as a decimal, three places, no division

Question

Solve in your head, no long division. What is 	frac{13}{17} as a decimal, correct to three decimal places? Aim: under 45 seconds.

Accepted Answer

Anchor the reciprocal: \frac{1}{17} \approx 0.05882 (from \frac{1}{17} = \frac{1}{16}	imes\frac{16}{17} \approx 0.0625 	imes 0.941). Route 1, complement. \frac{13}{17} = 1 - \frac{4}{17} \approx 1 - 4 	imes 0.05882 = 1 - 0.2353 = 0.765. Route 2, peel a round chunk. 17 	imes 0.7 = 11.9, so \frac{13}{17} = 0.7 + \frac{13 - 11.9}{17} = 0.7 + \frac{1.1}{17} = 0.7 + 1.1 	imes 0.05882 \approx 0.7 + 0.0647 = 0.765. True value: 0.764706\ldots, both routes give 0.765. Takeaway. Get \frac{1}{17} from the nearby \frac{1}{16} = 0.0625 and scale down a touch. With the reciprocal in hand, complement o…

13/17 as a decimal, three places, no division

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