Write 6/13 as a decimal

Question

Solve in your head, no long division. What is 	frac{6}{13} as a decimal (first six digits)? Aim: under 15 seconds.

Accepted Answer

The thirteenths fall into two families of rotations: - Cycle A, 076923: used by 	frac{1}{13}, 	frac{3}{13}, 	frac{4}{13}, 	frac{9}{13}, 	frac{10}{13}, 	frac{12}{13}. - Cycle B, 153846: used by 	frac{2}{13}, 	frac{5}{13}, 	frac{6}{13}, 	frac{7}{13}, 	frac{8}{13}, 	frac{11}{13}. Estimate the size to find the entry point: 	frac{6}{13} \approx 0.46 (since 6 	imes 7.7\% \approx 46\%). That is above 0.4, so it belongs to cycle B. Rotate 153846 to start near 46: 	frac{6}{13} = 0.\overline{461538}. Check: 13 	imes 0.461538 = 5.999994, correct to rounding. Takeaway. For 	frac{k}{13}: …

Write 6/13 as a decimal

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