Write 5/13 as a decimal

Question

Solve in your head, no long division. What is 	frac{5}{13} as a decimal (first six digits)? Aim: under 15 seconds.

Accepted Answer

Thirteenths split 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}. Locate the entry point by estimating the size: 	frac{5}{13} \approx 0.38 (since 5 	imes 7.7\% \approx 38\%). Rotating cycle B to start near 38: 	frac{5}{13} = 0.\overline{384615}. Check: 13 	imes 0.384615 = 4.999995, correct to rounding. Takeaway. For k/13: multiply k by 7.7\% to guess the first two digits,…

Write 5/13 as a decimal

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