19/23 as a decimal, three places, no division

Question

Solve in your head, no long division. What is 	frac{19}{23} as a decimal, correct to three decimal places? Aim: under 45 seconds.

Accepted Answer

Route 1, peel off a round chunk. 23 	imes 0.8 = 18.4, so \frac{19}{23} = 0.8 + \frac{19 - 18.4}{23} = 0.8 + \frac{0.6}{23}. Now \frac{0.6}{23} = 0.6 	imes \frac{1}{23}, and \frac{1}{23} \approx 0.04348 (since \frac{1}{23} = \frac{1}{25} 	imes \frac{25}{23} \approx 0.04 	imes 1.087). So 0.6 	imes 0.0435 \approx 0.0261: \frac{19}{23} \approx 0.8 + 0.0261 = 0.826. Route 2, complement: \frac{19}{23} = 1 - \frac{4}{23} \approx 1 - 4 	imes 0.0435 = 1 - 0.174 = 0.826. True value: 0.826086\ldots, both routes hit three decimals. Takeaway. Never long-divide an awkward fraction: peel a round multi…

19/23 as a decimal, three places, no division

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