Divide by 5, 25, 125 at Sight

Question

Compute 348 div 5 instantly. Then give the matching one-step moves for dividing by 25 and by 125, with an example of each.

Accepted Answer

Division by 5 grinds; multiplication by 2 is free. Since frac15 = frac210: 348 div 5 = frac348 times 210 = frac69610 = mathbf69.6. The same exchange works for every power of 5, because 5^k times 2^k = 10^k: - div,5: double, shift one place. 7.3 div 5 = 1.46. - div,25 = times,4 div 100: 1900 div 25 = frac7600100 = 76; 84 div 25 = 3.36. - div,125 = times,8 div 1000: 96 div 125 = frac7681000 = 0.768; 3 div 125 = 0.024. - div,50 = times,2 div 100: 87 div 50 = 1.74. Mirrored, the same idea makes the multiplications instant: times 5 = times 10 div 2 (86 times 5 = 430), times 25 = times 100 div 4 (44 times 25 = 1100), times 125 = times 1000 div 8 (56 times 125 = 7000, since 56 div 8 = 7). The principle: every divide-by-5^k is a multiply-by-2^k plus a decimal shift, and multiplying by a one-digit number always beats dividing by anything. Decimal shifts cost nothing, so spend your effort only on the 2^k move. Lot sizes of 25, per-unit conversions, and tick arithmetic make these divisions constant on Flow Traders and Akuna screens. The candidates who answer in one second are not dividing faster, they have stopped dividing altogether.

Divide by 5, 25, 125 at Sight

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