Multiply 386 by 47 in your head and state the decomposition you used. Which split costs fewer operations, 40 plus 7 or 50 minus 3?

Multiply 386 by 47 in your head and state the decomposition you used. Which split costs fewer operations, 40 plus 7 or 50 minus 3?

Approach: Round the two-digit multiplier to a friendly value and correct for the overshoot, rather than splitting it into tens and units.

18142. The fast decomposition rounds the multiplier to 50, since 386 * 50 = 386 * 100 / 2 = 19300. Correct by subtracting 386 * 3 = 1158, which leaves 19300 - 1158 = 18142. The distributive law is what licenses the split. Rounding to 50 costs one halving and one small product, while the 40 plus 7 split costs 15440 plus 2702 and a carry across the thousands column. The general rule on a desk is to round the multiplier to a multiple of 25 or 50 whenever it sits within 3 of one, because halving and doubling are the two operations that survive time pressure.

Follow-up: How would you compute 386 * 47 * 1.05 to the nearest whole number without writing anything down?

Key concepts: decomposition, distributive law, rounding correction.