97 × 96 Without Grinding

Question

Compute 97 times 96 mentally. State the general rule for multiplying two numbers just below 100, and say where the carry hides.

Accepted Answer

Measure each factor's deficit from 100 (97 to 3, 96 to 4) and expand: (100-a)(100-b) = 100,(100 - a - b) + ab. Recipe: cross-subtract for the leading digits, multiply the deficits for the trailing pair: 97 times 96: quad 97 - 4 = 96 - 3 = 93, qquad 3 times 4 = 12 ;to; mathbf9312. Cross-subtraction works from either side because 100 - a - b = (100-a) - b = (100-b) - a. Calibration: 98 times 93 = 91,|,14 = 9114; 95 times 97 = 92,|,15 = 9215; 89 times 94 = 83,|,66 = 8366. The carry: the deficit product owns exactly two digit slots. When it spills past 99, the excess carries into the leading pair: 88 times 86 to 74,|,168 to 7400 + 168 = 7568. The identity is signed, so it runs above 100 with surpluses, 103 times 106 = 109,|,18 = 10918, and handles straddles: 102 times 97 = 99,|,(-6) = 9900 - 6 = 9894. With a = b it degenerates into squaring near 100: 96^2 = 92,|,16 = 9216, the same machinery as the 50-anchor for squares. Prices in the 90s and low 100s dominate trading arithmetic, which is why this rule earns its keep: it converts the ugliest common products on an Optiver screen into a subtraction, a one-digit multiplication, and a glue step.

97 × 96 Without Grinding

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