The 11-Times Slide

Question

Compute 11 times 72 and then 11 times 87 mentally. One of the two hides a wrinkle, state the general rule, the wrinkle, and enter the second product as your answer.

Accepted Answer

Rule: to multiply a two-digit number by 11, slide its digits apart and drop their sum in the middle: 11 times 72: quad 7 + 2 = 9 ;to; 7,|,9,|,2 = mathbf792. The wrinkle, carries: when the digit sum exceeds 9, the middle slot keeps only the units and the 1 carries left: 11 times 87: quad 8 + 7 = 15 ;to; (8+1),|,5,|,7 = mathbf957. One more with a carry that grows a digit: 11 times 94 = (9+1),|,3,|,4 = 1034. Why it works: 11(10a + b) = 100a + 10(a+b) + b, the middle slot literally holds a+b, and anything past 9 overflows into the hundreds. Longer numbers chain the same way (each slot is the sum of adjacent digits): 11 times 792 = 7,|,16,|,11,|,2 to 8712. The slide travels well: times 1.1 is the same digit pattern through the decimal point (11 times 3.4 = 37.4), times 22 is the slide then a doubling, and times 111 drops the digit sum into two middle slots (111 times 72 = 7992). On arithmetic screens ×11 appears both raw and disguised, "add 10%" is exactly times 1.1, so a 87 stock rallying 10% is the same three digits, 95.70, read off the slide rather than computed. Train the reflex: see 11, think "shift and add", never "multiply".

The 11-Times Slide

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