# The Secrets of Mental Math by Arthur T. Benjamin PDF

By Arthur T. Benjamin

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When we reach the remainder step, we have to make sure to add 15 + 10, rather than 15 + 0. The result here is 50 with a remainder of 25. If the divisor ends in 8, 7, 6, or 5, the procedure is almost the same. For the problem 123,456 ÷ 78, we go up 2 to get to 80 and use 8 as our divisor. Then, as we go through the procedure, we double the previous quotient at each step. If the original divisor ends in 7, we would add 3 to reach a round number, so at each step, we add 3 times the previous quotient.

For example, to do the problem 43 × 28, it is easier to do 43 × 7 × 4 = 301 × 4 = 1204 than to do 43 × 4 × 7 = 172 × 7. squaring: Multiplying a number by itself. For example, the square of 5 is 25. subtraction method: A method for multiplying numbers by turning the original problem into a subtraction problem. For example, 9 × 79 = (9 × 80) – (9 × 1) = 720 – 9 = 711, or 19 × 37 = (20 × 37) – (1 × 37) = 740 – 37 = 703. Suggested Reading Benjamin and Shermer, Secrets of Mental Math: The Mathemagician’s Guide to Lightning Calculation and Amazing Math Tricks, chapter 3.

Consider the people can handle. problem 2001 ÷ 23. We start with a 2-by-1 multiplication problem: 23 × 8 = 184; thus, 23 × 80 = 1840. We know that 80 will be part of the answer; now we subtract 2001 – 1840. Using complements, we ¿nd that 1840 is 160 away from 2000. Finally, we do 161 ÷ 23, and 23 × 7 = 161 exactly, which gives us 87 as the answer. The problem 2012 ÷ 24 is easier. Both numbers here are divisible by 4; speci¿cally, 2012 = 503 × 4, and 24 = 6 × 4. We simplify the problem to 503 ÷ 6, which reduces the 2-digit problem to a 1-digit division problem.