Product of numbers close to 100

Say, you have to multiply 94 and 98. Take their differences to 100: 100 - 94 = 6 and 100 - 98 = 2. Note that 94 - 2 = 98 - 6 so that for the next step it is not important which one you use, but you'll need the result: 92. These will be the first two digits of the product. The last two are just 2×6 = 12. Therefore, 94×98 = 9212.

The same trick works with numbers above 100: 93×102 = 9486. First find the differences: 100 - 93 = 7 and 100 - 102 = -2. Then subtract one of the differences from the other number, e.g. 93 - (-2) = 95. This intends to show the first two digits of the product, i.e., the number 9500. Add to this the product of 7 and -2, or -14: 9500 - 14 = 9486.

Why does this work?

(100 - a)(100 - b)= (100 - a)·100 - (100 - a)·b
 = (100 - a)·100 - 100b + ab
 = (100 - a - b)·100 + ab.

The method works for numbers larger than 100 and when the two straddle the 100 divide. In general

 
(100 + x)(100 + y)= (100 + x)·100 + (100 + x)·y
 = (100 + x)·100 + 100y + xy
 = (100 + x + y)·100 + xy,

where x and y are arbitrary (even real, not necessarily integer) numbers. If A = 100 + x and B = 100 + y, the formula says that

A×B= (A + y)·100 + xy, or
A×B= (B + x)·100 + xy.

For example, if A = 115 and B = 98 then x = 15 and y = -2 so that

115×98= (115 - 2)·100 + 15×(-2), or
 = 11300 - 30
 = 11270.


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