Chapter 4.5, Problem 17P

There is an interesting mental technique for multiplying certain pairs of numbers. The following examples will illustrate how it works:

To calculate 32 . 28, notice that the two factors 28 and 32 are both 2 away—in opposite directions—from 30. To calculate 32 . 28, do the following:

   $\begin{array}{l}30.30-2.2=900-4\\ =896\end{array}$

Notice that you can do this calculation in your head.

• Similarly, to calculate 59 . 61, notice that both factors are 1 away—in opposite directions—from 60. Then 59 61 is
$\begin{array}{l}60.60-1.1=3600-1\\ =3599\end{array}$

Once again, notice that you can do this mentally.

a. Use the method just shown to calculate 83. 77, 195. 205, and one other multiplication problem like this that you make up.

b. Now explain why this method works. Hint: A drawing might be helpful. Another approach is to notice that the technique applies to multiplication problems of the form
$\left(A+B\right).\left(A-B\right).$