1.2.10 Multiplying Two Numbers Equidistant from a Third Number

To illustrate this concept, let’s look at an example of this type of problem: 83 × 87.

Notice that both 83 and 87 are 2 away from 85. So:

83×87=(85−2)×(85+2)83 \times 87 = (85 - 2) \times (85 + 2)

Which notice this is just the difference of two squares:

(85−2)×(85+2)=852−22=7225−4=7221(85 - 2) \times (85 + 2) = 85^2 - 2^2 = 7225 - 4 = 7221

So the procedure is:

  1. Find the middle number between the two numbers being multiplied and square it.
  2. Subtract from that the difference between the middle number and one of two numbers squared.

For most of these types of problems, the center number will be a multiple of 5, making the computation of its square relatively simple (See Section 1.2.7, Square’s Ending in 5 Trick). The following illustrates another example:

61×69=652−42=4225−16=420961 \times 69 = 65^2 - 4^2 = 4225 - 16 = 4209

Problem Set 1.2.10

Practice these problems. Type your answer and press Enter to check:

84 × 86
53 × 57
48 × 52
62 × 58
6.8 × 7.2
88 × 82
36 × 24
7.6 × 8.4
5.3 × 4.7
51 × 59 + 16
96 × 104
81 × 89 + 16
34 × 36 + 1
73 × 77 + 4
62 × 68 + 9
32 × 38 + 9
18 × 24 + 9
61 × 69 + 16
43 × 47 + 4
88 × 82 + 9
57 × 53 + 4
38 × 28
41 × 49 - 9
77 × 73 + 4
65 × 75 - 33
33 × 27 + 9
71 × 79 + 16
72 × 78 + 9
53 × 57 + 4
105 × 95
62 × 68 - 16
36 × 26
83 × 87 - 21
23 × 27 + 4
29 × 37
21 - 83 × 87
112 × 88
52 × 48 + 49 × 51
4.9 × 3.3
72 × 68 + 71 × 69
42 × 38 + 41 × 39
4.8 × 6.3
4000 + 322 × 318
118 × 122 + 4
5.1 × 7.9
34 × 36 × 34 × 36