When multiplying numbers below a power of 10, we subtract one number’s deficiency from the base from the other number and then add the product of the deficiencies.
Example 18.1: Find 98 x 97
= (100 – 2)(100 – 3)
= (100 – 2 – 3)(100) + (-2 x -3)
= (98 – 3) x 100 + 2 * 3
= 95 | 06
= 9,506
Remember that the right hand side has the same number of digits as the number of zeroes in the base.
Example 18.2: Find 89 x 87
89 x 87
= (100 – 11)(100 – 13)
= 87 – 11 | 13 * 11
= 76 | 143
= 77 | 43
It is good practice to use the smaller deficiency as the subtrahend.
Example 18.3: Find 6,879 x 9,998
6,879 x 9,998
= 6,879 – 2 | 3,121 * 2
= 6,877 | 6242
= 68,776,242
It is definitely easier to subtract 2 from 6,879 than to deduct 3,121 from 9,998.
Here is the algebraic proof of the method used:
Let x = base; a, b = deficiency from the base
(x – a)(x – b) |
Exercise 18: Find the following products using the base method.
- 6 x 9 =
- 99 x 98 =
- 98 x 93 =
- 88 x 98 =
- 75 x 97 =
- 87 x 88 =
- 97 x 67 =
- 94 x 91 =
- 995 x 975 =
- 997 x 778 =
Answers to all exercises are found in the answer key.
Discover the 25 Math Short Cuts ( 25 MSC )!