# MSC 18: Base Multiplication – Numbers Below the Base

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)  = x2 – (a + b)x + a*b  = (x – a – b)x + a*b  =[(x – a) – b]x + a*b

Exercise 18:      Find the following products using the base method.

1. 6 x 9 =
2. 99 x 98 =
3. 98 x 93 =
4. 88 x 98 =
5. 75 x 97 =
6. 87 x 88 =
7. 97 x 67 =
8. 94 x 91 =
9. 995 x 975 =
10. 997 x 778 =