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 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!

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