When one number is above and the other number is below the base, we either

• a) add the excess of the number above the base to the other number or
• b) deduct the deficiency of the number below the base from the other number.

Example 19.1:     Find 104 x 98
Our base is 100, 104 is 4 above the base, and 98 is 2 below the base.
104 x 98
= (100 + 4) *(100 – 2)
= (104 – 2) * 100 + 4 *(-2)

Note that 104 – 2 and 98 + 4 will both yield 102
= 10,200 – 08
= 10,192

Example 19.2:     Find 102 x 93
102 x 93
= (100 + 2)(100 – 7)
= 93 + 2 | 2 * (-7) Easier than 102 – 7
= 95 | -14
= 9500 – 14
= 9,486

Example 19.3:     Find 109 x 97
109 x 92
= 109 – 3 | 9*(-3)
= 106 | -27
= 10,573

Here we just deducted 1 from 106 (109 – 3) and affixed the ten’s complement of 27 = (9 x 3) at the end.

Example 19.4:     Find 114 x 88
114 x 88
= (100 + 14) (100 – 12)
= 114 – 12 | 14(-12)
= 102 | -168
= 100 | 200 – 168
= 10,032

Here the right hand part is -168, so we deducted 2 from the left hand part.

Proof: Let x = base; a, b = excess / deficiency from the base (x + a)(x – b) = x2 + (a – b)x + a(-b) = (x + a – b)x – ab = [(x + a) – b]x – ab or = [(x – b) + a]x – ab

Exercise 19: Find the following products using the base method

1. ) 12 x 9 =
2. ) 103 x 98 =
3. ) 102 x 97 =
4. ) 102 x 98 =
5. ) 103 x 97 =
6. ) 105 x 93 =
7. ) 75 x 103 =
8. ) 112 x 89 =
9. ) 1012 x 991 =
10. ) 1125 x 995 =

Answers to all exercises are found in the answer key.

Discover the 25 Math Short Cuts ( 25 MSC )!