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 – b)x + a*b =[(x – a) – b]x + a*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.

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