# MSC 19: Base Multiplication: One Number Above and One Number Below the Base

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 =

Discover the 25 Math Short Cuts ( 25 MSC )!

# MSC 17 – Base Multiplication: Multiplying “Teen” Numbers and Others

Most of us have learned by heart the multiplication table up to 10 x 10. A simple technique will enable us to extend our multiplication power up to 20 x 20.

Example 17.1:     Compute 14 x 12

• Cover one of the ten’s digit and add what remains to the other number: 14 + 2 or 4 + 12 will both give 16. This will be the first or left hand part of the product.
• Cover both ten’s digit and multiply what remains: 2 x 4 = 8. This will be the second or right hand part of the answer.
• Thus 14 x 12 = 16 | 8 or 168
 What we actually did in the first step is to add the excess over the base (10) of a number to the other.In the second step we multiplied the excesses. The algebraic proof of this method, known as base multiplication, is shown below: (x + a) * (x + b) = x2 + (a + b) * x + a * b = (x + a + b) * x + a * b = [ (x + a) + b] * x + a * b

In our example above, x = 10, a = 4 and b = 2. So we have
14 x 12
= [  (10 + 4) + 2 ] * 10 + 4 * 2

= 160 + 8
= 168

In this case both multiplicands are above the base.

Example 17.2:     Compute 16 x 13

(10 + 6) x (10 + 3)

• Add 3 (excess of 13 over 10) to 16 to get 19
= ( 10 + 6) + 3 |
= 19 |
• Multiply the excesses 6 x 3 = 18.
= 19 | 6 x 3
= 19 | 18
• Since there is only one zero in the base, only one place is allotted for the right hand part. Therefore the 1 of 18 must be carried or added to 19 to get a final answer of 208
= 20 | 8 = 208

Example 17.3:     Find 107 x 104

• The left part is 107 + 4 = 111, and the right part is 7 * 4 = 28
(100 + 7) * (100 + 4)
= 107 + 4 | 7 x 4

= 111 | 28

= 11,128Note that in this example we did not write the two zeroes after 111 but two places are reserved for the 28.

Example 17.4:     Find 1,025 x 1,012

• The left part is (1025 + 12 ) = 1037 and the right part is 25 * 12 = 300
1,025 x 1,012
= (1000 + 25) * (1000 + 12)
= 1025 + 12 | 25 * 12

= 1037 | 300
= 1,037,300

Example 17.5:     Find 115 x 111

• The left part is (115 + 11) and the right part 165
115 x 111
= (100 + 15) (100 + 11)
= 115 + 11 | 15 * 11
= 126 | 165
• Since we have only two zeroes in our base, we can allot only two spaces for the right hand part of the answer, so we must “carry” the 1 of 165 into the left side.
= 12,765

Example 17.6:     Find 102 x 104

• This is pretty straightforward with the left 102 + 4 = 106 and the right is 2 * 4 = 8
102 * 104
= (100 + 2) (100 + 4)
= 102 + 4 | 2(4)
= 106 | 08
• In this case, the product of the excesses is 8 but since the base is 100, two spaces are allotted to it. We write it as 08.
= 10,608

Example 17.7:     103 x 119

• The left side is 103 + 19 = 3 + 119 = 122 and the right side is 3 * 19 = 57
(100 + 3) * (100 + 19)
= 119 + 3 | 3 * 19
= 12,257
• Note that in this example we chose to add the smaller excess to the other number. This always leads to a simpler calculation.

Exercise 17: Find the following products using base multiplication

1. 12 x 13 =
2. 14 x 17 =
3. 15 x 18 =
4. 108 x 101=
5. 116 x 102 =
6. 108 x 112 =
7. 112 x 114=
8. 123 x 106 =
9. 1021 x 1006 =
10. 1432 x 1002 =