Method
We'll multiply 105 by 107. How does the method differ from earlier methods? Instead of subtracting the numbers from 100, we subtract 100 from then, and instead of subtracting diagonally, we add. (See table below.)| Numbers to Multiply | Left hand column minus 100 |
|---|---|
105
|
5
|
107
|
7
|
ADD the number diagonally opposite. It doesn't matter which pair we use as we get the same answer. The pairs are shown in red and blue.
|
Multiply the numbers in this column
|
105 + 7 or 107 + 5 = 112
|
5 x 7 = 35
|
Multiply this by 100
| |
112 x 100 = 11200
|
35
|
Finally, we add the two numbers.
| |
11200 + 35 = 11235
| |
When is it useful?
If you can multiply the two digits after 100 by each other, then this method works.Example 1
For instance, 102 x 145 since 2 x 45 = 90 is easy if you're comfortable with doubling. Once you've multiplied those, all that is left to do is add 102 and 45 which is 147, put a couple of zeroes on the end and add the 90 you got earlier. That is,
147 x 100 = 14700,
14700 + 90 = 14790,
and so
102 x 145 = 14790.
Example 2
Similarly, 120 x 145 is as follows. Do 20 x 45, which is just 2 x 45 = 90 with a zero on the end, that is
20 x 45 = 900.
Then
120 + 45 = 165,
165 x 100 = 16500,
16500 + 900 = 17400,
and so
120 x 145 = 17400.
Proof
The proof is similar to those I've given for the last three posts on this topic.We will multiply a and b, which are numbers greater than 100. Next we subtract 100 from each of them to give a - 100 and b - 100, and then multiply them together.
(a - 100)(b - 100) = ab - 100a - 100b + 10000. (1)
Moving on to the other part of the calculation, we add a - 100 to b to get a + b - 100. Next we multiply by 100 which gives
100(a + b - 100) = 100a + 100b - 10000. (2)
Adding (1) and (2), we obtain
ab - 100a - 100b + 10000 + 100a + 100b - 10000 = ab,
again showing that our method does indeed multiply a and b together.
Source: http://www.vedicmaths.org/Introduction/Tutorial/Tutorial.asp
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