Vedic Mathematics: Quick multiplication of numbers which are nearer to bases like 10, 100, 1000. | Examite

Today we are going to learn quick multiplication method of the numbers which are nearer to bases like 10, 100, 1000 i.e., to the powers of 10. 

The numbers taken can be either less or more than the base considered. The difference between the number and the base is termed as deviation. Deviation may be positive or negative. Now observe the following Chart.
 

Number	Base	Number – Base	Deviation
16	10	16-10	+6
7	10	7-10	-3
114	100	114-100	+14
92	100	92-100	-8
998	1000	998-1000	-2
1012	1000	1012-1000	+12

Now consider the following examples: 

Case 1: Both the numbers are lower than the base.

Example1: Multiply 7 by 8. 

Here the base is 10, since it is near to both the numbers. Let’s go through the method: 

1. Write the numbers one below the other.

7      X  8

?

2. Take the deviations of both the numbers from the base and place at right hand side.

7    (-3)    X  8   (-2)

?

3. The product or answer will have two parts, one on the left side and the other on the right. A vertical slash (‘/’) may be drawn for the separating the two parts i.e.,

7    (-3)    X  8   (-2)

/

4. The R.H.S. of the answer is the product of the deviations of the numbers. It shall contain the number of digits equal to number of zeros in the base.

7    (-3)    X  8   (-2)

/ 6

 

5. L.H.S of the answer is the sum of one number with the deviation of the other. It can be arrived at in any one of the following ways.

i) Cross-subtract deviation 2 on the second row from the original  number7 in the first row i.e., 7-2 = 5.

Or,

ii) Cross–subtract deviation 3 on the first row from the original number8 in the second row (converse way of(i)) i.e., 8 - 3 = 5.

7            (-3)      X  8            (-2)

(7-2) or (8-3) / -3 X -2

 

So,  7 x 8 =56

Example2: Multiply 107 by 104.

107    (+7)       X  104    (+4)

107+4  / 7 X 4

So, 107 X 104 = 11128

Case 2 : Both the numbers are higher than the base.

Example1: Multiply 97 by 94.

Here the base is 100, since it is near to both the numbers. Let’s go through the method:

1. Write the numbers one below the other.

97      X  94

?

 

2. Take the deviations of both the numbers from the base and place at right hand side.

97    (-3)      X  94    (-6)

?

3. A vertical slash (‘/’) may be drawn for the separating the two parts of the answer.

97    (-3)       X  94    (-6)

97-6  / -3 X -6

 

So, 97 X 94 = 9118

Example2: 986 X 989. Base is 1000.

986    (-14)       X  989   (-11)

986-11  / -14 X -11

So, 986 X 989 =975154

Case 3: One number is more and the other is less than the base.

Example1: 13X7. Base is 10

In this situation one deviation is positive and the other is negative. So the

product of deviations becomes negative. So the right hand side of the answer

obtained will therefore have to be subtracted.

So the complement of 9 must be taken on base of 100 which is (100-9)= 4.

13          (+3)       X    7           (-3 )

_                         /   9

                            _

Note:RHS will be   9 = (100-9)= 1  and LHS will subtracted by 1 due to the use of this complement.

13          (+3)       X    7           (-3)

(13-3)-1    /  1

So, 13X7 = 91

Example2: Multiply 103 by 98.

103    (+3)       X  98      (-2)

?

Here (+3) X (-2) = -6, which comes as negative RHS counterpart. So the complement of 6 must be taken on base of 100 which is (100-6)= 94.

103    (+3)      X  98      (-2)

103- 2 -1 / (100-6)

So, 103 by 98= 10094

 Exercises:
    

1.	994 X 991= ?
2.	996 X 1006= ?
3.	98 X 92= ?
4.	105 X 103= ?
5.	107 X 92= ?