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Multiplication (Vedic Mathematics)

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1. Multiplication (AB × AC)

This method of multiplication is derived from the second sutra of Vedic Mathematics, known as the “Nikhilam Sutra”. It is a special case of multiplication of two 2-digit numbers, where the sum of the last digits of the two numbers is equal to 10 and the first digits of the two numbers are same.
For example, if there are two numbers ab and aq such that b + q = 10, then this Vedic method can be applied.
Watch the following video to understand the method:

Solved Examples
Example 1:
Calculate 43 × 47
Solution:
Tens digits: Same (4)
Units digits: Sum = 10 (3 + 7)
∴43 × 47 = {Tens digit × (Tens digit + 1)} (Product of units digits) = (4 × 5) (3 × 7) = 2021
Example 2:
Calculate 66 × 64
Solution:
Tens digits: Same (6)
Units digits: Sum = 10 (6 + 4)
∴66 × 64 = {Tens digit × (Tens digit + 1)} (Product of units digits) = (6 × 7) (6 × 4) = 4224

2. Multiplication Method – I (Deficit Method of Multiplication)

The Deficit method of multiplication is derived from the second sutra of the 16 sutras given in Vedic Mathematics, known as the “Nikhilam Sutra”.
This method is very helpful and brief for multiplying numbers, which are close to 10n, i.e., close to 10, 100, 1000 etc.
Four probable cases are possible:
Case I: When both the numbers are less than 10n. (For example, in 98 × 93, both the numbers are less than and close to 100)
Case II: When both the numbers are greater than 10n. (For example, in 1001 × 1008, both the numbers are greater than and close to 1000)
Case III: When one number is greater than 10n and the other number is less than 10n. (For example, in 8 × 13, 8 is less than and close to 10 and 13 is greater than and close to 10)
Case IV: When the numbers are not close to 10n. (For example, in 45 × 49, both the numbers are not close to either 10 or 100)
Watch the following video to understand case I, where both the numbers are less than 10n
Solved Examples
Example 1:
Calculate 93 × 99
Solution:
Base = 100
Ist deficit = 100 − 93 = 7
IInd deficit = 100 − 99 = 1

∴93 × 99 = 9207
Example 2:
Calculate 88 × 94
Solution:
Base = 100
Ist deficit = 100 − 88 = 12
IInd deficit = 100 − 94 = 6

∴88 × 94 = 8272
In the above examples, the product of the deficits was always less than the base. What if the product of the deficits is greater than the base?
Let us watch the following video to understand what to do in such cases.
Solved Example
Example 1:
Calculate 92 × 81
Solution:
Base = 100
Ist deficit = 100 − 92 = 8
IInd deficit = 100 − 81 = 19
Product of the deficits = 8 × 19 = 152

∴92 × 81 = 7452
Now, let us go to case II, where both the numbers are greater than 10n. To understand this case, let us watch the following video.

Solved Examples
Example 1:
Calculate 105 × 109
Solution:
Base = 100
Ist surplus = 105 − 100 = 5
IInd surplus = 109 − 100 = 9

∴105 × 109 = 11445
Example 2:
Calculate 1012 × 1015
Solution:
Base = 1000
Ist surplus = 1012 − 1000 = 12
IInd surplus = 1015 − 1000 = 15
Product of the deficits = 12 × 15 = 180

∴1012 × 1015 = 1027180
Now, let us come to case III, where one of the numbers is greater than the base and the other one is less than the base. For example, let us try and calculate 108 × 92. Here, the base is 100 and 108 > 100 and 92 < 100.
Let us watch the following video to understand this method:
Solved Examples
Example 1:
Calculate 92 × 105
Solution:
Base = 100
Deficit = 92 − 100 = −8
Surplus = 105 − 100 = +5
Step I:

Step II:

∴92 × 105 = 9660
Example 2:
Calculate 994 × 1021
Solution:
Base = 1000
Deficit = 994 − 1000 = −6
Surplus = 1021 − 1000 = +21
Step I:

Step II:

∴994 × 1021 = 1014874
Now, only the last case is left, where the numbers are not close to the base. This case comes under a sub sutra of Vedic Mathematics, known as the “Anurupyena”, which simply means “Proportionality”.
Let us watch the following video to learn this method:
Solved Example
Example 1:
Calculate 42 × 54
Solution:
Base = 50 =
Deficit = 42 − 50 = −8
Surplus = 54 − 50 = +4
Step I:

Step II:

Step III:

∴42 × 54 = 2268
This completes all the possible cases for multiplying numbers close to either 10, 100, 1000, etc or to their multiples or sub-multiples.

3. Multiplication Method – II (Vertical and Cross-wise Multiplication)

This method of multiplication is derived from the third sutra of Vedic Mathematics, known as the “Urdhva − tiryagbhyam”, which translates into English as “vertical and cross-wise”.
 This is a general multiplication method and can be applied to all the possible arrangements of numbers.
 Four concepts need to be learnt in this method:
  • Multiplication of 2-digit numbers
  • Carry concept
  • Alternate method of multiplication
  • Multiplication of higher-digit numbers
To understand the method of multiplication of 2-digit numbers, watch the following video.
Solved Examples
Example 1:
Calculate 71 × 21
Solution:

∴71 × 21 = 1491
Example 2:
Calculate 62 × 11
Solution:

∴62 × 11 = 682
In all the above examples, the cross multiplication was between single-digit numbers. What if the product is a double-digit number?
To understand this method, watch the following video:

Solved Examples
Example 1:
Calculate 95 × 87
Solution:

∴95 × 87 = 8265
Example 2:
Calculate 74 × 88
Solution:

∴74 × 88 = 6512
Carrying a number and adding it at the next step increases the calculation time. Hence, instead of starting from left to right, multiplication can be done from right to left.
Look at the following video below to understand the alternate method.

Solved Examples:
Example 1:
Calculate 87 × 42
Solution:

∴87 × 42 = 3654
Example 2:
Calculate 64 × 98
Solution:

∴64 × 98 = 6272
Here, the carry step has been removed and it was done mentally. Hence, the time taken for calculation gets reduced.
The final concept deals with general multiplication of 3-digit numbers. This method can be further extended to any number having higher number of digits.
Look at the following video to understand the method.

Solved Example
Example 1:
Calculate 942 × 466
Solution:

∴ 942 × 466 = 438972
Hence, the multiplication method can be generalized as follows:

Note:
Though Vedic Mathematics suggests doing multiplication from left to right, one can also multiply from right to left in order to avoid the extra carry step.
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