Tips and Tricks Multiplication - 2 - Improve Your Calculations Vedic Maths

Table of Contents
1. Sub-base Method
2. Doubling and Making halves
3. Multiplication of Complimentary numbers
4. Multiplication by numbers consisting of all 9's
5. Multiplication by 11
6. General Method of Multiplication
7. Number Split Method
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Sub-base Method

Till now we have all the numbers which are either less than or more than base numbers. (i.e.10, 100, 1000, 10000 etc. , now we will consider the numbers which are nearer to the multiple of 10, 100, 10000 etc. i.e. 50, 600, 7000 etc. these are called sub-base.

Example: 213 × 202
Step1: Here the sub base is 200 obtained by multiplying base 100 by 2
Step 2: R. H. S. and L.H.S. of answer is obtained using base- method.

Step 3: Multiply L.H.S. of answer by 2 to get 215 × 2 = 430
∴ 213 × 202 = 43026

Example 2: 497 × 493 
Step 1: The Sub-base here is 500 obtained by multiplying base 100 by 5.
Step 2: The right hand and left hand sides of the answer are obtained by using base method.
Step 3: Multiplying the left hand side of the answer by 5.

490 × 5
= 2450
∴ 497 × 493 = 245021

Example 3: 206 × 197 
Sub-base here is 200 so, multiply L.H.S. by 2

Example 4: 212 × 188 
Sub - base here is 200

Checking: (11 - check method)
+ - +
2 1 2 = 2 + 2 - 1 = 3
+ - +
1 8 8 = 1 - 8 + 8 = 1
L.H.S. = 3 × 1 = 3
+ - + - +
R.H.S. = 3 9 8 5 6 = 3
As L.H.S = R.H.S. So, answer is correct.

Doubling and Making halves

Sometimes while doing calculations we observe that we can calculate easily by multiplying the number by 2 than the larger number (which is again a multiple of 2). This procedure in called doubling:
35 × 4 = 35 × 2 + 2 × 35 = 70 + 70 = 140
26 × 8 = 26 × 2 + 26 × 2 + 26 × 2 + 26 × 2 = 52 + 52 + 52 + 52
= 52 × 2 + 52 × 2 = 104 × 2 = 208
53 × 4 = 53 × 2 + 53 × 2 = 106 × 2 = 212
Sometimes situation is reverse and we observe that it is easier to find half of the number than calculating 5 times or multiples of 5. This process is called Making halves:
(1) 87 × 5 = 87 × 5 × 2/2 = 870/2 = 435
(2) 27 × 50 = 27 × 50 × 2/2 = 2700/2 = 1350
(3) 82 × 25 = 82 × 25 × 4/4 = 8200/4 = 2050

Multiplication of Complimentary numbers 

Sutra: By one more than the previous one.
This special type of multiplication is for multiplying numbers whose first digits(figure) are same and whose last digits(figures)add up to 10,100 etc.

Example 1: 45 × 45
Step I: 5 × 5 = 25 which form R.H.S. part of answer
Step II: 4 × (next consecutive number)
i.e. 4 ×5 = 20, which form L.H.S. part of answer ∴ 45 × 45 = 2025

Example 2: 95 × 95 = 9 × 10 = 90/25 → (5²)
i.e. 95 × 95 = 9025 

Example 3: 42 × 48 = 4 × 5 = 20/16 → (8 × 2)
∴ 42 × 48 = 2016

Example 4: 304 × 306 = 30 × 31 = 930/24 → (4 × 6)
∴ 304 × 306 = 93024

Multiplication by numbers consisting of all 9's

Sutras: 'By one less than the previous one' and 'All from 9 and the last from 10' 

When number of 9's in the multiplier is same as the number of digits in the multiplicand.

Example 1: 765 × 999 
Step I: The number being multiplied by 9's is first reduced by 1
i.e. 765 - 1 = 764 This is first part of the answer
Step II: "All from 9 and the last from 10" is applied to 765 to
get 235, which is the second part of the answer.
∴  765 × 999 = 764235

When 9's in the multiplier are more than multiplicand 

Example 2: 1863 × 99999
Step I: Here 1863 has 4 digits and 99999 have 5-digits, we suppose 1863 to be as 01863. Reduce this by one to get 1862 which form the first part of answer.
Step II: Apply 'All from 9 and last from 10' to 01863 gives 98137 which form the last part of answer
∴ 1863 x 99999 = 186298137 

When 9's in the multiplier are less than multiplicand 

Example 3: 537 x 99 
Step I: Mark off two figures on the right of 537 as 5/37, one more than the L.H.S. of it i.e. (5 + 1) is to be subtracted from the whole number, 537 - 6 = 531 this forms first part of the answer
Step II: Now applying "all from 9 last from 10" to R.H.S. part of 5/37 to get 63 (100 - 37 = 63)
∴ 537 x 99 = 53163

Multiplication by 11

Example 1: 23 × 11
Step 1: Write the digit on L.H.S. of the number first. Here the number is 23 so, 2 is written first.
Step 2: Add the two digits of the given number and write it in between. Here 2 + 3 = 5 Step 3 : Now write the second digit on extreme right. Here the digit is 3. So, 23 × 11 = 253
OR
23 × 11 = 2 / 2+3 / 3 = 253
(Here base is 10 so only 2 digits can be added at a time) 

Example 2: 243 × 11
Step 1: Mark the first, second and last digit of given number
First digit = 2, second digit = 4, last digit = 3
Now first and last digits of the number 243 form the first and last digits of the answer.
Step 2: For second digit (from left) add first two digits of the number i.e. 2 + 4 = 6
Step 3: For third digit add second and last digits of the number i.e. 3 + 4 = 7
So, 243 × 11 = 2673
OR
243 × 11 = 2 / 2 + 4 / 4 + 3 / 3 = 2673
Similarly, we can multiply any bigger number by 11 easily.

Example 3: 42431 × 11
42431 × 11 = 4 / 4 + 2 / 2 + 4 / 4 + 3 / 3 + 1 / 1 = 466741

If we have to multiply the given number by 111 

Example 1: 189 × 111
Step 1: Mark the first, second and last digit of given number
First digit = 1, second digit = 8, last digit = 9
Now first and last digits of the number 189 may form the first and last digits of the answer
Step 2: For second digit (from left) add first two digits of the number i.e. 1 + 8 = 9
Step 3: For third digit add first, second and last digits of the number to get 1 + 8 + 9 = 18 (multiplying by 111, so three digits are added at a time)
Step 4: For fourth digit from left add second and last digit to get, 8 + 9 = 17
As we cannot have two digits at one place so 1 is shifted and added to the next digit so as to get 189 × 111 = 20979

∴189 × 111 = 20979

Example 2:  2891 × 111

General Method of Multiplication

Sutra: Vertically and cross-wise.
Till now we have learned various methods of multiplication but these are all special cases, where numbers should satisfy certain conditions like near base, or sub-base, complimentary to each other etc. Now we are going to learn about a general method of multiplication, by which we can multiply any two numbers in a line. Vertically and cross-wise sutra can be used for multiplying any number.

For different figure numbers the sutra works as follows:

Two digit - multiplication 

Example: Multiply 21 and 23
Step 1: Vertical (one at a time)

Step 2: Cross-wise (two at a time)

Step 3: Vertical (one at a time)

∴  21 × 23 = 483

Multiplication with carry

Example: Multiply 42 and 26
Step 1: Vertical

Step 2: Cross-wise

Step 3: Vertical

∴ 42 × 26 = 1092

Three digit multiplication

Example: 212 × 112
Step 1: Vertical (one at a time)

Step 2: Cross-wise (two at a time)

Step 3: Vertical and cross-wise (three at a time)

2 × 2 + 2 × 1 + 1 × 1 = 4 + 2 + 1 = 7

Step 4: cross-wise (Two at a time)

Step 5: vertical (one at a time)

∴ 212 × 112 = 23744

Three digits Multiplication with carry

Example: 816 × 223

∴ 816 × 223 = 181968
Checking by 11 - check method 
+ - +                - +
8 1 6 = 14 - 1 = 1 3 = 3 - 1 = 2
+ - +
2 2 3 = 3
∴ L.H.S. = 3 × 2 = 6
-  + - + - + - +
1 8 1 9 6 8 = 1 7 = 7 - 1 = 6
As L.H.S. = R.H.S.
∴ Answer is correct

Number Split Method

As you have earlier used this method for addition and subtraction, the same may be done for multiplication also.
For example:

Note: The split allows us to add 36 + 24 and 42 + 39 both of which can be done mentally.

Multiplication of algebraic expressions

Sutra: Vertically and cross-wise 

Example 1: (x + 3) (x + 4)

Example 2: (2x + 5) (3x + 2)

Example 3: (x² + 2x + 5) (x² - 3x + 1)