Tips and Tricks: Multiplication - 2 | Improve Your Calculations: Vedic Maths (English) - Class 6 PDF Download

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

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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

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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

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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)