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Multiplication Techniques

Example

Find  42 × 48 ?

Solution
Both the numbers here start with 4 and the unit digits (2 and 8) add up to 10.
 
Steps:
  1. First multiply the unit digit of the number: 2 × 8 = 16
  2. Then multiply 4 by 5 (the succeeding number) to get 20 for the first part of the answer.
  3. So, Required answer is 2016.

 

Multiplying 2-digit numbers where the Ten’s digit add up to 10 and Unit digits are same.

Suppose, we want to find the multiplication of two numbers AB & CB (where A, B, C represent various digits of the numbers) and these two numbers satisfy a condition which says A + C = 10, then we can follow the following steps to find the answers –

  1. First find the multiplication of last two digits  of both the numbers ie. Find B × B. It will give us last 2 digits of the answer.
  2. Multiply the Ten’s digits and Add the common digit to the multiplication ie. find  A × C + B. It will give us the initial digits of the multiplication

 

Sol:  Steps-
  1. Here ten’s digit are 4 and 6 and their sum is 10 and unit digits of both numbers are same. Multiplying the unit digits, we get 4 × 4 = 16. Put it at right hand side.
  2. Again multiplying the ten’s digits of numbers and adding common digit, we get (4 × 6) + 4 = 24 + 4 = 28. Put it at the left hand side. So, we get required answer as 2816.
 
 
 
Multiplying numbers just over / below 100.
 
Example

108 × 109 ?

Solution
108 × 109 = 11772. The answer is in two parts: 117 and 72, 117 is just 108 + 9 (or 109 + 8), and 72 is just 8 × 9.
 
Now, check for 107 × 106 = 11342. As before, the surpluses above the base of 100 are set down on the right.
 
100 + 15 + 7 = 122 or 115 + 07 = 122    or 115 + 7 = 122 or 107 + 15 = 122
 
7 × 15 = 105, but since the right-hand portion has only two digits we must carry the 1 of 105 to the left. So, the answer is 12305.
 

Note: The above method can be applied to numbers which are just above 10000 as well, by changing the base to 1000 instead of 100.

Special Multiplication Technique for 5, 25, 125 & 625

Number Method Example:
Multiplication by 5 Put one 0 at the end of the number and divide by 2 512 × 5 = 5120/2 = 2560
Multiplication by 25 Put two 0 at the end of the number and divide by 4 512 × 25 = 51200/4 = 12800
Multiplication by 125 Put three 0 at the end of the number and divide by 8 512 × 125 = 512000/8 = 64000
Multiplication by 625 Put Four 0 at the end of the number and divide by 16 512 × 625 = 5120000/16 = 320000
 
Multiplication of digits near 100
 
Our approach should be this:

 
Whenever we have more than 100 on the right hand side we add the digits at the 100th place to the left hand side (as shown above). In the above case the digit at the 100th place is 1 so we will be required to add 1 to the left and side of the digit, thus our answer comes to 7743 which is same as obtained by conventional method.
 
Multiplication of digit near 50
 
We can state 50 = 100/2
Therefore, we will divide the number obtained after crosswise operation by 2.

­­
Example



33/39     = 3339 or 66 × 50 (Base) + 39 = 3300 + 39 = 3339
 

 
Example


Cross wise operation (47 + 14) or (64 – 3) given us 61

 

To get 3050, you can simply multiply the left-hand (61) side by 100 and divide by 2 to get the desired result.

 

 

Special Technique for multiplying any two/three digit number with a two/three digit number

Multiplication of a 2-digit number by a 2-digit number

Example

12 × 13?

 

Solution

Steps:

  1. Multiply the right-hand digits of multiplicand and multiplier (unit-digit of multiplicand with unit-digit of the multiplier).
     
  2. Now, do cross-multiplication, i.e., multiply 3 by 1 and 1 by 2. And the two products and write down the left of 6.
     
  3. In the last step we multiply the left-hand figures of both multiplicand and multiplier (ten’s digit of multiplicand with ten’s digit of multiplier).
     

So, the answer is 156.

 

Example

325 × 17 = ?

 
Solution

Steps:

 

Step 1


(5 × 7 = 35, put down 5 and carry over 3)
 
Step 2


(2 × 7 + 5 × 1 + 3 = 22, put down 2 and carry over 2)

Step 3


(3 × 7 + 2 × 1 + 2 = 25, put down 5 and carry over 2)
 
Step 4


So, answer is 5525

Multiplication of a 3-digit number by a 3-digit number

Example

321 × 132 = ?

Solution
  1.  
    2 (1 × 2 = 2)
  2.  
    72 (2 × 2 + 3 × 1 = 7)
  3.  
    (2 × 3 + 3 × 2 + 1 × 1 = 13, write down 3 and carry over 1)
  4.  (3 × 3 ++ 1 × 2 + 1 = 12, write down 2 and carry over 1)
  5.  
    4 2 3     7     2 (1 × 3 + 1 = 4) So, answer is 42372.
 
Three digit number multiplied by three digit number.
 
Now let us try to see the Magical method. We will explain this method using our old friends a, b, c and x, y, z.
 
Example

How to gain speed?

The key to speed is to do away with the intermediate steps:
 
This is being explained with the help of the following example.

 
To get better results, you should try to do all intermediate steps mentally and directly write down the answers in each step.

 

Multiplying a two digit number with 11
  1. Let us say, we have to multiply AB with 11 (where AB is the two digit number)
  2. Separate AB and create one digit space between them ie. A_B
  3. Add A & B and put the result in the above space created.
  4. If the addition of A & B is a two digit number, then put the unit digit in the space created in step 2 and add the carry digit to A.

 

Example

Multiply 43 × 11

Solution

We can do the multiplication in the following steps:

  1. Create a one digit space between 4 and 3 ie we get 4_3
  2. We add 4 & 3 to get 7 and put the 7 in the space created above
  3. We get 473 as answer.

Multiplying a two digit number with 11

Multiplying a two digit number with 11
  1. Let us say, we have to multiply AB with 11 (where AB is the two digit number)
  2. Separate AB and create one digit space between them ie. A_B
  3. Add A & B and put the result in the above space created.
  4. If the addition of A & B is a two digit number, then put the unit digit in the space created in step 2 and add the carry digit to A.

 

Example

Multiply 43 × 11

Solution
We can do the multiplication in the following steps:
  1. Create a one digit space between 4 and 3 ie we get 4_3
  2. We add 4 & 3 to get 7 and put the 7 in the space created above
  3. We get 473 as answer.
 
Multiplying any number with 11
 
Example

13,423 × 11

Write down the number, as shown on the right, with a zero placed at both ends. This is a zero sandwich.

0134230


Add the last two digits, 3 + 0 = 3, and write the answer below the 0.

For the ten’s digit, add the hundred place digit and ten’s digit, that is 2 + 3 = 5.

Continue to add adjacent digits, that is, 4 + 2 = 6, 3 + 4 = 7 and 1 + 3 = 4, 0 + 1 = 1. The answer is 147,653.

0134230 × 11
147653


With a little practice method can be applied without writing but some care must be taken when carry overs are involved.

 

 

Squaring Any 2 Digit Number – Boundary Method
  1. Look for the nearest 10 boundary. Eg. 3 from 47 to 50
  2. Since we went up 3 to 50, now go down 3 from 47 to 44.
  3. Now mentally multiply 44 × 50 = 2200 - 1st answer.
  4. 47 is 3 away from the 10 boundary 50, Square 3 as 9 - 2nd answer.
  5. Add the first and second answer, 2200 + 9
Answer: 2209
 

Multiplied with 12

Example

65214 × 12

Again we start with the zero sandwich. 0652140.

 

The ultimate digit is 0 and the penultimate digit is 4. Adding the ultimate digit and twice the penultimate digit, we get 0 + 8 = 8.

 

For the ten’s column, the ultimate digit is 4 and the penultimate digit is 1, so 4 + 2 = 6.

 

Likewise, 1 + 4 = 5 and 2 + 10 = 12. With 12 we set down 2 and carry 1.

 

5 + 12 + carry 1 = 18 and again we carry 1.

 

The final step is 6 + 0 + carry 1 = 7. So, the answer is 782568.
 





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