Shortcuts for multiplying

General Shortcut for multiplying any two numbers m and n  : 

 m x n = [(m+n)/2] ² - [(m-n)/2] ²

Example :

96 x 42 = [(138)/2] ² - [(54)/2] ² = 4761 - 729 = 4032.

However, as you see, the best way of multiplication is the normal multiplication method we were taught since school. This method is however useful for multiplying numbers like

155 x 45 = [(200)/2] ² - [(110)/2] ² = 10000 - 3025 = 6975.

Other ways of multiplication are :

Two digit numbers :

Say we have two numbers : 56 x 48.
First, multiply the unit's digit of both the numbers. Here, we have 6 x 8 = 48. 4 is the carry. Write down 8.

Now cross multiply both the numbers and add them. Thus, we have 5 x 8 + 6 x 4 = 64. Add the carry obtained in the previous step. Hence, we get 64 + 4 = 68. 6 is the carry. Write down 8.

Multiply left hand side of the numbers and add them with the carry in previous step. Thus, we have 5 x 4 + 6 = 26.
Hence,  we have, 56 x 48 = 2688.

Three digit numbers :
Say we have two three digit numbers : 275 x 354.
Firstly, multiply 5 and 4. 5 x 4 = 20. Write down 0 and carry 2.
Now cross multiply the last two digits and add the carry in the previous step. So, we have 5 x 5 + 4 x 7 + 2 = 55. Write down 5. 5 is the carry.
Next, cross multiply the first and last digits and add carry in previous step. Thus, we have 4 x 2 + 3 x 5 + 5 x 7 + 5 = 63. Carry 6 and write down 3.
Now cross multiply the last two digits and add the carry in the previous step. Thus, 5 x 2 + 3 x 7 + 6 = 37. Write down 7. 3 is the carry.
Finally, multiply the leftmost numbers and add the carry. Thus, we have, 3 x 2 + 3 = 9.
Hence, 275 x 354 = 97350.


Multiplying two numbers close to 100 :

Say we have two numbers : 89 x 83.
Firstly, subtract both the numbers from 100. 100 - 89 = 11. 100 - 83 = 17.
Multiply 11 x 17 = 187. Note 87 and carry 1.
Add 11 and 17. 11 + 17 = 28. Subtract it from 100. Hence 100 - 28 = 72. Add carry with 72. Therefore, 72 + 1 = 73.
Hence, final result = 7387.

Multiplying two numbers between 50 & 100 :
Say we have two numbers : 68 x 57.
Firstly, multiply 68 & 5. 68 x 5 = 340. Put a zero at the end. Thus, we have 3400.
Multiply 68 & 7. 68 x 7 = 476.Add 476 and 3400. Thus, 68 x 57 = 3876.