Vedic Maths India
Worlds fastest mental calculation system
Courtesy :  www.vedicmathsindia.org

How to find
Squares of Numbers ending in 5
                          in less than 3 seconds?                                   

 35²     45²     85²    etc. 

 Lets take for example 

    35
X   35
______________

That the units place (first column on the right) adds up to 5 + 5 = 10 and
that the Ten's place (second column from the right) is the same ( 3 and 3)

When this is true what we do is

           3 5

    x   3 5
    ________ 
              12       

Apply the word formula ‘By one more than the One before’
to the ten’s place.

 So what’s 1 more than 3? It is 4. So we multiply
3 with 4 to get 12. That’s the first part of our answer.

To get the second part of our answer

         3 5

   x    3 5
________
        1225

We multiply 5 with 5 to get 25.
That’s the second part of our answer.

So our answer becomes 1225.

Now try

       85

x      85
__________

Solution

  85

x 85
________
7225

We multiply 5 with 5 to get 25.  And
multiply 8 with the next number 9 to
get 72. So our answer becomes 7225.
Simple.

Now try 45²  55²    65²  95²  105²  125²  etc.
in the same way.

Now what do we do with these sums?

26        48        67

                             x24      x 42     x 63
                          _____   _____  _____

26      48         67

                            x 24     x 42    x 63
                          _____   _____  _____

Note that the units add up to 10. (Numbers in red colour
7 + 3 = 10.   8 + 2 = 10.   6 + 4 = 10.) and the ten's place (second
column from the right) is same (6 and 6. 4 and 4. 2 and 2.)

So we can apply the same rule 'By one more than the One before'.
Lets see what we get.

26                   48                  67

                                       x 24               x 42              x 63           
                                     ______         ______         ______
                                       6|24       20|16     42|21

In 48x42we have 4 multiplied with the next
number 5 which gives us 20 and 8x2 which
gives us 16. Therefore, our answer is 2016.

How to multiply numbers with
11-19 in less than 5 seconds?

Multiply numbers with 11

Lets take for example 32 x 11   

Step 1 : Spli 3 and 2 into left and right parts.

3  2
              
3  _       2

Step 2 : Simply add 3 + 2 and place 5 in between

3  2
       \/      
3 5         2

Step 3 : Our final answer is 352.
3     2
              
3      5           2

Try with other numbers now.

Now lets take for example 1234 x 11

The method is called Adding to the Neighbour.

In 1234 the neighbours of 2 are 1 & 3 and the
neighbours of 3 are 2 & 4 respectively.

Step 1 : We place two strokes beside
1234 and place two zeros.

0 | 1 2 3 4 | 0 

Step 2 : Now we start adding to the neighbour.
First we add 4 to 0

0 | 1 2 3 4 | 0
                    
(4 + 0 = 4)

         Gives us  :                                                  4

Step 3 : Add 3 to 4 gives us 7.

( 4 + 0 = 4 )
( 3 + 4 = 7 )

  0 | 1 2 3 4 |  0
                  

         Gives us  :                                                74         

Step 4 : Add 2 to 3 gives us 5.
( 4 + 0 = 4 )
( 3 + 4 = 7 )
( 2 + 3 = 5 )

                             0 | 1 2 3 4 |  0
                                                          

         Gives us  :                                             5 74     

Step 5 : Add 1 to 2 gives us 3.
( 4 + 0 = 4 )
( 3 + 4 = 7 )
( 2 + 3 = 5 )
( 1 + 2 = 3 )

                             0 | 1 2 3 4 |  0
                                                     

         Gives us  :                                         3   5 74  

Step 6 : Finally Add 0 to 1 gives us 1.
( 4 + 0 = 4 )
( 3 + 4 = 7 )
( 2 + 3 = 5 )
( 1 + 2 = 3 )
( 0 + 1 = 1 )

                             0 | 1 2  3 4 |  0
                                         

         Gives us  our answer:               1  3   5 74  

         Simple isn't it? Now try 12345 x 11 on your own.

Now lets take for example 777 x 11

This sum will involve a carryover

Step 1 : We place two strokes beside
777 and place two zeros.

0 | 777 | 0 

         Gives us  :                                                 7      

Step 2 : Now we start adding to the neighbour.
First we add 7 to 0.
( 7 + 0 = 7 )

0 | 777 | 0 
                

Step 3 : Add 7 to 7 gives us 14. We write
4 and take 1 as carry to the next step.
( 7 + 0 = 7 )
( 7 + 7 = ¹4 )

0 | 7 7 7 | 0 
                  

         Gives us  :                                           ¹ 4  7 

Step 4 : Add 7 to 7 gives us 14. We add 1 from carryover
gives us 15. We write 5 and again take 1 as carry over to
the next step.
( 7 + 0 = 7 )
( 7 + 7 = ¹4 )
( 7 + 7 = 14 + 1 = ¹5 )

         Gives us  :                                        ¹ 5 4  7 

Step 5 : Finally we add 0 to 7   and then add 1 of the carry
over to give us 8.

( 7 + 0 = 7 )
( 7 + 7 = ¹4 )
( 7 + 7 = 14 + 1 = ¹5 )
( 0 + 7 = 7 + 1 = 8 )

         Gives us  our final answer :       8 5 4  7

Simple isn't it? Now any number multiplied with 11.

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