matha namo namaha


ĀNURŨPYENA
The upa-Sutra 'anurupyena' means 'proportionality'. This Sutra is highly useful to find products of two numbers when both of them are near the Common bases i.e powers of base 10 . It is very clear that in such cases the expected 'Simplicity ' in doing problems is absent.
Example 1: 46 X 43
As per the previous methods, if we select 100 as base we get

                    46   -54    This is much more difficult and of no use.
                    43   -57
                   ¯¯¯¯¯¯¯¯
Now by ‘anurupyena’ we consider a working base In three ways. We can solve the problem.
Method 1: Take the nearest higher multiple of 10. In this case it is 50.

        Treat it as 100 / 2 = 50. Now the steps are as follows:

i) Choose the working base near to the numbers under consideration.
i.e., working base is 100 / 2 = 50

ii) Write the numbers one below the other

                i.e.     4    6
                         4    3
                       ¯¯¯¯¯¯¯

iii) Write the differences of the two numbers respectively from 50 against each number on right side

                i.e.    46    -04
                        43    -07
                       ¯¯¯¯¯¯¯¯¯

iv) Write cross-subtraction or cross- addition as the case may be under the line drawn.
                        
v) Multiply the differences and write the product in the left side of the answer.

                           46    -04
                           43    -07
                       ____________
                        39  / -4 x –7

                             = 28

vi) Since base is 100 / 2 = 50 , 39 in the answer represents 39X50.

    Hence divide 39 by 2 because 50 = 100 / 2
Thus 39 ÷ 2 gives 19½ where 19 is quotient and 1 is remainder . This 1 as Reminder gives one 50 making the L.H.S of the answer 28 + 50 = 78(or Remainder ½ x 100 + 28 )
i.e. R.H.S 19 and L.H.S 78 together give the answer 1978 We represent it as

                    46    -04
                    43    -07
                   ¯¯¯¯¯¯¯¯¯
               2)  39  /   28
                   ¯¯¯¯¯¯¯¯¯                 
                   19½ /  28

                = 19 / 78 = 1978
Example 2:  42 X 48.
With 100 / 2 = 50 as working base, the problem is as follows:

                    42    -08
                    48    -02
                   ¯¯¯¯¯¯¯¯¯
                2) 40   /  16
                   ¯¯¯¯¯¯¯¯¯
                    20  /   16

                 42 x 48 = 2016

Method 2: For the example 1: 46X43. We take the same working base 50. We treat it as 50=5X10. i.e. we operate with 10 but not with 100 as in method

            now
                       
                                 (195 + 2) / 8  =  1978

    [Since we operate with 10, the R.H.S portion shall have only unit place .Hence out of the product 28, 2 is carried over to left side. The L.H.S portion of the answer shall be multiplied by 5, since we have taken 50 = 5 X 10.]

Now in the example 2: 42 x 48 we can carry as follows by treating 50 = 5 x 10


                       

Method 3: We take the nearest lower multiple of 10 since the numbers are 46 and 43 as in the first example, We consider 40 as working base and treat it as 4 X 10.

                    

     Since 10 is in operation 1 is carried out digit in 18.
Since 4 X 10 is working base we consider 49 X 4 on L.H.S of answer i.e. 196 and 1 carried over the left side, giving L.H.S. of answer as 1978. Hence the answer is 1978.

We proceed in the same method for 42 X 48

                    

Let us see the all the three methods for a problem at a glance

      Example 3: 24 X 23

      Method - 1:     Working base = 100 / 5 = 20

                    24    04
                    23    03
                   ¯¯¯¯¯¯¯¯
               5)  27 /  12
                  ¯¯¯¯¯¯¯¯
                  5 2/5 / 12    =  5 / 52  =  552

        [Since 2 / 5 of 100 is 2 / 5 x 100 = 40 and 40 + 12 = 52]

Method - 2:  Working base 2 X 10 = 20

                           

Now as 20 itself is nearest lower multiple of 10 for the problem under consideration, the case of method – 3 shall not arise.
Let us take another example and try all the three methods.
Example 4: 492 X 404
Method - 1 :  working base = 1000 / 2 = 500

                        492    -008
                        404    -096
                       ¯¯¯¯¯¯¯¯¯¯¯
                   2)  396  /  768      since 1000 is in operation
                       ¯¯¯¯¯¯¯¯¯¯¯
                       198  /   768       =  198768

Method 2: working base = 5 x 100 = 500

                        


Method - 3.
          Since 400 can also be taken as working base, treat 400 = 4 X 100 as working base.

            Thus
                            
        No need to repeat that practice in these methods finally takes us to work out all these mentally and getting the answers straight away in a single line.
Example 5:   3998 X 4998

                Working base = 10000 / 2 = 5000

                    3998    -1002
                    4998    -0002
                   ¯¯¯¯¯¯¯¯¯¯¯¯
                2) 3996  /  2004       since 10,000 is in operation

                   1998  /  2004        =  19982004

or taking working base = 5 x 1000 = 5,000 and

                        
   

    What happens if we take 4000 i.e. 4 X 1000 as working base?
                      _____
            3998    0002
            4998    0998        Since 1000 is an operation
           ¯¯¯¯¯¯¯¯¯¯¯¯
            4996 / 1996
                                            ___                                   ___
     As 1000 is in operation, 1996 has to be written as 1996 and 4000 as base, the L.H.S portion 5000 has to be multiplied by 4. i. e. the answer is

                      
A simpler example for better understanding.

Example 6: 58 x 48

          Working base 50 = 5 x 10 gives

                       

          Since 10 is in operation.

Use anurupyena by selecting appropriate working base and method.

      Find the following product.

        1.   46 x 46            2. 57 x 57                    3.  54 x 45

        4.   18 x 18            5. 62 x 48                    6. 229 x 230

        7.   47 x 96            8. 87965 x 99996        9. 49x499

        10. 389 x 512