Vedic Mathematics | Upa-Sutras anurupyena

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

                            

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

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