Anurupyena

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
‾‾‾‾‾‾‾‾
76

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
77

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 answer1978 We represent it as
46 -04
43 -07
‾‾‾‾‾‾‾‾‾
2) 39 / 28
‾‾‾‾‾‾‾‾‾
19½ / 28
= 19 / 78 = 1978

Ekanyunena Purvena

The Sutra Ekanyunena purvena comes as a Sub-sutra to Nikhilam which gives

the meaning 'One less than the previous' or 'One less than the one before'.

1) The use of this sutra in case of multiplication by 9,99,999.. is as follows .

Method :

a) The left hand side digit (digits) is ( are) obtained by applying the ekanyunena

purvena i.e. by deduction 1 from the left side digit (digits) .

e.g. ( i ) 7 x 9; 7 – 1 = 6 ( L.H.S. digit )

b) The right hand side digit is the complement or difference between the

multiplier and the left hand side digit (digits) . i.e. 7 X 9 R.H.S is 9 - 6 = 3.

c) The two numbers give the answer; i.e. 7 X 9 = 63.

Puranapuranabhyam

The Sutra can be taken as Purana - Apuranabhyam which means by the

completion or non - completion. Purana is well known in the present

system. We can see its application in solving the roots for general form of

quadratic equation.

We have : ax2 + bx + c = 0

x2 + (b/a)x + c/a = 0 ( dividing by a )

x2 + (b/a)x = - c/a

completing the square ( i.e.,, purana ) on the L.H.S.

x2 + (b/a)x + (b2/4a2) = -c/a + (b2/4a2)

[x + (b/2a)]2 = (b2 - 4ac) / 4a2
________

- b ± √ b2 – 4ac

Proceeding in this way we finally get x = _______________

2a

Anurupye - Sunyamanyat

The Sutra Anurupye Sunyamanyat says : "If one is in ratio, the other one is zero".

We use this Sutra in solving a special type of simultaneous simple equations
in which the coefficients of 'one' variable are in the same ratio to each other
as the independent terms are to each other. In such a context the Sutra says
the 'other' variable is zero from which we get two simple equations in the first
variable (already considered) and of course give the same value for the variable.

Example 1: 

 3x + 7y = 2

 4x + 21y = 6

Observe that the y-coefficients are in the ratio 7 : 21 i.e., 1 : 3, which is

 same as the ratio of independent terms i.e., 2 : 6 i.e., 1 : 3. Hence the other

 variable x = 0 and 7y = 2 or 21y = 6 gives y = 2 / 7


Example 2:

323x + 147y = 1615

969x + 321y = 4845

The very appearance of the problem is frightening. But just an observation

and anurupye sunyamanyat give the solution x = 5, because coefficient of x

ratio is

323 : 969 = 1 : 3 and constant terms ratio is 1615 : 4845 = 1 : 3.

y = 0 and 323 x = 1615 or 969 x = 4845 gives x = 5.

Sunyam Samya Samuccaye

The Sutra 'Sunyam Samyasamuccaye' says the 'Samuccaya is the same, that
Samuccaya is Zero.' i.e., it should be equated to zero. The term 'Samuccaya'
has several meanings under different contexts.

i) We interpret, 'Samuccaya' as a term which occurs as a common factor in all
the terms concerned and proceed as follows.

Example 1: The equation 7x + 3x = 4x + 5x has the same factor ‘ x ‘ in all its terms. Hence by the sutra it is zero,i.e., x = 0.

 Otherwise we have to work like this:
 7x + 3x = 4x + 5x
 10x = 9x
 10x – 9x = 0
 x = 0
 This is applicable not only for ‘x’ but also any such unknown quantity as follows.

Paravartya Yojayet

'Paravartya – Yojayet' means 'transpose and apply'

(i) Consider the division by divisors of more than one digit, and when the
divisors are slightly greater than powers of 10.

Example 1 : Divide 1225 by 12.

Step 1 : (From left to right ) write the Divisor leaving the first digit, write the
other digit or digits using negative (-) sign and place them below the divisor
as shown.
12
-2
‾‾‾‾

Step 2 : Write down the dividend to the right. Set apart the last digit for the
remainder.
42
i.e.,, 12 122 5
- 2

Step 3 : Write the 1st digit below the horizontal line drawn under
thedividend. Multiply the digit by –2, write the product below the 2nd digit
and add.
i.e.,, 12 122 5
-2 -2
‾‾‾‾‾ ‾‾‾‾
10
Since 1 x –2 = -2and 2 + (-2) = 0

Step 4 : We get second digits’ sum as ‘0’. Multiply the second digits’ sum
thus obtained by –2 and writes the product under 3rd digit and add.
12 122 5
- 2 -20
‾‾‾‾ ‾‾‾‾‾‾‾‾‾‾
102 5

Step 5 : Continue the process to the last digit.
i.e., 12 122 5
- 2 -20 -4
‾‾‾‾‾ ‾‾‾‾‾‾‾‾‾‾
102 1

Step 6: The sum of the last digit is the Remainder and the result to its left is
Quotient.
Thus Q = 102 andR = 1
Example 2 : Divide 1697 by 14.
14 1 6 9 7
- 4 -4–8–4
‾‾‾‾ ‾‾‾‾‾‾‾
1 2 1 3
Q = 121, R = 3.

Example 3 : Divide 2598 by 123.

Note that the divisor has 3 digits. So we have to set up the last two
43
digits of the dividend for the remainder.
1 2 3 25 98 Step ( 1 ) & Step ( 2 )
-2-3
‾‾‾‾‾ ‾‾‾‾‾‾‾‾
Now proceed the sequence of steps write –2 and –3 as follows :
1 2 3 25 98
-2-3 -4 -6
‾‾‾‾‾ -2–3
‾‾‾‾‾‾‾‾‾‾
21 1 5

Since 2 X (-2, -3)= -4 , -6;5 – 4 = 1
and (1 X (-2,-3); 9 – 6 – 2 = 1; 8 – 3 = 5.
Hence Q = 21 and R = 15.

Urdhva - tiryagbhyam

Urdhva –  tiryagbhyam is the general formula applicable to all cases of
multiplication and also in the division of a large number by another large
number. It means


(a) Multiplication of two 2 digit numbers.

Ex.1: Find the product 14 X 12

i) The right hand most digit of the multiplicand, the first number (14) i.e.,4 is
multiplied by the right hand most digit of the multiplier, the second number
(12) i.e., 2. The product 4 X 2 = 8 forms the right hand most part of the answer.

ii) Now, diagonally multiply the first digit of the multiplicand (14) i.e., 4 and
second digit of the multiplier (12)i.e., 1 (answer 4 X 1=4); then multiply the
second digit of the multiplicand i.e.,1 and first digit of the multiplier i.e., 2
(answer 1 X 2 = 2); add these two i.e.,4 + 2 = 6. It gives the next, i.e., second
digit of the answer. Hence second digit of the answer is 6.

iii) Now, multiply the second digit of the multiplicand i.e., 1 and second digit of
the multiplieri.e., 1 vertically, i.e., 1 X 1 = 1. It gives the left hand most part of
the answer.

Thus the answer is 16 8.