Sutra 1: "By one more than the one before"

provides a simple way of multiplication of numbers with the same first digit and the sum of their last unit digits is 10.

1 An interesting application of this formula is in computing squares of numbers ending in five. Consider

35 × 35 = ((3 × 3) + 3)*10 + 25 = 1225 or
= (3 x 4) *10 +(5 x 5) = 1225
125 x 125 = (12 x 13)*10 + 25 = 15625

Explanation: The latter portion is multiplied by itself (5 by 5) and the previous portion is multiplication of first digit and the digit higher to the first digit resulting in the answer 1225

Example 2 :
----------------------

Method 1
45*45 = (( 4*4) + 4 )*10 + 25
200
25
------
2025
or

Method 2 :
45 * 45 = (4*5 ) * 100 + 5*5
= 20 * 100 + 25
= 2000 + 25
= 2025


2 It can also be applied in multiplications when the last digit is not 5 but the sum of the last
digits is the base (10) and the previous parts are the same. Consider

37 × 33 = (3 × 4)*100 + (7 × 3) = 1221


Example 2 :

29 × 21 = (2 × 3)*100 + (9 × 1) = 609


Example 3 :

32 * 38 = ( 3 * 4) * 100 + 8 * 2

1200
16
-------
1216

3 The Sutra is very useful in its application to convert fractions into their equivalent
decimal form.

Consider fraction 1/19. Using this formula, this can be converted into a decimal form in a single step. This can be done by applying the formula for either a multiplication or division operation, thus yielding two methods.

a) ( Multiplication Method )


Using multiplication. The sutra "one more than the one before" provides a simple way of calculating values like 1/x9 (e.g 1/19, 1/29, etc). Let's take one 1/x9 and calculate e.g. 1/19. In this case, x=1. To convert 1/19 to decimals, since 19 is not divisible by 2 or 5, the fractional result is a purely circulating (recurring) decimal.

Example 1

Steps :

1) start with the numerator digit.Ie 1 apply the sutra 1. ie By one more than the one before
multiply the first digit by the number one more than before.
let the number be x = 1 , this number should be multiplied by numbr 1 more than before. ie x+1 = 2.

b) Place the number before the original number.

2x = 21 ( where x=1 ). Here is should not be consired to moultiply. Just prefix 2 before the original number.

3) multiply the first number from 2 & prefix the number to preivios number.
now the number is
(2*2)21
= 421
(4*2)421
= 8421

(8*2) 8421
= (16)8421 , if the number obtained after multiplying by 2 is greater than 10.keep the number as carryover & add it to the next digit
= 68421 ( carryover is 1 )

(6*2)68421
= ( 12 ) 68421 , add the carryover (1) to the number 12
= ( 13 ) 68421
= 368421 carryover 1

= (3*2) 68421
= 768421

= (7*2) 768421
= 4768421 carry over 1

= 9768421

Now we have 9 digits of the answer. There are a total of 18 digits (= denominator − numerator i.e 19-1 = 18) in the answer. The last 9 digits can be computed by complementing the lower half (with its complement from nine i.e number + complement = 9):

052631578
947368421

Thus the result is 1/19 = 0.052631578,947368421 repeating.

If you picked up 1/29, you'll have to do it till 28 digits (i.e. 29-1). You'll get the following
1/29 = 03448275862068,
96551724137931
Run this on your calculator and check the result!

Example 2 :

1/19 decimal value =

x = 1

1+1 = 2

= 1
= 21
= 421
= 8421
= 68421
= 368421 1
= 7368421
= 47368421 1
= 947368421
= 8947368421 1
= 78947368421 1
= 578947368421 1
= 1578947368421 1
= 31578947368421
= 631578947368421
= 2631578947368421 1
= 52631578947368421
= 052631578947368421 1
= 1052631578947368421
= 21052631578947368421
= 421052631578947368421
= 8421052631578947368421
= 68421052631578947368421 1
= 368421052631578947368421 1
= 7368421052631578947368421

7368421
052631578947368421

The value od 1/29 = 0.052631578947368421

b) ( Division Method )

Using division. The earlier process can also be done using division instead of multiplication.

For A/X9, We divide A by (1+X). Incase of 1/19, we divide 1/(1+1), the answer is 0 (lets say N) with remainder 1 (lets say D)


1 Divide the number "A" by "1+X".where X value is the first number in the denominator.
2 0.0: D (in this case 1) carries forward and become (D*10 + N) i.e. 10. This is then divided by
2 for N = 5. postfix the value of N to the existng value.
3 0.05 : next dive 5 by 2 , the N = 2 & D =1
4 0.052 Next 12 ( D*10 + N = 1*10 + 2 , 1 from earlier remiander and 2 from the answer) is
divided by 2 for answer 6 ie N = 3 , D = 0
5 0.0526 Next divide 6 by 2 , N = 3 ,D = 0
6 0.05263 Next Divide 3 by 2 , N = 1 D = 1
7 0.052631 next ( D*10 + N = 1*10 + 1 , 1 from earlier remiander and 1 from the answer) is
divided by 2 , ie N = 5 , D = 1
8 0.0526315 next ( 1*10 + 5 = 15 ) by 2 , ie N = 7 D =1
9 0.05263157 next ( 1*10 + 7 ) N 8 , D = 1
10 0.052631578

Value of 1/19 = 0.052631578 , use the calculator to verify the answer.


Example 2 :

For A/X9For A/X9, We divide A by (1+X). Incase of 1/19, we divide 1/(1+1), the answer is 0 (lets say N) with remainder 1 ( lets say D)

x= 2
1/29 = 1/1+2 - = 1/3 N = 0 , D = 1

0.0 1*10 + 0 ( D*10 + N ) , Divide by 3 N=3 D=1

0.03 1*10 + 3 = 13 / 3 Divide by 3 , N = 4 , D = 1
0.034 1*10 + 4 = 14 / 3 ie N = 4 . D = 2
0.0344 2*10 + 4 = 24 / 3 ie N = 8 , D = 0
0.03448 0 + 8 = 8 / 3 = N = 2 , D = 2 0.034482 20 + 2 = 22 / 3 ie N = 7 , D = 1
0.0344827