One Less Than The Previous

One less than the previous or One less than the before :As the name indicates this sutra involves subtracting one from the given number to get our final result.This sutra is highly helpful in case of multiplication by 9,99,999....to any other number and in solving fractions of certain numbers like (1/7),(1/13),(1/17)..etc.

PART 1: For solving Multiplications

Example 1:

6 * 9 = ?

 

Step 1 : Minus one from the number on L.H.S digit  above

Step 2 : Minus the answer(result) got from step 1 from R.H.S digit (i.e  5 from number 9)

6-1 = 5

9-5 = 4

Combine them LHS,RHS ,to get 54

Therefore,The Result for 6 * 9 = 54

Example 2:

Now lets try the same with 99

75 * 99 =?

Step 1 : 75 -1 = 74

Step 2 : 99 -74 = 25

Combining the above result, 7425

Therefore,The Result : 75 * 99 = 7425

Example 3:

Now lets try the same again with 999

899 * 999 =?

Step 1: 899-1 = 898

Step 2: 999-898 = 101

Combining the above, 898101

Therefore,The Result :899 * 999 = 898101

Example 4:

Try 9765 * 9999 =?

Step 1 : 9765 - 1 = 9764

Step 2 : 9999 - 9764  = 0235

Therefore,The Result : 9765 * 9999 = 97640235

Note:

This can be done to any number of digits of 9...is'nt  it wonderful ! .Wondered why?Thats the Beauty of number '9'.Check  here to get the answer.

Algebraic Proof:

Take any two digit number in form of (10 x + y)

 Now,multiplying it with (10x + y ) x 99
                 = ( 10x + y ) x ( 100 – 1 )

                 = 10x . 100 - 10x +100y - y
                 = 10x . 10² – 10x + 10² .y – y
                 = x . 10³ + y . 10² – ( 10x + y )
                 = x . 10³ + ( y – 1 ) . 10² + [ 10² – ( 10x + y )]

Look at our answer.We can that it is a four digit number whose 1000^th place is x, 100^th place is ( y - 1 ) and the two digit number which makes up the 10^th and unit place is the number obtained by subtracting the multiplicand from 100.

Thus in Example 2  ,

75 * 99. 

The 1000^th place is x i.e. 7
The 100^th place is ( y - 1 ) i.e. (5 - 1 ) = 4
Number in the last two places 100-75=25.
We got the answer as 7425

Giving, 75 * 99 =7425

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