6. Trick - Division by numbers ending in 9 (Part I)

The sutra (formula) that empowers me to divide by numbers ending in 9 is again "Ekadhikena Purvena" meaning by one more than the previous one

The division can be done in two ways - by division and by multiplication .

I will take up by multiplication in this article. I will illustrate by taking two examples, 1/19 and 1/7 = 7/49

The numerator in 1/19 is 1. So, we start with 1.

Digit previous to 9 is 1 and one greater than 1 is 2. We will deal with multiplications by 2 only.

Multiply 1 by 2 to get 2

Now, we have 21.

Multiply 2 by 2 to get 4

Now, we have 421.

Multiply 4 by 2 to get 8

Now, we have 8421.

Multiply 8 by 2 to get 6 as base-number and 1 as carry over.

Now, we have ₁68421.

Multiply base-number 6 by 2 to get 2 as base-number and 1 as carry over. Add the previous carry over 1 to get 3 as base number.

Now, we have ₁368421.

Carry on till you have a total of (19-1)/2 = 9 digits in place. And they would be 947368421. We call is Lower Half. To calculate the Upper Half, take 9's complement.

  999999999
 -947368421
------------
  052631578
------------

The result, therefore, is:

 0.052631578947368421

Enjoy! Now the example of 1/7. It will contain a total of 7 - 1 = 6 digits that will recur.

1/7 = 7/49. The number previous to 9 is 4 and one next to 4 is 5.

First, write 7. Multiply by 5, yielding 5 as base-number and 3 as carry overy - ₃57. The full chain is as below:

7 => 357 => 2857
  => 42857 => 142857

Therefore,

 1/7 = 0.142857