Vedic Mathematics : Recurring decimal Expression of Vulgar Fraction with denominator ending with '9'

 

Vedic Mathematics

Suthra 1     :        Ekadhikena Purvena

Application : Recurring Decimal Expansion of Vulgar Fractions with denominator ending with ‘9’.

Meaning    :        By One More than the Previous One

We can apply the same suthra to find out the decimal representation(Recurring Decimal)  of VULGAR fractions with denominator ending with the digit ‘9’.

There is another method to find out the result of the same operation. In the previous post, you have seen by using ‘Division Method’. Here we use a simple technique, ‘Multiplication Method’. Here it goes…..

 

Let us use this suthra to find out the recurring decimal of

In, 1/19 – PREVIOUS of 9 is ‘1’.

And ONE MORE THAN of it is ‘1+1=2’.

Hence ‘2’ is the multiplier of the conversion.

Write the last digit in the numerator as 1

And follow the following steps…..

Step 1 :           1

Step 2 :           21 (multiply 1 by 2, write the result on left)

Step 3 :           421 (multiply 2 by 2, write it on left)

Step 4             :           8421 (multiply 4 by 2, write it on left)

Step 5 :           ₁68421(multiply 8 by 2, write 6 on left, carry over 1)

Step 6 :           ₁368421(multiply 6 by 2,add the carry over,with carry over 1)

Step 7 :           7368421(3 x 2 + 1 = 7)

Step 8 :           ₁47368421(7x2=14, 4 with carry over 1)

Step 9             :           947368421(4x2+1=9)

Step 10:          ₁8947368421(9x2=18, 8 carry over 1)

Step 11:          ₁78947368421(8x2+1=17, 7 with carry over 1)

Step 12:          ₁578947368421(7x2+1=15, 5 carry over 1)

Step 13:          ₁1578947368421(5x2+1=11, 1 carry over 1)

Step 14:          31578947368421(1x2+1=2)

Step 15:          631578947368421(3x2=6)

Step 16:          ₁2631578947368421(6x2=12, 2 carry over 1)

Step 17:          52631578947368421(2x2+1=5)

Step 18:          ₁052631578947368421

Next step onwards, these 18 digits repeat……..like recurring decimal…

Hence 1/19 = 0. 05231578947368421…..