### Vulgar fractions whose denominators are numbers ending in NINE

We now take examples of 1 / a9, where a = 1, 2, -----, 9. In the conversion of such vulgar fractions into recurring decimals,

*process can be effectively used both in division and multiplication.*

**Ekadhikena Purvena**Multiplication Method: Value of 1 / 19

First we recognize the last digit of the denominator of the type 1 / a9. Here the last digit is 9. For a fraction of the form in whose denominator 9 is the last digit, we take the case of 1 / 19 as follows: For 1 / 19, 'previous' of 19 is 1. And one more than of it is 1 + 1 = 2. Therefore 2 is the multiplier for the conversion.

We write the last digit in the numerator as 1 and follow the steps leftwards.

Step. 1 : 1

Step. 2 : 21(multiply 1 by 2, put to left)

Step. 3 : 421(multiply 2 by 2, put to left)

Step. 4 : 8421(multiply 4 by 2, put to left)

Step. 5 :

*1*68421 (multiply 8 by 2=16,

*1*carried over, 6 put to left)

Step. 6 :

*1*368421 ( 6 X 2 =12,+1 = 13,

*1*carried over, 3 put to left )

Step. 7 : 7368421 ( 3 X 2, = 6 +1 = 7, put to left)

Step. 8 :

*1*47368421 (as in the same process)

Step. 9 : 947368421 ( Do – continue to step 18)

Step. 10 :

*1*8947368421

Step. 11 :

*1*78947368421

Step. 12 :

*1*578947368421

Step. 13 :

*1*1578947368421

Step. 14 : 31578947368421

Step. 15 : 631578947368421

Step. 16 :

*1*2631578947368421

Step. 17 : 52631578947368421

Step. 18 :

*1*052631578947368421

Now from step 18 onwards the same numbers and order towards left continue.

Thus 1 / 19 = 0.052631578947368421

Find the recurring decimal form of the fractions 1 / 29, 1 / 59, 1 / 69, 1 / 79, 1 / 89 using Ekadhika process

## 1 Comments:

This method is little lengthy.

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