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, Ekadhikena Purvena process can be effectively used both in division and multiplication.

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 : 168421 (multiply 8 by 2=16, 1 carried over, 6 put to left)
Step. 6 : 1368421 ( 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 : 147368421 (as in the same process)
Step. 9 : 947368421 ( Do – continue to step 18)
Step. 10 : 18947368421
Step. 11 : 178947368421
Step. 12 : 1578947368421
Step. 13 : 11578947368421
Step. 14 : 31578947368421
Step. 15 : 631578947368421
Step. 16 : 12631578947368421
Step. 17 : 52631578947368421
Step. 18 : 1052631578947368421
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