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Question
Hello:

How is a repeating decimal like 0.272727... changed to the fraction 3/11 by using a geometric series?

Please to not omit any steps in your explanation/calculation!

I thank you for your reply.

Answer
Let x = 0.2727...
The repeating digits are 27.

Multiply x by 100 so there is one repeat to the left of the decimal point
100x = 27.2727...

subtract the first equation from the second equation
100x = 27.2727...
  -x = -0.2727...
---------------
  99x = 27

x = 27/99 = 3/11

I just realized that I didn't use a geometric series to answer the question. I've never tried it that way, but if I find out how, I'll revise this.

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