Let’s say you wanted to convert 0.83838383 … to a decimal. There is a very simple process to doing this. Let me walk you through it:

set x = 0.838383 …, then multiply x by the power of 10 equal to the number of digits that your decimal repeats. In this case the 83 repeats so you will multiply x by 10^{2} or 100. When you do this you will get 100x = 83.838383 …

Now subtract x from 100x. When you do this you get 99x = 83. The 0.838383 … goes away! Now solve for x by dividing both sides by 99 and you get x = 83/99.

Final step, check if you can simplify it … in this case no.

Let’s take a look at another one. Lets say I have 0.5617617617617 …

In this case the 617 repeats so I first set x = 0.5617617617617 and then multiply x by 10^{3} or by 1000. 1000x = 561.7617617617 …

O.k. so now subtract x from 1000x to get 999x = 561.2 then multiply both sides by 10 to get rid of the decimal so you have 9990x = 5612. Solving for x you get x = 5612/9990 then simplifying you get 2806/4995 and it’s as simple as that!

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