Hi had a question on how to solve this. The answer is 10^w -1. When watching the video I understood how to convert a repeating decimal into a fraction. I tried to apply that same process here but got it wrong and did not understand what to do with the “w” or why they raised 10^w in the explanation.

What is 0.\overline{1} and 0.\overline{12} written as a fraction? What is w in these two cases?

1/9 and 12/99. Would “w” be 1 and 2 since it is the number of digits after the decimal point?

Correct. Do you now see what the correct option is, and why?

Is it because the minus 1 will always make it some form of 9 no matter what 10^w is ?

Correct. In fact it would *only* contain 9s.

When initially solving the problem I solved it how you would convert it to a fraction. Set it to x multiply by 10 then subtract x and then divide. But what was the point of raising it to “w”? Or I guess how would you have known to raise it to the “w”. I didn’t know what to do with the w.

Because w is the number of digits after the decimal point, and the repeating decimal to fraction process involves multiplying by 10^w, as you saw when doing the two examples I presented earlier.

It’s just applying the process to a theoretical problem.

A general tip for problems like this is to try a couple of examples to see what’s going on.