Repeating Decimals in Reverse

These problems give you a fraction and ask for the first few digits of its decimal representation. This requires working the repeating decimal formulas in reverse.

For a fraction like $\frac{23}{90}$, you know the denominator 90 corresponds to a decimal of the form $.abbb...$. You need to find digits $a$ and $b$ such that the two-digit number $ab - a = 23$. It's quick to see that $25 - 2 = 23$, so the decimal is $.2555...$.

If the fraction has been reduced (e.g., $\frac{17}{45}$), you must first un-reduce it to a denominator of 90, 99, 990, etc. In this case, $\frac{17}{45} = \frac{34}{90}$. Then solve $ab - a = 34$, which gives $ab=37$. The decimal is $.3777...$.