Converting repeating decimals of other bases (base 6 etc.) into fractions of that base

Question

My question is about figuring out how to write a fraction in base 6 from the repeating decimal: $0.\overline{113}$ from base 6.

Answer 1

For $x = 0.\overline{113}$ use $1000x = 113.\overline{113}$

Do the subtraction in base $6$, to get $555_6x = 113_6$

So $x = \frac{113_6}{555_6}$. Which reduces to $\frac{5\cdot13_6}{5\cdot111_6} = \frac{13_6}{111_6}$ (base 6).

To see that the reduction is possible, unless you are very familiar with multiplication and factoring in base 6, it is helpful to convert to base 10.

Answer 2

While the posted answer gives the direct solution, the following answers this side question.

tried converting from base 6 to base 10 (...) but it didn't lead to the answer

This does also work, albeit at the expense of a double conversion to and from base $10\,$:

$$ 0.\overline{113}_6=0.113113113\cdots_{\,6}=\sum_{n \ge 1} 113_6 \cdot 10_6^{-3n} = \sum_{n \ge 1} 45_{10} \cdot 6_{10}^{-3n} \\ = 45_{10}\cdot 6_{10}^{-3} \sum_{n \ge 0} 6_{10}^{-3n} = \frac{45_{10}}{6_{10}^3} \cdot \frac{1}{1 - 6_{10}^{-3}} = \frac{45_{10}}{6_{10}^3-1} = \frac{45_{10}}{215_{10}} = \frac{9_{10}}{43_{10}} = \frac{13_6}{111_6} $$