Question

What is 0.482 repeating as a simplified fraction?

Answer 1

the idea being, we multiply the value by some power of 10 just to move the repeating digits to the left of the decimal point.  We have 3 digits, we'll use 1000, anyhow, without further adieu,

$\bf \begin{array}{|ccl|ll} \cline{1-3} \stackrel{\qquad}{x=0.\overline{482}}&&\\ \cline{1-3} &&\\ \stackrel{\textit{3 digits, we use 3 zeros}}{1000\cdot x}&=&482.\overline{482}\\ &&482+0.\overline{482}\\ &&482+x\\ &&\\ \cline{1-3} \end{array}\implies 1000x=482+x \\\\\\ 999x=482\implies x=\cfrac{482}{999}$