How to convert a repeating decimal into a ratio of two integers

We know that $\frac{2}{3}$ is $0.666\bar{6}$, but given the number $0.666\bar{6}$ how could we know its ratio of two integers value is $\frac{2}{3}$?

Let $x = 0.666\bar{6}$

Multiply $x$ by the appropriate powers of $10$, then subtract to eliminate the repeating part.

$\begin{aligned} 10x &= 6.666\bar{6} \\ x &= 0.666\bar{6} \\ 9x &= 6 \\ x &= \frac{6}{9} \\ x &= \frac{2}{3} \end{aligned}$