Question

What is the number with decimal expansion $0.00\overline{266}$?

Answer 1

Let $r=0.00\overline{266}$. Then we obtain ${10}^{2}r=0.\overline{266}$ and ${10}^{5}r=2.66\overline{266}$. Hence $({10}^5-{10}^2)r=266$. This gives $r=\frac{266}{99900}.$
We see that every time we get a rational number. we can make a general statement.