jmc

First let's convert $0.\overline{05}$ to a fraction. Let's define $x=0.\overline{05}$. If we multiply both sides by $100$ we get $100x=5.\overline{05}$ so $99x=5$ and $x=0.\overline{05}=\frac{5}{99}$. Since $1.8=\frac{9}{5}$ we get $$\frac{\cancel{5}}{99}\cdot \frac{9}{\cancel{5}}=\frac{9}{99}=\boxed{\frac{1}{11}}.$$