jmc

It is almost always easier to use fractions than decimals when dividing. So the first task is to convert these repeating decimals to fractions. First, $.0\overline{3}$: $$\begin{gathered} 10\cdot .0\overline{3}=.\overline{3}=\frac{1}{3} \\ \Rightarrow .0\overline{3}=\frac{1}{3}÷10=\frac{1}{3}\cdot \frac{1}{10}=\frac{1}{30}. \end{gathered}$$Next, $.\overline{03}$: $$\begin{gathered} 99\cdot .\overline{03}=(100-1)\cdot .\overline{03}=3.\overline{03}-.\overline{03}=3 \\ \Rightarrow .\overline{03}=\frac{3}{99}=\frac{3}{3\cdot 33}=\frac{1}{33}. \end{gathered}$$We now have the tools to make our calculation: $$\begin{array}{rl}.0\overline{3}÷.\overline{03}& =\frac{1}{30}÷\frac{1}{33}=\frac{1}{30}\cdot \frac{33}{1}\\ & =\frac{33}{30}=\frac{3\cdot 11}{3\cdot 10}=\frac{11}{10}\\ & =\frac{10+1}{10}=\boxed{1\frac{1}{10}}.\end{array}$$