G4.3 Given that $m, c$ are positive integers less than 10 . If $m=2 c$ and $0 . \dot{m} \dot{c}=\frac{c+4}{m+5}$, find the value of $c$.
$\begin{array}{l}\text { 0. } \dot{m} \dot{c}=\frac{10 m+c}{99}=\frac{c+4}{m+5} \\ \Rightarrow \frac{20 c+c}{99}=\frac{c+4}{2 c+5} \\ \Rightarrow \frac{7 c}{33}=\frac{c+4}{2 c+5} \\ \Rightarrow 14 c^{2}+35 c=33 c+132 \\ 14 c^{2}+2 c-132=0 \\ \Rightarrow 7 c^{2}+c-66=0 \\ \Rightarrow(7 c+22)(c-3)=0 \\ \Rightarrow c=3\end{array}$