smc

Here, we use Mass Points. Let $AF=x$. We then have $AE=2x$, $EC=16-2x$, and $FB=12-x$ Let $B$ have a mass of $2$. Since $M$ is the midpoint, $C$ also has a mass of $2$. Looking at segment $AB$, we have $$2\cdot (12-x)=\text{m}{A}_{AB}\cdot x$$ So $$\text{m}{A}_{AB}=\frac{24-2x}{x}$$ Looking at segment $AC$,we have $$2\cdot (16-2x)=\text{m}{A}_{AC}\cdot 2x$$ So $$\text{m}{A}_{AC}=\frac{16-2x}{x}$$ From this, we get $$\text{m}E=\frac{16-2x}{x}+2\Rrightarrow \text{m}E=\frac{16}{x}$$ and $$\text{m}F=\frac{24-2x}{x}+2\Rrightarrow \text{m}F=\frac{24}{x}$$ We want the value of $\frac{EG}{GF}$. This can be written as $$\frac{EG}{GF}=\frac{\text{m}F}{\text{m}E}$$ Thus $$\frac{\text{m}F}{\text{m}E}=\frac{\frac{24}{x}}{\frac{16}{x}}=\frac{3}{2}$$ $\boxed{\frac{3}{2}}$