Example 1. If the total surface area of a cone is $n$ times the area of its inscribed sphere, find the ratio of the volume of the sphere to the volume of the cone.
Let $\mathrm{r}$ be the radius of the sphere, $\mathrm{S}$ and $\mathrm{V}$ be the surface area and volume of the sphere, $\mathrm{S}^{\prime}$ and $\mathrm{V}^{\prime}$ be the total surface area and volume of the cone, then
$$
\mathrm{V}=\frac{1}{3} \mathrm{Sr}, \quad \mathrm{V}^{\prime}=\frac{1}{3} \mathrm{~S}^{\prime} \mathrm{r} .
$$

Therefore, $V: V^{\prime}=\frac{1}{3} S r: \frac{1}{3} S^{\prime} r=S: S^{\prime}=1: n$.