A large sphere has a volume of $288\pi$ cubic units.  A smaller sphere has a volume which is $12.5\%$ of the volume of the larger sphere.  What is the ratio of the radius of the smaller sphere to the radius of the larger sphere? Express your answer as a common fraction.
To solve this problem, I need to find the relationship between the volumes and the radii of the spheres. I recall that the volume of a sphere is given by the formula $V = \frac{4}{3}\pi r^3$, where $r$ is the radius. Let $r_1$ be the radius of the larger sphere and $r_2$ be the radius of the smaller sphere. Then, using the given information, I can write the equation $\frac{4}{3}\pi r_2^3 = 0.125 \cdot \frac{4}{3}\pi r_1^3$. Simplifying, I get $r_2^3 = 0.125 r_1^3$. Taking the cube root of both sides, I get $r_2 = 0.5 r_1$. Therefore, the ratio of the radius of the smaller sphere to the radius of the larger sphere is $\frac{r_2}{r_1} = 0.5 = \frac{1}{2}$. 