The diameter of the moon is approximately one-fourth of the diameter of the earth.

Solution:

Let the diameter of earth be $d$. Therefore, the radius of earth will be $\frac{d}{2}$.

Diameter of moon will be $\frac{d}{4}$ and the radius of moon will be $\frac{d}{8}$.

Volume of moon $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi\left(\frac{d}{8}\right)^{3}=\frac{1}{512} \times \frac{4}{3} \pi d^{3}$

Volume of earth $=\frac{4}{3} \pi r^{3}=\frac{4}{3} \pi\left(\frac{d}{2}\right)^{3}=\frac{1}{8} \times \frac{4}{3} \pi d^{3}$

$\frac{\text { Volume of moon }}{\text { Volume of earth }}=\frac{\frac{1}{512} \times \frac{4}{3} \pi d^{3}}{\frac{1}{8} \times \frac{4}{3} \pi d^{3}}$

$=\frac{1}{64}$

$\Rightarrow$ Volume of moon $=\frac{1}{64}$ Volume of earth

Therefore, the volume of moon is $\frac{1}{64}$ of the volume of earth.