The diameter of the moon is approximately one-fourth of the diameter of the earth. What fraction of the volume of the earth is the volume of the moon?

Let the diameter of earth be d.
Therefore, the diameter of moon will be \(\frac{d}{4}\).
Thus, Radius of earth = \(\frac{d}{2}\)
\(\therefore \) Radius of moon
= \(\frac{1}{2}\) × \(\frac{d}{4}\)
= \(\frac{d}{8}\)

Now, Volume of moon
= \(\frac{4}{3}{\pi}{r}^3 \)
= \(\frac{4}{3}{\pi} × {\frac{d}{8}}^3 \)
=\( \frac{1}{512} × \frac{4}{3}{\pi} × {d}^3\)

Similarly,
Volume of earth
= \(\frac{4}{3}{\pi}{r}^3 \)
=\( \frac{4}{3}{\pi} × {\frac{d}{2}}^3 \)
= \(\frac{1}{8} × \frac{4}{3}{\pi} × {d}^3\)

\(\therefore \) \(\frac{Volume of moon}{Volume of earth} \)
=\( \frac{\frac{1}{512} × \frac{4}{3}{\pi} × {d}^3}{\frac{1}{8} × \frac{4}{3}{\pi} × {d}^3} \)
= \(\frac{1}{64}\)

Therefore, the volume of moon = \(\frac{1}{64}\) is of the volume of earth.