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 diameter of moon will be $\frac{d}{4}$.

Radius of earth $=\frac{d}{2}$

Radius of moon $=\frac{1}{2} \times \frac{d}{4}=\frac{d}{8}$

Surface area of moon $=4 \pi\left(\frac{d}{8}\right)^{2}$

Surface area of earth $=4 \pi\left(\frac{d}{2}\right)^{2}$

Required ratio $=\frac{4 \pi\left(\frac{d}{8}\right)^{2}}{4 \pi\left(\frac{d}{2}\right)^{2}}$

$=\frac{4}{64}=\frac{1}{16}$

Therefore, the ratio between their surface areas will be 1:16.