Question

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

Answer 1

Let the diameter of the earth be $x$, then the diameter of the moon will be $\frac{x}{4}$.

The radius of earth will be $\frac{x}{2}$ and the radius of moon will be $\frac{1}{2}\times \frac{x}{4}=\frac{x}{8}$.

Now,

$\frac{\text{Volume of moon}}{\text{Volume of earth}}=\frac{\frac{4}{3}\pi {\left(\text{Radius of moon}\right)}^3}{\frac{4}{3}\pi (\text{Radius of earth}{)}^3}$

$=\frac{(\text{Radius of moon}{)}^3}{(\text{Radius of earth}{)}^3}$

$=\frac{{\left(\frac{x}{8}\right)}^3}{{\left(\frac{x}{2}\right)}^3}$

$=\frac{x^3}{8^3}\times \frac{2^3}{x^3}$

$=\frac{2^3}{8^3}$

$=\frac{2\times 2\times 2}{8\times 8\times 8}$

$=\frac{1}{64}$

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