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?

Given,

⇒ Diameter of the moon (d) = $\dfrac{1}{4}$ × diameter of the earth (D)

⇒ $\dfrac{\text{d}}{2} = \dfrac{1}{4} \times \dfrac{D}{2}$

⇒ The radius of the moon = $\dfrac{1}{4}$ × radius of the earth

Let radius of moon be r and radius of earth be R.

⇒ r = $\dfrac{1}{4}\times R$

⇒ r = $\dfrac{R}{4}$ …..(1)

Since, earth and moon are spherical.

Volume of the earth (V) = $\dfrac{4}{3}πR^3$

Volume of the moon (v) = $\dfrac{4}{3}πr^3$

Substituting value of r from equation (1) in above equation, we get :

$\Rightarrow v = \dfrac{4}{3}\times π \times \Big(\dfrac{R}{4}\Big)^3 (\text{From equation (1)}) \\[1em] \Rightarrow v = \dfrac{4}{3} \times π \times \dfrac{R}{4} \times \dfrac{R}{4} \times \dfrac{R}{4} \\[1em] \Rightarrow v = \dfrac{1}{64} \times \dfrac{4}{3}π R^3 \\[1em] \Rightarrow v = \dfrac{1}{64} \times V.$

Hence, the volume of the moon is $\dfrac{1}{64}$ times the volume of the earth.