If the surface area of two spheres are in the ratio of 4 : 25, then the ratio of their volumes is :

Answer: Option D

Let their radii be R and r
Then,
$$\begin{array}{rl}& \frac{4\pi R^2}{4\pi r^2}=\frac{4}{25}\\ & \Rightarrow {\left(\frac{R}{r}\right)}^2={\left(\frac{2}{5}\right)}^2\\ & \Rightarrow \frac{R}{r}=\frac{2}{5}\end{array}$$
∴ Ratio of volumes :
$$\begin{array}{rl}& =\frac{\frac{4}{3}\pi R^3}{\frac{4}{3}\pi r^3}\\ & ={\left(\frac{R}{r}\right)}^3\\ & ={\left(\frac{2}{5}\right)}^3\\ & =\frac{8}{125}\\ & =8:125\end{array}$$