Establish the relationship between volume and radius

Since the volume $V$ of a sphere is proportional to the cube of the radius $r$ ($V \propto r^3$), if $V_1:V_2 = 1:8$, then the ratio of their radii is the cube root of the volumes.

$\frac{r_1}{r_2} = \sqrt[3]{\frac{1}{8}} = \frac{1}{2}$

Establish the relationship between radius and surface area

The surface area $A$ of a sphere is proportional to the square of its radius ($A \propto r^2$). Thus, the ratio of the surface areas is the square of the ratio of the radii.

$\frac{A_1}{A_2} = \left(\frac{r_1}{r_2}\right)^2 = \left(\frac{1}{2}\right)^2$

Calculate the final ratio

Squaring the ratio of the radii gives the final ratio of the surface areas.

$\frac{A_1}{A_2} = \frac{1}{4}$