Explanation:

- The volume of a sphere is proportional to the cube of its radius: $V\propto r^3$.

- Given the volume ratio $\frac{V_1}{V_2}=\frac{27}{64}$, we have ${\left(\frac{r_1}{r_2}\right)}^3=\frac{27}{64}$.

- This simplifies to $\frac{r_1}{r_2}=\frac{3}{4}$.

- The surface area of a sphere is proportional to the square of its radius: $A\propto r^2$.

- Using $\frac{r_1}{r_2}=\frac{3}{4}$, the ratio of surface areas $\frac{A_1}{A_2}={\left(\frac{3}{4}\right)}^2=\frac{9}{16}$.

Correct Answer: Option 1, 9 : 16