The surface areas of two spheres are in the ratio 16:9. The ratio of their volumes is :

It is given that the surface areas of two spheres are in the ratio 16:9 and we need to find the ratio of their volumes.

It is known that the surface of the sphere is given by $\mathrm{S}=4{\mathrm{\pi r}}^2$ and the volume is given by 

$\mathrm{V}=\frac{4}{3}{\mathrm{\pi r}}^3$

Let r₁ and r₂ be the radius of the first and second sphere respectively.

It is given that the ratio of the surface areas is 16:9. Therefore, it can be written as

$\begin{array}{c}\frac{4{\mathrm{\pi r}}_1^2}{4{\mathrm{\pi r}}_2^2}=\frac{16}{9}\\ \Rightarrow \frac{{\mathrm{r}}_1^2}{{\mathrm{r}}_2^2}=\frac{16}{9}\\ \Rightarrow \frac{{\mathrm{r}}_1}{{\mathrm{r}}_2}=\frac{4}{3}\end{array}$

Now, the ratio between the volumes of spheres is given as

$\begin{array}{r}\Rightarrow \frac{\frac{4}{3}{\mathrm{\pi r}}_1^3}{\frac{4}{3}{\mathrm{\pi r}}_2^3}={\left(\frac{4}{3}\right)}^3\\ \Rightarrow \frac{\frac{4}{3}{\mathrm{\pi r}}_1^3}{\frac{4}{3}{\mathrm{\pi r}}_2^3}=\frac{64}{27}\end{array}$

Hence, the ratio of their volumes is $\frac{64}{27}$.