The ratio of the radii of the two cones is 2 : 3 and the ratio of their heights is 1: 2. What is the ratio of the volumes of the two cones?

Let the radii of the two cones be 2r and 3r and their heights are h and 2h respectively.

Now, The ratio of volumes of cones = $\frac{{\displaystyle \frac{1}{3}\mathrm{\pi }{\left(2\mathrm{r}\right)}^2\mathrm{h}}}{\frac{1}{3}\mathrm{\pi }{\left(3\mathrm{r}\right)}^2\left(2\mathrm{h}\right)}=\frac{4{\mathrm{r}}^2\mathrm{h}}{9{\mathrm{r}}^2\times 2\mathrm{h}}=\frac{2}{9}$

Thus, the ratio of the volumes of the two cones is 2: 9.