The ratio of radii of a cylinder and cone is 1: 3 and the ratio of their heights is 3: 4. What is the ratio of the volumes of the cylinder and the cone?

Let R and H respectively be the radius and the height of the cylinder.

Let r and h respectively be the radius and the height of the cone.

According to the given information:

$$\begin{gathered} \frac{R}{r}=\frac{1}{3} \\ \frac{H}{h}=\frac{3}{4} \end{gathered}$$

Volume of the cylinder = πR²H

Volume of the cone = $\frac{1}{3}{\mathrm{\pi r}}^2\mathrm{h}$

$\begin{array}{rcl}\therefore \frac{\mathrm{Volume} \mathrm{of} \mathrm{cylinder}}{\mathrm{Volume} \mathrm{of} \mathrm{cone}}& =& \frac{\mathrm{\pi }{R}^{2}H}{{\displaystyle \frac{1}{3}}{\mathrm{\pi r}}^2\mathrm{h}}\\ & =& 3{\left(\frac{R}{r}\right)}^2\left(\frac{H}{h}\right)\\ & =& 3\times {\left(\frac{1}{3}\right)}^2\times \frac{3}{4}\\ & =& \frac{1}{4}\end{array}$
Thus, the ratio of the volumes of the cylinder and the cone is 1 : 4.