The radii of two cylinders are in the ratio of 2 : 3 and their heights are in the ratio of 5 : 3.

Let the radii of the cylinders be $r_{1}$ and $r_{2} ;$ and their heights be $h_{1}$ and $h_{2}$.

We have,

$r_{1}: r_{2}=2: 3$ or $\frac{r_{1}}{r_{2}}=\frac{2}{3}$   ..............(i)

and $h_{1}: h_{2}=5: 3$ or $\frac{h_{1}}{h_{2}}=\frac{5}{3}$    ...........(ii)

Now,

The ratio of the volumes of the cylinders $=\frac{\text { Volume of the first cylinder }}{\text { Volume of the second cylinder }}$

$=\frac{\pi r_{1}{ }^{2} h_{1}}{\pi r_{2}{ }^{2} h_{2}}$

$=\left(\frac{r_{1}}{r_{2}}\right)^{2} \times \frac{h_{1}}{h_{2}}$

$=\left(\frac{2}{3}\right)^{2} \times \frac{5}{3} \quad[$ Using (i) and (ii) $]$

$=\frac{20}{27}$

$=20: 27$

So, the ratio of the volumes of the given cylinders is 20 : 27.