Find the ratio of the area of two triangles on the same base if the ratio of their heights is 5 : 3.

Take the base of both triangles is b and height of the first triangle is h₁, and the height of the second triangle is h₂.

$\frac{\mathrm{Area} \mathrm{of} 1^{\mathrm{st} }\mathrm{triangle}}{\mathrm{Area} \mathrm{of} 2^{\mathrm{st} }\mathrm{triangle}}=\frac{{\displaystyle \frac{1}{2}\times \mathrm{b}\times {\mathrm{h}}_1}}{\frac{1}{2}\times \mathrm{b}\times {\mathrm{h}}_2}=\frac{{\mathrm{h}}_1}{{\mathrm{h}}_2}$

So, the ratio of the area of two triangles with the same base is equal to the ratio of their heights.

Hence, the required ratio is 5 : 3.