Two planets A and B of radii $$R$$ and $$1.5R$$ have densities $$\rho$$ and $$\frac{\rho}{2}$$ respectively. The ratio of acceleration due to gravity at the surface of B to A is:

Acceleration due to gravity at the surface of a planet: $$g = \frac{GM}{R^2} = \frac{G \cdot \frac{4}{3}\pi R^3 \rho}{R^2} = \frac{4}{3}G\pi R\rho$$

For planet A: $$g_A = \frac{4}{3}G\pi R\rho$$

For planet B: $$g_B = \frac{4}{3}G\pi (1.5R)\left(\frac{\rho}{2}\right) = \frac{4}{3}G\pi \times \frac{1.5R\rho}{2} = \frac{4}{3}G\pi \times \frac{3R\rho}{4}$$

$$\frac{g_B}{g_A} = \frac{3R\rho/4}{R\rho} = \frac{3}{4}$$

The ratio $$g_B : g_A = 3 : 4$$.

This matches option 3.