Solution:

The formula for acceleration due to gravity \( g \) on a planet is:

\[
g = \frac{GM}{R^2}
\]

Where:
- \( G \) is the gravitational constant,
- \( M \) is the mass of the planet,
- \( R \) is the radius of the planet.

Mass \( M \) is related to density \( \rho \) and volume \( V \):

\[
M = \rho V = \rho \frac{4}{3} \pi R^3
\]

Substituting into the equation for \( g \):

\[
g = \frac{G \rho \frac{4}{3} \pi R^3}{R^2} = \frac{4}{3} \pi G \rho R
\]

Thus, \( g \propto \rho R \).

Given:
- Diameter ratio \( R_1 : R_2 = 4:1 \), so \( R_1 : R_2 = 4:1 \),
- Density ratio \( \rho_1 : \rho_2 = 1:2 \).

Now,

\[
g_1 : g_2 = (\rho_1 R_1) : (\rho_2 R_2) = (1 \times 4) : (2 \times 1) = 4:2 = 2:1
\]

Thus, the ratio of acceleration due to gravity on the planets is \( 2:1 \).