Explanation

(b) Acceleration due to gravity on earth is given by
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ g = \frac{{GM}_{e}}{R_{e}^{2}}$
$or\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ g \propto \frac{M_{e}}{R_{e}^{2}}$
$Hence,\ \ \ \ \ \ \ \ \ \ \ \ \frac{g_{p_{1}}}{g_{p_{2}}} = \frac{M_{p_{1}}}{M_{p_{2}}} \times \left( \frac{R_{p_{1}}}{R_{p_{2}}} \right)^{2}\ \ \ \ \ \ \ \ \ \ \ \ \ldots(i)$
$Given,\ \ \ \ \ \ \ \ \ \ \ \ \ \frac{M_{p_{1}}}{M_{p_{2}}} = \frac{1}{2}\ \ \ and\ \frac{R_{p_{1}}}{R_{p_{2}}} = \frac{1}{2}$
Substituting the given value in Eq. (i), we get
$\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \frac{g_{p_{1}}}{g_{p_{2}}} = \frac{1}{2} \times \left( \frac{2}{1} \right)^{2} = \frac{2}{1}$
$\therefore\ \ \ \ \ \ \ \ \ \ \ \ g_{p_{1}}:g_{p_{2}} = 2\ :1$