If the densities of the moon and the earth are related by $\rho_m / \rho_e =3/5$, and if $g_m /g_e = 1/6$, then what is $R_m /R_e$? ($g=$ acceleration due to gravity and $R=$ Radius)
You need to use Gauss's law for the inverse square field, in this case gravity. This tells you that for a spherically symmetric system:
- The gravitational force is always directed towards the centre of the sphere.
- The force at radius $r$ from the centre is given by the force owing to a point mass at the centre when that point mass equals the total, spherically-symmetrically-distributed mass enclosed in a sphere of radius $r$ centred at the centre of spherical symmetry.
Details of the reasoning can be found in my answer here. This should let you get an expression for $R_m/R_e$.