Question

The acceleration due to gravity on planet a is one-sixth what it is on planet b, and the radius of the planet a is one-fourth that of planet
b. the mass of planet a is what fraction of the mass of planet b?

Answer 1

The acceleration due to gravity on a large body is given by:
acceleration = (gravitational constant * mass of body) / radius²

For b, this is:
g = GM/r²
M = gr²/G

For a,
g(a) = 1/6 g
r(a) = 1/4 r
M(a) = g(a)r(a)²/G
M(a) = 1/6 g * (1/4 r)²/G
M(a) = gr²/96G

The fraction is given as:
M(a) / M = (gr²/96G) / (gr²/G)
M(a) / M = 1/96

The mass of planet a is 1/96 of the mass of planet b.