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?

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.