jmc

Let $d$ be the distance from Bill to the center of the Earth and $f$ be the gravitational force that the Earth exerts on him. Since $f$ is inversely proportional to $d^2$, $f\cdot d^2=k$ for some constant $k$. Since the force when Bill is on the surface of Earth is 600 Newtons, $k=600\cdot 4000^2=9,600,000,000$. Therefore, if we let $x$ be the force that the Earth acts on Bill with when he is on the Moon, $x\cdot 240,000^2=960,000,000$ so $x=\boxed{\frac{1}{6}}$.

Alternatively, the distance between Bill and the center of the Earth has been increased by a factor of 60, so the force has to be decreased by a factor of $60^2=3600$. Since $\frac{600}{3600}=\boxed{\frac{1}{6}}$, we get the same answer.