An astronaut is in equilibrium when he is positioned 140 km from planet X and 581 km from planey Y, along the straight line joining the planets' centers. What is the ration of the masses X/Y? I know the answer is 0.0581, I just do not know how to solve it.

Well, yeah, sometimes we have that weird feeling that our guts know the answer but we don't know why. Let me show you how I got mine.

Gravitational force obeys an inverse square law - as in the gravitational force depends on the distance as F = GMm / r ²

Small m is the mass of astronaut, M the planet, etc. equate the two forces to have the astronaut in equilibrium. G, m cancel, then arrange to get the Mass ratio: Mx / My = 140 ²/581²