Interesting question.

I did some algebra. Do me a favor and check it for me. Thanks. I wish reddit had a way to typeset mathematical expressions.

We start with the Tsiolkovsky Equation: dV = Isp * g * ln( mi / mf ) where dV is delta-v in m/s, Isp is specific impulse in seconds, g is Earth gravity = 9.8 m/s/s, ln is the natural logarithm function, mi is the initial mass of the vehicle in kg and mf is the final mass of the vehicle in kg.

The first thing we do is get rid of the mi and mf because that's too complicated and we don't care. From now on, mi/mf = 1/fm where fm is the mass fraction in kg/kg (unitless). This gives us dv = Isp * g * ln( 1/fm )

Okay, now solve for 1/fm because that's got the info we want hidden inside it. This gives us 1/fm = e ^{dV / Isp / g} We'll call this equation 1.

Remember how 1/fm = mi/mf ? Well let's return to that for a sec and think about what we mean by "initial mass" and "final mass". Initial mass is the mass of payload plus the engine plus the metal of the fuel tank plus the fuel inside it. Now, you've already designed your rocket, so you know what this total mass is. You can't change it or you would have to re-design your lower stage to re-balance the rocket. So initial mass is a fixed constant from now on. I'm gonna call it "upper stage mass" or Mus for short. As for final mass mf that's the payload mass plus the mass of the engine plus the mass of the empty tanks. All this gives me 1/fm = Mus / (Mpay + Mdry) where fm is the mass fraction (unitless) Mus is upper stage total mass, including payload, engine, tank metal and fuel, all in kg, Mpay is payload mass in kg and Mdry is the mass of the empty tank plus the engine in kg.

Now solve this equation for the payload mass Mpay I got: Mpay = Mus * fm - Mdry. I call this "equation 2"

Now you can either solve this in two steps, first equation 1 and then substitute the answer into equation 2, or you can sub equation 1 into equation 2 directly and solve.

Either way, I think this will get you your answer. Have fun.