Algebraic Derivation

The formatting of the equations is terrible but what can you do? It's plain text. Just follow order of operations carefully.

Equation 1, the Tsiolkovsky Equation: dV= Isp * g * ln( mi/mf ) where dV is delta-V in m/s, Isp is specific impulse in s, g is Earth gravitational acceleration = 9.8 m/s, ln is the natural log function, mi is initial mass in kg, mf is final mass in kg.

Equation 2, The Mass Fraction: mi / mf = 1 / fm where fm is the mass fraction in kg/kg (unitless)

Substitute Equation 2 into Equation 1 and solve for 1/fn to produce Equation 3

Equation 3, Calculating Mass Fraction 1/fn = e^(dV/Isp/g) where the right side of the equation is Euler's Number raised to the power of (dV / Isp / g) in case the superscript is not obvious.

Take Equation 2 and re-write it in terms of payload mass, upper-stage fuel mass, and upper-stage dry mass, where dry mass is the mass of everything that is not payload or fuel, e.g. empty tank metal and engine. This is Equation 4

Equation 4, Mass Breakdown 1/fm = mi/mf = (Mp + Md + Mf) / (Mp + Md) where Mp is payload mass in kg, Md is upper stage dry mass, which is everything not payload or fuel, in kg, and Mf is upper stage fuel mass in kg.

Solve Equation 4 for payload mass Mp to get Equation 5

Equation 5, Calculating Payload Mass Mp = Mf / ( 1/fm - 1 ) - Md

To calculate your answer, first solve Equation 3 to find 1/fm, then plug it into Equation 5 to calculate Mp.

Don't take my word for it. Practice your algebra skills and perform the derivation yourself!