According to Kepler's third law, $T^2 \propto a^3$, where a is the orbital radius. $a_A = R+r$ and $a_B = R+2r$. So, $(T_A/T_B)^2 = ((R+r)/(R+2r))^3$. Thus, $T_A/T_B = (\frac{R+r}{R+2r})^{3/2}$. The mass of the satellite is irrelevant.