We calculate the phase diagrams of mixtures between hard-sphere colloids and star-polymers of arm numbers f=2,6,32 for different star-polymer-colloid size ratios 0.2less than or equal toqless than or equal to0.6 using an effective one-component description for the colloids in the presence of the stars. We map the full two-component system onto an effective one-component system by inverting numerically the Ornstein-Zernike equation for binary mixtures, supplemented by the Rogers-Young closure, in the low-colloid density limit. The free energy for the fluid and crystalline phase is calculated by using both hard-sphere perturbation theory and thermodynamic integration of simulation data. We find stable fluid-fluid demixing transitions for low arm numbers f=2,6 above a critical value of the size ratio q(c) below preempted by a fcc-solid. For the linear polymer limit, f=2, the critical size ratio is found to be q(c)approximate to0.4, in agreement with other approaches to colloid-polymer mixtures. Increasing the arm number, the region of stability of the demixing transition with respect to crystallization of the colloids shrinks, and q(c) grows. A comparison between the one- and two-component descriptions that demonstrates the consistency between the two routes is also carried out.