We present experimental investigations on the mutual influence of fermionic 40K and bosonic 87Rb atoms during the various trapping and cooling stages needed to reach quantum degeneracy of the sample. In particular we present particle number optimization strategies, which have led to large 40K–87Rb mixtures in the quantum degenerate regime. In fact the achievable particle numbers are only limited by a collapse due to an attractive overall mean-field interaction for the mixture. We discuss the influence of dynamical effects on the collapse parameters and find our critical particle numbers consistent with predictions based on recent Feshbach resonance measurements. We also investigate loss mechanisms of the mixture where we have identified three-body K–Rb–Rb losses as the dominant loss process and have measured the loss coefficient to be.