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We show that an analogy between crowding in fluid and jammed phases of hard spheres captures the density dependence of the kissing number for a family of numerically generated jammed states. We extend this analogy to jams of mixtures of hard spheres in $d=3$ dimensions, and thus obtain an estimate of the random close packing (RCP) volume fraction, $\phi_{\textrm{RCP}}$, as a function of size polydispersity. We first consider mixtures of particle sizes with discrete distributions. For binary systems, we show agreement between our predictions and simulations, using both our own and results reported in previous works, as well as agreement with recent experiments from the literature. We then apply our approach to systems with continuous polydispersity, using three different particle size distributions, namely the log-normal, Gamma, and truncated power-law distributions. In all cases, we observe agreement between our theoretical findings and numerical results up to rather large polydispersities for all particle size distributions, when using as reference our own simulations and results from the literature. In particular, we find $\phi_{\textrm{RCP}}$ to increase monotonically with the relative standard deviation, $s_{\sigma}$, of the distribution, and to saturate at a value that always remains below 1. A perturbative expansion yields a closed-form expression for $\phi_{\textrm{RCP}}$ that quantitatively captures a distribution-independent regime for $s_{\sigma} < 0.5$. Beyond that regime, we show that the gradual loss in agreement is tied to the growth of the skewness of size distributions.