We propose a new approach to determining the random packing densities of superballs via binary mixtures with spheres. The main idea of the approach is to suppress order formations in non-spherical particle packings via the polydispersity of particle shapes, which avoids using order metrics. The packing density of superballs in a mixture can be segregated using a linear fitting method with the concept of equivalent packing size (or size ratio with unit spheres) which represents the effective size (or volume) of a non-spherical particle in a binary mixture with spheres. We systemically study the packing properties of binary mixtures consisting of spheres and superballs and obtain the equivalent packing sizes of superballs. Our results show that the equivalent packing size ratio always corresponds to the minimal packing density or specific volume (reciprocal of packing densities) variation, and is independent of the solid volume fraction. The specific volumes of mixtures with the equivalent packing size ratio are always the upper bound for all the solid volume fractions. The linear relationship between the specific volume and solid volume fraction is only observed in the mixtures with superballs of small surface shape parameters (shapes close to a sphere), which results from the highly disordered nature in the mixtures. Moreover, the ideal random packing densities of mono-sized superballs obtained via the linear fitting method are surprisingly close to those of the MDRPs (maximally dense random packings), further verifying that the MDRPs of non-spherical particles correspond to the ideal random packings whose degrees of order are at the same level with that of the random close packing of spheres. Our work leads to a better understanding towards the random and binary packings and sheds new light on the essence of the MDRP. Our work also guides the optimal particle size distributions of powders in chemical engineering process. (C) 2019 Elsevier Ltd. All rights reserved.