  The majority of classical Cepheids are binary stars, yet the contribution of
companions' light to the total brightness of the system has been assumed
negligible and lacked a thorough, quantitative evaluation. We present an
extensive study of synthetic populations of binary Cepheids, that aims to
characterize Cepheids' companions (e.g. masses, evolutionary and spectral
types), quantify their contribution to the brightness and color of Cepheid
binaries, and assess the relevance of input parameters on the results. We
introduce a collection of synthetic populations, which vary in metal content,
initial parameter distribution, location of the instability strip edges, and
star formation history. Our synthetic populations are free from the selection
bias, while the percentage of Cepheid binaries is controlled by the binarity
parameter. We successfully reproduce recent theoretical and empirical results:
the percentage of binary Cepheids with main sequence (MS) companions, the
contrast-mass ratio relation for binary Cepheids with MS companions, the
appearance of binary Cepheids with giant evolved companions as outlier data
points above the period-luminosity relation. Moreover, we present the first
estimation of the percentage of binary Cepheids in the Large Magellanic Cloud
and announce the quantification of the effect of binarity on the slope and
zero-point of multiband period-luminosity relations, which will be reported in
the next paper of this series.
