We combine molecular gas masses inferred from CO emission in 500 star forming galaxies (SFGs) between z=0 and 3, from the IRAM-COLDGASS, PHIBSS1/2 and other surveys, with gas masses derived from Herschel far-IR dust measurements in 512 galaxy stacks over the same stellar mass/redshift range. We constrain the scaling relations of molecular gas depletion time scale (tdepl) and gas to stellar mass ratio (Mmolgas/M*) of SFGs near the star formation main-sequence with redshift, specific star formation rate (sSFR) and stellar mass (M*). The CO- and dust-based scaling relations agree remarkably well. This suggests that the CO-H2 mass conversion factor varies little within 0.6dex of the main sequence (sSFR(ms,z,M*)), and less than 0.3dex throughout this redshift range. This study builds on and strengthens the results of earlier work. We find that tdepl scales as (1+z)^-0.3 *(sSFR/sSFR(ms,z,M*))^-0.5, with little dependence on M*. The resulting steep redshift dependence of Mmolgas/M* ~(1+z)^3 mirrors that of the sSFR and probably reflects the gas supply rate. The decreasing gas fractions at high M* are driven by the flattening of the SFR-M* relation. Throughout the redshift range probed a larger sSFR at constant M* is due to a combination of an increasing gas fraction and a decreasing depletion time scale. As a result galaxy integrated samples of the Mmolgas-SFR rate relation exhibit a super-linear slope, which increases with the range of sSFR. With these new relations it is now possible to determine Mmolgas with an accuracy of 0.1dex in relative terms, and 0.2dex including systematic uncertainties.
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