We use the eagle simulations to study the connection between the quenching time-scale, τQ, and the physical mechanisms that transform star-forming galaxies into passive galaxies. By quantifying τQ in two complementary ways-as the time over which (i) galaxies traverse the green valley on the colour-mass diagram, or (ii) leave the main sequence of star formation and subsequently arrive on the passive cloud in specific star formation rate (SSFR)-mass space-we find that the τQ distribution of high-mass centrals, low-mass centrals, and satellites are divergent. In the low stellar mass regime where M <109.6 M·, centrals exhibit systematically longer quenching time-scales than satellites (≈4 Gyr compared to ≈2 Gyr). Satellites with low stellar mass relative to their halo mass cause this disparity, with ram pressure stripping quenching these galaxies rapidly. Low-mass centrals are quenched as a result of stellar feedback, associated with long τQ 3 Gyr. At intermediate stellar masses where $109.7<M☆ <1010.3M⊙, τQ are the longest for both centrals and satellites, particularly for galaxies with higher gas fractions. At M☆1010.3M⊙, galaxy merger counts and black hole activity increase steeply for all galaxies. Quenching time-scales for centrals and satellites decrease with stellar mass in this regime to τQ2 Gyr. In anticipation of new intermediate redshift observational galaxy surveys, we analyse the passive and star-forming fractions of galaxies across redshift, and find that the τQ peak at intermediate stellar masses is responsible for a peak (inflection point) in the fraction of green valley central (satellite) galaxies at z ≈ 0.5-0.7.