  We find that the mass ratio q in quiescent black hole (BH) X-ray transients
is tightly correlated with the ratio of the double peak separation (DP) to the
full-width-half maximum (FWHM) of the Halpha emission line, log q = -6.88 -23.2
log (DP/FWHM). The correlation is explained through the efficient truncation of
the outer disc radius by the 3:1 resonance with the companion star. This is the
dominant tidal interaction for extreme mass ratios q=M2/M1<~0.25, the realm of
BH (and some neutron star) X-ray transients. Mass ratios can thus be estimated
with a typical uncertainty of ~32%, provided that the Halpha profile used to
measure DP/FWHM is an orbital phase average. We apply the DP/FWHM-q relation to
the three faint BH transients XTE J1650-500, XTE J1859+226 and Swift J1357-0933
and predict q=0.026 (+0.038-0.007), 0.049 (+0.023-0.012) and 0.040
(+0.003-0.005), respectively. This new relation, together with the FWHM-K2
correlation presented in Paper I (casares 2015) allows the extraction of
fundamental parameters from very faint targets and, therefore, the extension of
dynamical BH studies to much deeper limits than was previously possible. As an
example, we combine our mass ratio determination for Swift J1357-0933 with
previous results to yield a BH mass of 12.4+-3.6 Msun. This confirms Swift
J1357-0933 as one of the most massive BH low-mass X-ray binaries in the Galaxy.
