  We propose a new dynamical method to estimate binary mass ratios by using the
period of superhumps in SU UMa-type dwarf novae during the growing stage (the
stage A superhumps). This method is based on a working hypothesis in which the
period of the superhumps at the growing stage is determined by the dynamical
precession rate at the 3:1 resonance radius, a picture suggested in our new
interpretation of the superhump period evolution during the superoutburst
(Osaki, Kato 2013, arXiv:1305.5877). By comparison with the objects with known
mass ratios, we show that our method can provide sufficiently accurate mass
ratios comparable to those obtained by quiescent eclipse observations. This
method is very advantageous in that it requires neither eclipses, nor an
experimental calibration. It is particularly suited for exploring the low
mass-ratio end of the evolution of cataclysmic variables, where the secondary
is undetectable by conventional methods. Our analysis suggests that previous
estimates of mass ratios using superhump periods during superoutburst were
systematically underestimated for low mass-ratio systems and we provided a new
calibration. It suggests that most of WZ Sge-type dwarf novae have secondaries
close to the border of the lower main-sequence and brown dwarfs, and most of
the objects have not yet reached the evolutionary stage of period bouncers. Our
result is not in contradiction with an assumption that the observed minimum
period (~77 min) of ordinary hydrogen-rich cataclysmic variables is indeed the
period minimum. We highlight the importance of early observation of stage A
superhumps and propose a future desirable strategy of observation.
