  While molecular gas mass is usually derived from $^{12}$CO($J$=1-0) - the
most fundamental line to explore molecular gas - it is often derived from
$^{12}$CO($J$=2-1) assuming a constant $^{12}$CO($J$=2-1)/$^{12}$CO($J$=1-0)
line ratio ($R_{2/1}$). We present variations of $R_{2/1}$ and effects of the
assumption that $R_{2/1}$ is a constant in 24 nearby galaxies using $^{12}$CO
data obtained with the Nobeyama 45-m radio telescope and IRAM 30-m telescope.
The median of $R_{2/1}$ for all galaxies is 0.61, and the weighted mean of
$R_{2/1}$ by $^{12}$CO($J$=1-0) integrated-intensity is 0.66 with a standard
deviation of 0.19. The radial variation of $R_{2/1}$ shows that it is high
(~0.8) in the inner ~1 kpc while its median in disks is nearly constant at 0.60
when all galaxies are compiled. In the case that the constant $R_{2/1}$ of 0.7
is adopted, we found that the total molecular gas mass derived from
$^{12}$CO($J$=2-1) is underestimated/overestimated by ~20%, and at most by 35%.
The scatter of a molecular gas surface density within each galaxy becomes
larger by ~30%, and at most by 120%. Indices of the spatially resolved
Kennicutt-Schmidt relation by $^{12}$CO($J$=2-1) are underestimated by 10-20%,
at most 39% in 17 out of 24 galaxies. $R_{2/1}$ has good positive correlations
with star-formation rate and infrared color, and a negative correlation with
molecular gas depletion time. There is a clear tendency of increasing $R_{2/1}$
with increasing kinetic temperature ($T_{\rm kin}$). Further, we found that not
only $T_{\rm kin}$ but also pressure of molecular gas is important to
understand variations of $R_{2/1}$. Special considerations should be made when
discussing molecular gas mass and molecular gas properties inferred from
$^{12}$CO($J$=2-1) instead of $^{12}$CO($J$=1-0).
