$\rho =\frac{{{m}_{1}}+{{m}_{2}}}{{{v}_{1}}+{{v}_{2}}}$
$15=\frac{{{m}_{1}}+{{m}_{2}}}{\frac{{{m}_{1}}+{{m}_{2}}}{{{\rho }_{1}}+{{\rho }_{2}}}}$
$15=\frac{{{m}_{1}}+{{m}_{2}}}{\underset{20}{\mathop{\cancel{{{m}_{1}}}}}\,+\underset{12}{\mathop{\cancel{{{m}_{2}}}}}\,}\to 15=\frac{{{m}_{1}}+{{m}_{2}}}{\frac{3{{m}_{1}}}{60}+\frac{5{{m}_{2}}}{60}}\to \overset{1}{\mathop{\cancel{15}}}\,=\frac{\overset{4}{\mathop{\cancel{60}}}\,({{m}_{1}}+{{m}_{2}})}{3{{m}_{1}}+5{{m}_{2}}}$
$3{{m}_{1}}+5{{m}_{2}}=4{{m}_{1}}+4{{m}_{2}}$
$5{{m}_{2}}-4{{m}_{2}}=4{{m}_{1}}-3{{m}_{1}}\to {{m}_{2}}={{m}_{1}}\Rightarrow \frac{{{m}_{2}}}{{{m}_{1}}}=1$