The relationship between kinetic energy ($K$), linear momentum ($p$), and mass ($m$) is given by $K = \frac{p^2}{2m}$, which can be rearranged as $m = \frac{p^2}{2K}$. Given that the linear momenta of both bodies are equal ($p_1 = p_2$), the mass is inversely proportional to the kinetic energy ($m \propto \frac{1}{K}$). Therefore, the ratio of their masses is: $\frac{m_1}{m_2} = \frac{K_2}{K_1} = \frac{1}{4}$ So, the ratio of their masses is $1 : 4$.