A stationary object explodes into two particles of masses m1 and m2 which move in opposite directions with velocities v1 and v2. The ratio of their kinetic energy is what???

I found that depends on the ratio of the two masses only.
I tried considerng ##p## (along ##x##):
Before: ##0##
After: ##m_1v_1+m_2v_2##
and together:
##0=m_1v_1+m_2v_2##
so: ##m_1v_1=-m_2v_2##
squaring both and rearranging:
##v_1^2/v_2^2=m_2^2/m_1^2##
Kinetic is ##K=1/2mv^2##
So:
##K_1/K_2=(cancel(1/2)m_1v_1^2)/(cancel(1/2)m_2v_2^2)##
substituting: ##v_1^2/v_2^2=m_2^2/m_1^2##
##K_1/K_2=(cancel(m_1))/(cancel(m_2))*m_2^cancel(2)/m_1^cancel(2)=m_2/m_1##