A body initially at rest, breaks up into two pieces of masses 2M and 3M respectively, together having a total kinetic energy E. The piece of mass 2M, after breaking up, has a kinetic energy: 

According to law of conservation of Linear Momentum:

$\begin{array}{l}0=2{\mathrm{MV}}_1-3{\mathrm{MV}}_2----\left(1\right)\\ \mathrm{E}=\frac{1}{2}2{\mathrm{MV}}_1^2+\frac{1}{2}3{\mathrm{MV}}_2^2\\ ={\mathrm{K}}_1+\frac{1}{2}3{\mathrm{MV}}_2^2-----\left(2\right)\\ \mathrm{from} \left(1\right),\hspace{0.17em}\hspace{0.17em}\mathrm{\hspace{0.17em}}{\mathrm{V}}_2\mathrm{\hspace{0.17em}}=\hspace{0.17em}\mathrm{\hspace{0.17em}}\frac{2{\mathrm{V}}_1}{3}\hspace{0.17em}\mathrm{\hspace{0.17em}}-----\left(3\right)\end{array}$

putting value from (3) in (2) and solve we get:

${\mathrm{K}}_1=\frac{3\mathrm{E}}{5}$