An object of mass  $m_1$ collides with another object of mass  $m_2$, which is at rest. After the collision the objects move with equal speeds in opposite direction. The ratio of the masses $m_2:m_1$  is

Before collision,  $u_1=u;u_2=0$,  

After collision $v_1=-v;v_2=v$  
If the collision is elastic, then we can write

  $$\begin{gathered} u_1-u_2=v_2-v_1 \\ \Rightarrow u=2v -----\left(1\right) \end{gathered}$$
Applying principle of conservation of momentum

  $m_{1}u=\left(m_2-m_1\right)v -----\left(2\right)$
From (1) & (2) we get  $\frac{m_2}{m_1}=3$