A vehicle travels half the distance with speed $v$ and the remaining distance with speed $2v$ its average speed is:

Let the total distance travelled be 2x.  

$\begin{array}{rl} & \text{Average speed}=\frac{\text{total distance}}{\text{total time}}\\ & =\frac{x+x}{\frac{x}{v}+\frac{x}{2v}}=\frac{2x}{\frac{x}{v}(1+\frac{1}{2})}=\frac{2v}{\left(\frac{3}{2}\right)}=\frac{4v}{3}.\end{array}$