A vehicle travels half the distance with speed v and the remaining...

Let the total distance travelled be $d$.

Time taken in both part is,

{t}_{1} = \dfrac{\left(\dfrac{d}{2}\right)}{v}$and${t}_{2} = \dfrac{\left(\dfrac{d}{2}\right)}{2v}$.</p><p>Now, average speed can be written as</p><p>${v}_{avg} = \dfrac{d}{{t}_{1} + {t}_{2}} = \dfrac{d}{\dfrac{d}{2v} + \dfrac{d}{4v}} = \dfrac{4v}{3}