A car travels from a place X to place Y at an average speed of v km/hr, from Y to X at an average speed of 2v km/hr, again from X to Y at an average speed of 3v km/hr and again from Y to X at an average speed of 4v km/hr. Then the average speed of the car for the entire journey

Correct Answer: (B) lies between v and 2v km/hr

Let the distance between X and Y be $d$. The total distance for the four trips is $4d$. The total time taken is $\frac{d}{v}+\frac{d}{2v}+\frac{d}{3v}+\frac{d}{4v}=\frac{d}{v}(1+\frac{1}{2}+\frac{1}{3}+\frac{1}{4})=\frac{25d}{12v}$$\frac{d}{v}$ + $\frac{d}{2v}$ + $\frac{d}{3v}$ + $\frac{d}{4v}$ = $\frac{d}{v}$ \left(1 + $\frac{1}{2}$ + $\frac{1}{3}$ + $\frac{1}{4}$\right) = $\frac{25d}{12v}$. The average speed is $\frac{4d}{25d\mathrm{/}(12v)}=\frac{48}{25}v=1.92v\text{ km/hr}$$\frac{4d}{25d/(12v)}$ = $\frac{48}{25}$ v = 1.92v $\text{ km/hr}$, which lies strictly between $v$ and $2v$.