Question

A taxi driver wants to go from Mumbai to Pune at an average speed of $60km/hr$ . But due to a traffic jam, he could cover $1/3$ of the distance at $40km/hr$ . How fast must he drive on the rest of the highway to reach Pune in time?
(A) $60km/hr$
(B) $70km/hr$
(C) $80km/hr$
(D) $90km/hr$

Answer 1

First, we have to calculate the time taken to complete the journey with average speed $60km/hr$. The second step is to calculate the time taken to complete $1/3$ of the total distance with speed $40km/hr$ and the rest of the journey be completed with a speed let $vkm/hr$. Since the time taken to complete the journey is the same. We have to equate them.
Formula used
$speed = \dfrac{{dis\tan ce}}{{time}}$

Complete Step-by-step solution
Let the distance covered in the entire journey be $xkm$ . if the driver drives with an average speed of$60km/hr$then the total time taken $T$ would be:
$\Rightarrow T = \dfrac{x}{{60}}$
But due to a traffic jam, the driver is covered $1/3$ of the distance with a speed of$40km/hr$.
Time taken to cover $1/3$of the distance is,
$\Rightarrow {t_1} = \dfrac{{\left( {\dfrac{x}{3}} \right)}}{{40}} = \dfrac{x}{{120}}$
Now, the rest of the distance to be covered is,
$\Rightarrow x - \left( {\dfrac{x}{3}} \right) = \dfrac{{2x}}{3}$
Let the rest of the journey be completed with the speed of$vkm/hr$.
Time taken to cover the rest of the journey is,
$\Rightarrow {t_2} = \dfrac{{\left( {\dfrac{{2x}}{3}} \right)}}{v} = \dfrac{{2x}}{{3v}}$
Now since the time taken to complete the entire journey is the same,
$\Rightarrow T = {t_1} + {t_2}$
On substituting the value of ${t_1}$ , ${t_2}$and $T$ in the above equation,
$\Rightarrow \dfrac{x}{{60}} = \dfrac{x}{{120}} + \dfrac{{2x}}{{3v}}$
Taking x common out and canceling it,
$\Rightarrow \dfrac{1}{{60}} = \dfrac{1}{{120}} + \dfrac{2}{{3v}}$
$\Rightarrow \dfrac{1}{{60}} = 3v + \dfrac{{240}}{{360v}}$
$\Rightarrow 3v + 240 = 6v$
$\Rightarrow 3v = 240$
$v = 80km/hr$

Hence, the correct option is (C) $80km/hr$.

Additional Information: Speed can be thought of as the rate at which an object covers distance. A fast-moving object has a high speed and covers a relatively large distance in a given amount of time, while a slow-moving object covers a relatively small amount of distance in the same amount of time.

Note: The given formula of speed is valid only when the speed is constant or when the acceleration is zero but when the acceleration is non zero then the given formula is not valid as the acceleration is the change of the rate of speed and the speed is not constant.