Question

A car travels the first one third of a certain distance with the speed of 10 km/hr, the next one third with the speed of 20km/hr and the last one third distance with a speed of 60 k/hr. The average speed of the car for the whole journey is
$
  (a){\text{ 18}}\dfrac{{km}}{{hr}} \\
  (b){\text{ 24}}\dfrac{{km}}{{hr}} \\
  (c){\text{ 30}}\dfrac{{km}}{{hr}} \\
  (d){\text{ 36}}\dfrac{{km}}{{hr}} \\
$

Answer 1

Hint: In this question the total distance which is to be travelled is x km, now this distance is being covered with different speeds in intervals of $\dfrac{x}{3}$km. Use the relation between distance, speed and time which is ${\text{time = }}\dfrac{{{\text{distance}}}}{{{\text{speed}}}}$, to calculate the total time and then average speed will simply be the velocity that is $\dfrac{x}{t}$.

Complete step-by-step answer:
Let the total distance = x km.
So one third of the distance is = $\dfrac{x}{3}$ km.
As we know that time, distance and speed is related as, ${\text{time = }}\dfrac{{{\text{distance}}}}{{{\text{speed}}}}$
Now a car travel one third of the total distance with a speed of 10 km/hr
So the time (t1) taken by the car to travel one third of the total distance,
 $ \Rightarrow {t_1} = \dfrac{{\dfrac{x}{3}}}{{10}} = \dfrac{x}{{30}}$ hr.
Now a car travel next one third of the total distance with a speed of 20 km/hr
So the time (t2) taken by the car to travel next one third of the total distance,
 $ \Rightarrow {t_2} = \dfrac{{\dfrac{x}{3}}}{{20}} = \dfrac{x}{{60}}$ hr.
Now a car travel last one third of the total distance with a speed of 60 km/hr
So the time (t3) taken by the car to travel last one third of the total distance,
 $ \Rightarrow {t_3} = \dfrac{{\dfrac{x}{3}}}{{60}} = \dfrac{x}{{180}}$ hr.
So the total time (t) taken by the car to travel total distance is
$ \Rightarrow t = {t_1} + {t_2} + {t_3}$
$ \Rightarrow t = \dfrac{x}{{30}} + \dfrac{x}{{60}} + \dfrac{x}{{180}}$
Now simplify the above equation we have,
$ \Rightarrow t = \dfrac{{6x + 3x + x}}{{180}} = \dfrac{{10x}}{{180}} = \dfrac{x}{{18}}$ hr.
So the average speed (vavg) of the car is the ratio of total distance to total time.
$ \Rightarrow {v_{avg}} = \dfrac{x}{t} = \dfrac{x}{{\dfrac{x}{{18}}}} = 18$ Km/hr.
So the average speed of the car is 18 km/hr.
Hence option (A) is correct.

Note: Whenever we face such types of problems the key concept is to evaluate the overall time taken for the journey by adding up the individual times of the different intervals of distance. It is advised to remember the direct distance, time and speed relationship used in the above solution as it helps getting on the right track for most distance, time problems.