Question

An automobile went up a hill at a speed of $10$ km an hour and down the same distance at a speed of $20$ km an hour. The average speed for the round trip was
A $12\dfrac{1}{2}$ km/hr
B $13\dfrac{1}{3}$ km/hr
C $14\dfrac{1}{2}$ km/hr
D $12$ km/hr

Answer 1

In this question first let us suppose that distance for one side is X. Hence for the average speed we know that the it is equal to $\dfrac{{Total{\text{ }}distance{\text{ }}travel}}{{Total{\text{ }}time{\text{ }}taken}}$. As total distance travel is $2X$ and total time will be $\dfrac{X}{{10}} + \dfrac{X}{{20}}$ hours

Complete step-by-step answer:
Let us suppose that the distance of going up to the hill is X km .
And it is given that the automobile went up a hill at a speed of $10$ km an hour
As we know that $Speed = \dfrac{{distance}}{{Time}}$
Hence time taken by the automobile going up to the hill is equal to distance divided by speed.
Time = $\dfrac{X}{{10}}$ hr
Now for automobile going down from the hill
It is given that the automobile covers the same distance at a speed of $20$ km an hour.
In this case time taken by the automobile is equal
Time = $\dfrac{X}{{20}}$ hr
So for the average speed for the automobile
Average speed = $\dfrac{{Total{\text{ }}distance{\text{ }}travel}}{{Total{\text{ }}time{\text{ }}taken}}$
As we know that the total distance travel by the automobile for going up, then up to down is $2X$.
Total time taken by the automobile for going up , then up to down is $\dfrac{X}{{10}} + \dfrac{X}{{20}}$ hours
Hence Average speed of automobile is = $\dfrac{{2X}}{{\dfrac{X}{{10}} + \dfrac{X}{{20}}}}$
Divide by X in both numerator and denominator we get
= $\dfrac{2}{{\dfrac{1}{{10}} + \dfrac{1}{{20}}}}$
= $\dfrac{2}{{\dfrac{2}{{20}} + \dfrac{1}{{20}}}}$
= $\dfrac{2}{{\dfrac{{2 + 1}}{{20}}}}$
= $\dfrac{{2 \times 20}}{{2 + 1}}$
= $\dfrac{{40}}{3}$ km/hour or $13\dfrac{1}{3}$ km/hr

So, the correct answer is “Option B”.

Note: As always remember that the average speed = $\dfrac{{Total{\text{ }}distance{\text{ }}travel}}{{Total{\text{ }}time{\text{ }}taken}}$ and $Speed = \dfrac{{distance}}{{Time}}$ as in this question it is given Speed of the automobile and the distance is same for both way. Sometimes in the question it is given like it travels x distance in $t_1$ time and y distance in $t_2$ time then simply uses the above formula for finding average speed.