Question

A automobile driver travel from plane to a hilly area 120 km distance at an average speed of 30 km per hour. After than makes the return trip at average speed of 25 km per hour. Also covers another 120 km distance on plane at an average speed of 50 km per hour. What is the average speed for whole distance 360 km ?
A. \[\dfrac{{30 + 25 + 50}}{3}km/hr\]
B. \[(30,25,50)\dfrac{1}{3}\]
C. \[\dfrac{3}{{\dfrac{1}{{30}} + \dfrac{1}{{25}} + \dfrac{1}{{50}}}}km/hr\]
D. None of these

Answer 1

Hint:In this question, we use the concept of distance, average, and speed. We can define the average speed in this way as the total distance traveled by the body in total time. So, with the help of the average speed formula, we will work on this question.

Formula used:
In this question, we use the formula for finding the time and average speed.
The formula for calculating time:
\[\text{Time} = \dfrac{\text{Distance}}{\text{Speed}}\]
And, the formula for average speed:
\[\text{Average speed} = \dfrac{\text{Total distance}}{\text{Total time taken}}\]
So, By using this formula we will calculate the average speed of the whole distance.

Complete step-by-step solution:In this question given, that first of all a driver covers 120 km distance from the plane to the hill at an average speed of 30km per hour. So, the time taken to cover this distance is \[{t_1}\].
\[\text{Time}({t_1}) = \dfrac{{\text{Distance}}}{\text{Speed}}\]
\[\text{Time}({t_1}) = \dfrac{{120}}{{30}}\]
Next given in the question that the driven when returning to plane from hilly area then he covers the distance 120 km at an average speed of 25 km per hour. So, the time taken to cover this distance is \[{t_2}\].
\[\text{Time}({t_2}) = \dfrac{\text{Distance}}{\text{Speed}}\]
\[\text{Time}({t_2}) = \dfrac{{120}}{{25}}\]
Now, next given in this question that the driver covers another 120 km distance on the plane at an average speed of 50 km per hour. So, the time taken to cover this distance is \[{t_3}\].
\[\text{Time}({t_3}) = \dfrac{\text{Distance}}{\text{Speed}}\]
\[Time({t_3}) = \dfrac{{120}}{{50}}\]
Now,we can find the average speed for whole distance 360 km by time \[{t_{1,}}{t_2}\]and \[{t_3}\] .So, the average speed for whole distance 360 km:
\[Average{\rm{ }}speed = \dfrac{\text{Total distance}}{\text{Total time taken}}\]
             \[ = \dfrac{\text{Total distance}}{{{t_1} + {t_2} + {t_3}}}\]
             \[ = \dfrac{{120 + 120 + 120}}{{\dfrac{{120}}{{30}} + \dfrac{{120}}{{25}} + \dfrac{{120}}{{50}}}}\]
             \[ = \dfrac{{120 + 120 + 120}}{{120(\dfrac{1}{{30}} + \dfrac{1}{{25}} + \dfrac{1}{{50}})}}\]
             \[ = \dfrac{{120 + 120 + 120}}{{120(\dfrac{1}{{30}} + \dfrac{1}{{25}} + \dfrac{1}{{50}})}}\]
             \[ = \dfrac{{360}}{{120(\dfrac{1}{{30}} + \dfrac{1}{{25}} + \dfrac{1}{{50}})}}\]
             \[ = \dfrac{3}{{\dfrac{1}{{30}} + \dfrac{1}{{25}} + \dfrac{1}{{50}}}}\]
So, The average speed for whole distance 360 km is \[\dfrac{3}{{\dfrac{1}{{30}} + \dfrac{1}{{25}} + \dfrac{1}{{50}}}}km/hr\].
Hence, the option (C) is the correct answer:

Note:In these types of questions the major problem with the students they did not understand the question statement and applied the wrong method. Most of the students in this question give the answer first option but this is wrong. So students need this question to understand the statement first and then do the solution.