Question

A man walks on a straight road from his home to a market $2.5{\text{km}}$ away with a speed of $5{\text{km/h}}$. Finding the market closed , he immediately turns and walks back home with a speed of $7.5{\text{km/h}}$. The average speed of the man over the interval of time zero to forty minutes is equal to:
A. $5{\text{km/h}}$
B. $\dfrac{{25}}{4}{\text{km/h}}$
C. $\dfrac{{30}}{4}{\text{km/h}}$
D. $\dfrac{{45}}{8}{\text{km/h}}$

Answer 1

Hint:In order to calculate the average speed of the man in time zero to forty minutes we need total time and total distance here total time is given $40{\text{min}}$and total distance is calculated by adding $2.5{\text{km}}$to the distance from market to home which can be calculated using distance formula.

Complete step by step answer:
Given is:
Distance from home to market $ = 2.5{\text{km}}$
Speed of travel $ = 5{\text{km/h}}$
Therefore time $\left( {{{\text{t}}_1}} \right)$ taken to travel $2.5{\text{km}}$ with a speed of $5{\text{km/h}}$:
${\text{Time = }}\dfrac{{{\text{Distance}}}}{{{\text{Speed}}}}$ $ = \left( {{{\text{t}}_1}} \right) = \dfrac{{2.5}}{5} = \dfrac{1}{2}{\text{hr = 30min}}$
It i given that total time of travel $ = 40{\text{min}}$
Remaining time $ = 40 - 30 = 10{\text{min}}$
$ \Rightarrow 10{\text{min = }}\dfrac{{10}}{{60}} = \dfrac{1}{6}{\text{hr}}$
It means that the man has taken thirty minutes or $\dfrac{1}{2}{\text{hr}}$ to reach from home to market now he will be travelling the remaining ten minute or $\dfrac{1}{6}{\text{hr}}$ of return journey with a speed of $7.5{\text{km/h}}$ which is given.
As we know that ${\text{distance = speed}} \times {\text{time}}$
Therefore the distance travelled by man in ten minutes from market to home: $ = 7.5 \times \dfrac{1}{6} = 1.25{\text{km}}$
Hence total distance is the distance from home to market which is given i.e. $2.5{\text{km}}$ plus distance from market to home which we have calculated above i.e. $1.25{\text{km}}$
Now total distance $ = 2.5 + 1.25 = 3.75{\text{km}}$
Total time given $ = 40{\text{min = }}\dfrac{{40}}{{60}} = \dfrac{2}{3}{\text{hr}}$
Therefore average speed of the man over the interval of time zero to forty minutes:
$
   = \dfrac{{{\text{total distance}}}}{{{\text{total time}}}} \\
   = \dfrac{{3.75}}{{\dfrac{2}{3}}} \\
   = \dfrac{{1125}}{{200}} \\
   = \dfrac{{45}}{8}{\text{km/hr}} \\
$
Hence the correct option is D.

Note: In this question first we calculated the time taken to travel $2.5{\text{km}}$ with a speed of $5{\text{km/h}}$ which is calculated to be thirty minutes after that we calculated the distance travelled by man from market to home which is $1.25{\text{km}}$ then we calculated the total distance and as total time is given we calculated the average speed of the man over the interval of time zero to forty minutes by the speed formula.