Question

An aeroplane flies from A to B at the rate of $ 500{\text{ km/hour}} $ and comes back from B to A at the rate of $ 700{\text{ km/hour}} $ . The average speed of the aeroplane is

Answer 1

Hint: In this question we have to find the average speed of the aeroplane traveling between two points A and B. to solve this question we use the relationship formula between the speed, time and distance of the aeroplane. The average speed of the aeroplane is the ratio of the total distance travelled by the aeroplane to the total time taken while traveling.

Complete step-by-step answer:
Given:
Let us assume the distance between A and B is $ x{\text{ km}} $ , the speed of the aeroplane while traveling from A to B is $ {S_{AB}}{\text{ km/hr}} $ and the speed of the aeroplane while returning from B to A is $ {S_{BA}}{\text{ km/hr}} $ .
Then we have given,
The speed of the aeroplane while traveling from A to B
 $\Rightarrow {S_{AB}} = 500{\text{ km/hr}} $
The speed of the aeroplane while returning from B to A
 $\Rightarrow {S_{BA}} = 700{\text{ km/hr}} $
We know the relationship formula between the speed, time and distance is given by –
 $\Rightarrow {\text{Time = }}\dfrac{{{\text{Distance}}}}{{{\text{Speed}}}} $
So, the time taken by the aeroplane while traveling from A to B
 $\Rightarrow {t_{AB}} = \dfrac{x}{{{S_{AB}}}} $
Substituting the value of speed $ {S_{AB}} = 500{\text{ km/hr}} $ in the formula we get,
 $ \Rightarrow {t_{AB}} = \dfrac{x}{{500}} $
And the time taken by the aeroplane while traveling from B to A
 $\Rightarrow {t_{BA}} = \dfrac{x}{{{S_{BA}}}} $
Substituting the value of speed $ {S_{BA}} = 700{\text{ km/hr}} $ in the formula we get,
 $\Rightarrow {t_{BA}} = \dfrac{x}{{700}} $
So, total time taken by the aeroplane while traveling from A to B then B to A
 $
\Rightarrow {T_{total}} = \left( {{t_{AB}} + {t_{BA}}} \right)\\
 = \left( {\dfrac{x}{{{S_{AB}}}} + \dfrac{x}{{{S_{BA}}}}} \right)\\
 = \left( {\dfrac{x}{{500}} + \dfrac{x}{{700}}} \right)\\
 = x\left( {\dfrac{1}{{500}} + \dfrac{1}{{700}}} \right)
 $
And, the total distance covered by the aeroplane
 $
\Rightarrow {D_{total}} = x + x\\
 = 2x
 $
Now using the relationship formula between the speed, time and distance we get,
Average speed of the aeroplane
 $
\Rightarrow {\text{Average Speed}} = \dfrac{{{\text{Total Distance}}}}{{{\text{Total Time}}}}\\
\Rightarrow {S_{avg}} = \dfrac{{{D_{total}}}}{{{T_{total}}}}\\
 = \dfrac{{2x}}{{x\left( {\dfrac{1}{{500}} + \dfrac{1}{{700}}} \right)}}
 $
Solving this we get,
 $
\Rightarrow {S_{avg}} = \dfrac{2}{{\left( {\dfrac{1}{{500}} + \dfrac{1}{{700}}} \right)}}\\
 = \dfrac{{2 \times 3500}}{{\left( {7 + 5} \right)}}\\
 = \dfrac{{7000}}{{12}}\\
 = \dfrac{{1750}}{3} {\text{ or 583}}{\text{.33 km/hr}}
 $
Therefore, the average speed of the aeroplane is $ \dfrac{{1750}}{3} {\text{ or 583}}{\text{.33 km/hr}} $ .

Note: It should be noted that the distance between the two points A and B remains constant while traveling from A to B and B to A but the speed and the time taken by the aeroplane could be different.