CA Foundation | CBSE Class 9/12 Maths | Average Speed Calculation The Harmonic Mean Approach

Have you ever wondered how to calculate average speed when an object travels the same distance at different speeds? It's not as simple as just averaging the two speeds! In this video, we'll solve a classic problem: an aeroplane flying from A to B at 500 km/hr and back at 700 km/hr. You'll learn the correct formula for average speed in such scenarios (which is actually the Harmonic Mean of the speeds) and see a step-by-step calculation. Essential for anyone preparing for competitive exams or looking to understand speed-time concepts deeply! #averagespeed #harmonicaverage #speeddistancetime #quantitativeaptitude #math #tutorial #physics

Problem: An aeroplane flies from A to B at the rate of 500 km/hr and comes back from B to A at the rate of 700 km/hr. The average speed of the aeroplane is :
Option (A): 600 km/hr
Option (B): 583.33 km/hr
Option (C): 100√35 km/hr
Option (D): 620 km/hr
*This video is suitable for students studying CA Foundation (Indian curriculum) and CBSE Class 9/12 Maths (Indian curriculum).*

Solution:
This problem can be solved using the formula for average speed when the distance traveled in each direction is the same.
Let 'd' be the distance between A and B.
Speed from A to B (v1) = 500 km/hr
Speed from B to A (v2) = 700 km/hr

Time taken to fly from A to B (t1) = Distance / Speed = d / v1 = d / 500 hours
Time taken to fly from B to A (t2) = Distance / Speed = d / v2 = d / 700 hours

Total distance traveled = d (from A to B) + d (from B to A) = 2d
Total time taken = t1 + t2 = (d / 500) + (d / 700)

Average Speed = Total Distance / Total Time
Average Speed = 2d / [(d / 500) + (d / 700)]

We can factor out 'd' from the denominator:
Average Speed = 2d / [d * (1/500 + 1/700)]
Average Speed = 2 / (1/500 + 1/700)

Now, find a common denominator for the fractions in the denominator:
LCM of 500 and 700 is 3500.
1/500 = 7/3500
1/700 = 5/3500

So, 1/500 + 1/700 = 7/3500 + 5/3500 = 12/3500

Substitute this back into the average speed formula:
Average Speed = 2 / (12/3500)
Average Speed = 2 * (3500 / 12)
Average Speed = 7000 / 12
Average Speed = 3500 / 6
Average Speed = 1750 / 3

Now, convert this to a decimal:
1750 / 3 ≈ 583.3333... km/hr

Alternatively, for equal distances, the average speed is the harmonic mean of the two speeds:
Average Speed = (2 * v1 * v2) / (v1 + v2)
Average Speed = (2 * 500 * 700) / (500 + 700)
Average Speed = (2 * 350000) / 1200
Average Speed = 700000 / 1200
Average Speed = 7000 / 12
Average Speed = 1750 / 3
Average Speed ≈ 583.33 km/hr

Comparing with the options:
Option (A): 600 km/hr
Option (B): 583.33 km/hr
Option (C): 100√35 km/hr (100 * 5.916 ≈ 591.6 km/hr)
Option (D): 620 km/hr

The calculated average speed matches Option (B).

The final answer is B

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