Shortcut Trick
When an object covers equal distances at different speeds, the Average Speed is the Harmonic Mean of those speeds.
Average Speed = 4 ÷ (1/250 + 1/450 + 1/650 + 1/850)
Average Speed = (4 × 50 × 9945) ÷ (1989 + 1105 + 765 + 585)
Average Speed = 1989000 ÷ 4444 ≈ 447.57 km/h
Rounding to the nearest integer gives 448 km/h.
∴ The correct answer is 448 km/h.
Alternate Method
Given:
Speeds of the helicopter along the sides: v1 = 250 km/h, v2 = 450 km/h, v3 = 650 km/h, v4 = 850 km/h.
Formula Used:
Average Speed = Total Distance ÷ Total Time
Calculations:
⇒ Let the distance of each side be d km.
⇒ Total distance covered = d + d + d + d = 4d km.
⇒ Total time taken = d/250 + d/450 + d/650 + d/850
⇒ To simplify, let d = LCM of (250, 450, 650, 850) = 497,250 km.
⇒ Time 1 = 497,250 ÷ 250 = 1,989 hours.
⇒ Time 2 = 497,250 ÷ 450 = 1,105 hours.
⇒ Time 3 = 497,250 ÷ 650 = 765 hours.
⇒ Time 4 = 497,250 ÷ 850 = 585 hours.
⇒ Total Time = 1,989 + 1,105 + 765 + 585 = 4,444 hours.
⇒ Average Speed = (4 × 497,250) ÷ 4,444 = 1,989,000 ÷ 4,444
⇒ Average Speed = 447.5697... km/h
⇒ Nearest integer = 448 km/h.
∴ The correct answer is 448 km/h.
Additional Information
Average Speed for Equal Distances
When an object travels two equal distances at speeds v1 and v2, the average speed is given by (2 × v1 × v2) ÷ (v1 + v2).
Harmonic Mean Relationship
The Average Speed is always the Harmonic Mean of individual speeds when the distances covered are equal. It is always less than or equal to the Arithmetic Mean.
General Average Speed Formula
For any journey with multiple segments, Average Speed = (Total Distance) ÷ (Total Time) = (d1 + d2 + ...) ÷ (t1 + t2 + ...).