Statistics
Harmonic Mean / Average Speed
Grade 11
Question:
<p>An automobile driver travels from a plain to a hill station 120 km away at an average speed of 30 km per hour. He then makes the return trip at an average speed of 25 km per hour. He covers another 120 km on the plain at an average speed of 50 km per hour. His average speed (in km/hr) over the entire distance of 360 km will be</p>
<p>\(\dfrac{30+25+50}{3}\)</p>
<p>\(\dfrac{\dfrac{1}{30}+\dfrac{1}{25}+\dfrac{1}{50}}{3}\)</p>
<p>\(\dfrac{3}{\dfrac{1}{30}+\dfrac{1}{25}+\dfrac{1}{50}}\)</p>
<p>None of these</p>
Step-by-Step Solution
Key Concept: Average speed is total distance divided by total time, NOT the arithmetic mean of individual speeds. You must calculate total time by summing time for each segment separately.
<p><strong>Step 1:</strong> Identify the three segments:</p><ul><li>Plain to hill: Distance = 120 km, Speed = 30 km/hr</li><li>Hill to plain (return): Distance = 120 km, Speed = 25 km/hr</li><li>Plain travel: Distance = 120 km, Speed = 50 km/hr</li></ul><p><strong>Step 2:</strong> Calculate time for each segment using Time = Distance/Speed:</p><ul><li>Segment 1: t₁ = 120/30 = 4 hours</li><li>Segment 2: t₂ = 120/25 = 4.8 hours</li><li>Segment 3: t₃ = 120/50 = 2.4 hours</li></ul><p><strong>Step 3:</strong> Calculate total time:</p><p>Total time = 4 + 4.8 + 2.4 = 11.2 hours</p><p><strong>Step 4:</strong> Apply average speed formula:</p><p>Average speed = Total distance/Total time = 360/11.2 = 3600/112 = 32.14 km/hr ≈ <strong>32.14 km/hr</strong></p><p>∴ Answer: C</p>
Correct Answer: C