Question

A car covers 4 successive stretches of 3 km each at speed of 10 kmph, 20 kmph, 30 kmph and 60 kmph respectively. The average speed of the car for the entire journey is:

• A. 15 kmph
• B. 20 kmph
• C. 25 kmph
• D. 35 kmph

Hint:

When the distance covered is the same for different speeds, the average speed is the Harmonic Mean of the individual speeds. For speeds \(s_1, s_2, ..., s_n\), the average speed is \( \frac{n}{\frac{1}{s_1} + \frac{1}{s_2} + ... + \frac{1}{s_n}} \). Here, it would be \( \frac{4}{\frac{1}{10} + \frac{1}{20} + \frac{1}{30} + \frac{1}{60}} = \frac{4}{(6+3+2+1)/60} = \frac{4}{12/60} = \frac{4 \times 60}{12} = 20 \).

Answer 1

The Correct Option is B

Step 1: Concept Definition:
Average speed is defined as total distance traveled divided by total time elapsed. It is distinct from the simple arithmetic mean of individual speeds.
Step 2: Calculation Formula:
\[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} \]Total Time is the sum of durations for each segment, where \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \).
Step 3: Detailed Calculation:
The car covers 4 segments, each measuring 3 km.
Total Distance = \( 4 \times 3 \text{ km} = 12 \text{ km} \).
Time taken for each segment:
- Segment 1 (\(t_1\)) = \( \frac{3 \text{ km}}{10 \text{ kmph}} = \frac{3}{10} \text{ hours} \)
- Segment 2 (\(t_2\)) = \( \frac{3 \text{ km}}{20 \text{ kmph}} = \frac{3}{20} \text{ hours} \)
- Segment 3 (\(t_3\)) = \( \frac{3 \text{ km}}{30 \text{ kmph}} = \frac{3}{30} \text{ hours} \)
- Segment 4 (\(t_4\)) = \( \frac{3 \text{ km}}{60 \text{ kmph}} = \frac{3}{60} \text{ hours} \)
Total Time = \( t_1 + t_2 + t_3 + t_4 \)
\[ \text{Total Time} = \frac{3}{10} + \frac{3}{20} + \frac{3}{30} + \frac{3}{60} \]Using a common denominator of 60:
\[ \text{Total Time} = \frac{18}{60} + \frac{9}{60} + \frac{6}{60} + \frac{3}{60} = \frac{36}{60} \text{ hours} \]\( \frac{36}{60} \) hours is equivalent to 0.6 hours.
Average speed calculation:
\[ \text{Average Speed} = \frac{12 \text{ km}}{0.6 \text{ hours}} = \frac{120}{6} = 20 \text{ kmph} \]Step 4: Conclusion:
The car's average speed for the entire trip is 20 kmph.