Question

A cyclist covers first five kilometers at an average speed of 10 k.m. per hour, another three kilometers at 8 k.m. per hour and the last two kilometers at 5 k.m. per hour. Then, the average speed of the cyclist during the whole journey, is

• A. 6.51 km/hr
• B. 8.40 km/hr
• C. 7.84 km/hr
• D. 7.05 km/hr

Hint:

A common mistake is to simply average the given speeds (10, 8, and 5). This is incorrect because the cyclist spends different amounts of time traveling at each speed. Always use the formula: Total Distance / Total Time.

Answer 1

The Correct Option is C

Step 1: Understanding the Concept:
Average speed is not the average of the speeds. It is calculated by dividing the total distance traveled by the total time taken for the journey.

Step 2: Key Formula or Approach:
1. Calculate the total distance traveled.
2. Calculate the time taken for each segment of the journey using the formula: Time = Distance / Speed.
3. Calculate the total time taken.
4. Calculate the average speed using the formula: Average Speed = Total Distance / Total Time.

Step 3: Detailed Explanation:
1. Total Distance:
The total distance is the sum of the distances of the three segments: \[ \text{Total Distance} = 5 \text{ km} + 3 \text{ km} + 2 \text{ km} = 10 \text{ km} \] 2. Time for each segment:


Time for the first segment (\(t_1\)): Distance = 5 km, Speed = 10 km/hr. \[ t_1 = \frac{5 \text{ km}}{10 \text{ km/hr}} = 0.5 \text{ hours} \]
Time for the second segment (\(t_2\)): Distance = 3 km, Speed = 8 km/hr. \[ t_2 = \frac{3 \text{ km}}{8 \text{ km/hr}} = 0.375 \text{ hours} \]
Time for the third segment (\(t_3\)): Distance = 2 km, Speed = 5 km/hr. \[ t_3 = \frac{2 \text{ km}}{5 \text{ km/hr}} = 0.4 \text{ hours} \] \end{itemize} 3. Total Time:
The total time is the sum of the times for each segment: \[ \text{Total Time} = t_1 + t_2 + t_3 = 0.5 + 0.375 + 0.4 = 1.275 \text{ hours} \] 4. Average Speed:
Now, we calculate the average speed: \[ \text{Average Speed} = \frac{\text{Total Distance}}{\text{Total Time}} = \frac{10 \text{ km}}{1.275 \text{ hours}} \approx 7.843 \text{ km/hr} \]
Step 4: Final Answer:
The average speed of the cyclist during the whole journey is approximately 7.84 km/hr.