A runner completes a three-part race. The speeds (in km/h) for Part I, Part II and Part III are 9, 8 and 7.5 respectively, and the corresponding times are 50, 80 and 100 minutes. What is the runner's overall average speed for the entire race (in km/h)?

Introduction / Context:
This question checks understanding of average speed when different parts of a journey are covered at different speeds for different durations. Many students incorrectly average the speeds directly, which gives the wrong result. The correct method is to compute the total distance and total time, then divide distance by time. Such race and travel based problems are very common in competitive exams and require careful handling of units and basic arithmetic.

Given Data / Assumptions:

Speed in Part I = 9 km/h, time in Part I = 50 minutes.
Speed in Part II = 8 km/h, time in Part II = 80 minutes.
Speed in Part III = 7.5 km/h, time in Part III = 100 minutes.
The runner moves at constant speed within each part, and there are no breaks between parts.
We need the overall average speed in km/h for the full race.

Concept / Approach:
Average speed over a journey is defined as total distance divided by total time. It is not the arithmetic average of different speeds unless the time spent at each speed is the same. Here, each stage has a different time duration, so we must calculate each distance separately. We convert times from minutes to hours to match the speed units in km/h. Then we find the total distance, the total time in hours, and finally compute the average speed as distance over time.

Step-by-Step Solution:
Step 1: Convert all times into hours. Part I: 50 minutes = 50 / 60 hours = 5 / 6 hours. Part II: 80 minutes = 80 / 60 hours = 4 / 3 hours. Part III: 100 minutes = 100 / 60 hours = 5 / 3 hours. Step 2: Compute the distance in each part. Distance I = 9 * (5 / 6) = 7.5 km. Distance II = 8 * (4 / 3) = 32 / 3 km ≈ 10.67 km. Distance III = 7.5 * (5 / 3) = 12.5 km. Step 3: Add distances and times. Total distance = 7.5 + 32 / 3 + 12.5 = 92 / 3 km ≈ 30.67 km. Total time = 5 / 6 + 4 / 3 + 5 / 3 = 23 / 6 hours ≈ 3.83 hours. Step 4: Compute average speed. Average speed = (92 / 3) / (23 / 6) = (92 / 3) * (6 / 23) = 8 km/h.

Verification / Alternative check:
As a quick check, note that all speeds are around 8 km/h, and the longest time segment (100 minutes) uses 7.5 km/h, which would drag the average slightly below 8 if times were symmetric. However, the other durations and distances balance such that the exact computation gives 8 km/h. Performing the fraction simplification carefully confirms that the ratio of total distance to total time is exactly 8.

Why Other Options Are Wrong:
Option a (8.17 kmph), option c (7.80 kmph), and option d (7.77 kmph) come from various incorrect ways of averaging, such as weighted incorrectly by distance, or by directly averaging the speeds without considering time. None of these values exactly match the correct ratio of total distance to total time. Only 8.00 kmph correctly reflects the proper computation.

Common Pitfalls:
Many candidates take the simple mean of 9, 8 and 7.5, which gives 8.17, and then choose that as the answer. This is wrong because the time spent in each segment is not equal. Another frequent error is forgetting to convert minutes to hours, which leads to very small or very large speeds. Always ensure consistent units and remember that average speed is total distance divided by total time, not an average of the given speeds.

Final Answer:
Therefore, the runner's overall average speed is 8.00 kmph.