A man can walk at a speed of 2 km/h and run at a speed of 6 km/h. He must cover a total distance of 20.5 km, walking exactly half the distance (10.25 km) and running the other half. How much total time will he take to complete the journey?

Introduction / Context:
This question combines two different speeds over equal distances and asks for the total time taken. It is a typical time and distance problem where a person covers half the distance at one speed and the remaining half at another speed. The problem checks whether the student can correctly divide the journey, compute the time for each part using time = distance / speed, and then add those times accurately.

Given Data / Assumptions:

• Total distance to be covered = 20.5 km.
• Half the distance is covered on foot (walking) and half by running.
• Half of 20.5 km is 10.25 km.
• Walking speed = 2 km/h.
• Running speed = 6 km/h.
• Speeds are constant throughout each part of the journey.

Concept / Approach:
When the total distance is split into two equal parts but with different speeds, we compute the time for each part separately. The formula used is time = distance / speed. The total time is the sum of the walking time and the running time. Since the question asks for the total time, we finally convert the answer from hours into hours and minutes for easier interpretation.

Step-by-Step Solution:
Half distance for walking = 20.5 / 2 = 10.25 km. Half distance for running = 10.25 km. Walking time = distance / speed = 10.25 / 2 hours. Walking time = 5.125 hours. Running time = distance / speed = 10.25 / 6 hours. Running time = 1.7083 hours approximately. Total time in hours = 5.125 + 1.7083 ≈ 6.8333 hours. Convert 0.8333 hours to minutes: 0.8333 * 60 ≈ 50 minutes. So total time ≈ 6 hours 50 minutes.

Verification / Alternative check:
We can calculate the exact value using fractions. 10.25 km is 41 / 4 km. Walking time = (41 / 4) / 2 = 41 / 8 hours. Running time = (41 / 4) / 6 = 41 / 24 hours. Total time = 41 / 8 + 41 / 24 = (123 + 41) / 24 = 164 / 24 = 41 / 6 hours. 41 / 6 hours equals 6 and 5 / 6 hours, and 5 / 6 of an hour is 50 minutes, again giving 6 hours 50 minutes. This confirms our approximate decimal calculation.

Why Other Options Are Wrong:
5 hours 40 minutes and 7 hours 20 minutes are too small or too large compared to the exact value of 6 hours 50 minutes. 8 hours 10 minutes and 9 hours are clearly overestimates and do not match the computed total time. Only 6 hours 50 minutes is consistent with the exact calculations based on the given speeds and distances.

Common Pitfalls:
A common mistake is to average the speeds (2 km/h and 6 km/h) and multiply by the total distance, which is not valid when the distances are equal but speeds differ. Another error is to split the total time instead of the total distance, which changes the structure of the problem. Carefully following time = distance / speed for each part of the journey avoids these issues.

Final Answer:
The man will require a total of approximately 6 hours 50 minutes to cover the 20.5 km journey.