A man on tour travels the first 160 km at 64 km/h and the next 160 km at 80 km/h. What is his average speed for the first 320 km of the journey?

Introduction / Context:
This question tests your ability to compute average speed when a journey is performed at different speeds over equal distances. Average speed is not the arithmetic mean of the speeds unless the time intervals are equal. Instead, when distances are equal, the average speed is based on total distance divided by total time.

Given Data / Assumptions:

• First part of journey: distance = 160 km at 64 km/h.
• Second part of journey: distance = 160 km at 80 km/h.
• Total distance considered = 160 + 160 = 320 km.
• We must find average speed over the entire 320 km.

Concept / Approach:
Average speed is always defined as:
Average speed = Total distance / Total time.
We first compute the time taken for each part of the journey using time = distance / speed, then add these times to obtain the total time. Finally, we divide the total distance by total time to get the average speed.

Step-by-Step Solution:
Step 1: Time taken for first 160 km at 64 km/h = 160 / 64 hours. Step 2: 160 / 64 = 2.5 hours. Step 3: Time taken for next 160 km at 80 km/h = 160 / 80 = 2 hours. Step 4: Total time = 2.5 + 2 = 4.5 hours. Step 5: Total distance = 160 + 160 = 320 km. Step 6: Average speed = Total distance / Total time = 320 / 4.5 km/h. Step 7: 320 / 4.5 ≈ 71.11 km/h.
Verification / Alternative check:
When distances are equal, the harmonic mean of the two speeds can be used. Average speed = 2 * v1 * v2 / (v1 + v2). Here v1 = 64, v2 = 80, so average speed = 2 * 64 * 80 / (64 + 80) = 10240 / 144 ≈ 71.11 km/h. This matches the detailed method result and confirms the correctness of the answer.

Why Other Options Are Wrong:

• 35.55 km/h and 36 km/h: These are far too low and would make the journey unrealistically slow.
• 71 km/h: A rounded value that slightly underestimates the exact average of about 71.11 km/h.

Common Pitfalls:

• Taking the simple average (64 + 80) / 2 = 72 km/h, which is incorrect because times are not equal.
• Using incorrect division when computing 320 / 4.5.
• Mixing up the formula for average of speeds with the formula for average speed over a distance.

Final Answer:
The average speed for the first 320 km of the tour is approximately 71.11 km/h.