A boat goes a certain distance at 40 km/h and comes back the same distance at 24 km/h. What is the average speed (in km/h) for the total journey?

Introduction / Context:
Here we are again dealing with average speed for a round trip where the distances are equal but the speeds are different. The boat travels one way at 40 km/h and returns at 24 km/h. To find the overall average speed, we must use total distance divided by total time, which for equal distances leads to the harmonic mean of the two speeds, not the simple arithmetic average.

Given Data / Assumptions:

• Speed going from starting point to destination = 40 km/h.
• Speed coming back = 24 km/h.
• Let the one way distance be d km.
• Total distance for the round trip = 2d km.
• We must find the average speed over the total journey.

Concept / Approach:
If two equal distances are covered at speeds v1 and v2, then: Average speed = 2 * v1 * v2 / (v1 + v2). This formula is derived from total distance divided by total time. Here v1 = 40 and v2 = 24. Alternatively, we can compute time for each leg separately, add them, and then use total distance / total time to get the same result.

Step-by-Step Solution:
Step 1: Let the one way distance be d km. Total distance = 2d km. Step 2: Compute the time for each leg. Time forward = d / 40 hours. Time return = d / 24 hours. Total time = d / 40 + d / 24. Step 3: Combine the times using a common denominator. d / 40 + d / 24 = (3d + 5d) / 120 = 8d / 120 = d / 15. Step 4: Compute average speed = total distance / total time. Average speed = (2d) / (d / 15) = 2d * 15 / d = 30 km/h.
Verification / Alternative check:
Using the harmonic mean formula directly: Average speed = 2 * 40 * 24 / (40 + 24) = 1920 / 64 = 30 km/h. Both methods confirm the same value.
Why Other Options Are Wrong:
The arithmetic mean (40 + 24) / 2 = 32 km/h is not correct because the boat spends longer time at the slower speed. Other values such as 28, 34, 36 km/h do not match the correct ratio of total distance to total time for the given speeds.
Common Pitfalls:
The most common mistake is using the simple average of 40 and 24 instead of total distance divided by total time. Another error is mishandling the fractions when adding times, leading to an incorrect denominator and hence wrong average speed.
Final Answer:
The average speed for the total journey is 30 km/h.