A boat goes a certain distance at 30 km/h and returns the same distance at 60 km/h. What is the average speed (in km/h) for the entire journey?

Introduction / Context:
This question is about average speed for a round trip where the distances in both directions are equal but the speeds are different. The boat goes a certain distance at 30 km/h and comes back the same distance at 60 km/h. We must find the average speed for the entire journey. This is a classic example of when the average speed is not the simple arithmetic mean of the two speeds, but instead the harmonic mean due to equal distances.

Given Data / Assumptions:

• Speed from starting point to destination = 30 km/h.
• Speed on the way back = 60 km/h.
• The distance one way is the same in both directions.
• Let the one way distance be d km.
• We must find the average speed over the whole trip.

Concept / Approach:
For two equal distances covered at speeds v1 and v2, the average speed is given by: Average speed = 2 * v1 * v2 / (v1 + v2). This formula comes from using total distance divided by total time. Here v1 = 30 and v2 = 60. Alternatively, we can compute the total distance as 2d and the total time as d / 30 + d / 60 and then divide. Both methods give the same result.

Step-by-Step Solution:
Step 1: Let the one way distance be d km. Total distance = 2d. Step 2: Compute the time for each leg. Time forward = d / 30 hours. Time return = d / 60 hours. Total time = d / 30 + d / 60. Step 3: Combine the times. d / 30 + d / 60 = (2d + d) / 60 = 3d / 60 = d / 20. Step 4: Use average speed = total distance / total time. Average speed = (2d) / (d / 20) = 2d * (20 / d) = 40 km/h.
Verification / Alternative check:
Use the harmonic mean formula directly: Average speed = 2 * 30 * 60 / (30 + 60) = 3600 / 90 = 40 km/h. Both methods yield the same average speed, confirming the result.
Why Other Options Are Wrong:
The arithmetic mean (30 + 60) / 2 = 45 km/h is not correct because the boat spends more time travelling at the slower speed. Speeds 50, 35 or 36 km/h do not match the total distance divided by the total time for the given legs.
Common Pitfalls:
A frequent error is to simply average the two speeds, forgetting that average speed is based on total distance and total time, not on the arithmetic mean of speeds. Another pitfall is mishandling the addition of the time fractions, leading to an incorrect total time and hence a wrong average speed.
Final Answer:
The average speed for the total journey is 40 km/h.