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

Introduction / Context:
This aptitude question focuses on the concept of average speed over a round trip when the distances are the same but the speeds differ. It is important to remember that the average speed is not the simple average of the two speeds. Instead, for equal distances, the correct formula is based on the harmonic mean.

Given Data / Assumptions:

• The boat travels from point 1 to point 2 at 30 km/h.
• It returns from point 2 to point 1 at 60 km/h.
• The distance one way is the same in both directions (say d km).
• We are asked to find the average speed for the entire journey.

Concept / Approach:
Average speed is defined as total distance travelled divided by total time taken. For a round trip with equal distances d at two different speeds v1 and v2, the formula simplifies to: average speed = (2 * v1 * v2) / (v1 + v2) This is the harmonic mean of the two speeds. We can derive this quickly by computing times for each leg and then using the definition of average speed.

Step-by-Step Solution:
Step 1: Let the one-way distance be d km. Step 2: Time taken for the onward journey at 30 km/h = d / 30 hours. Step 3: Time taken for the return journey at 60 km/h = d / 60 hours. Step 4: Total distance for the round trip = d + d = 2d km. Step 5: Total time = d / 30 + d / 60. Step 6: Compute total time: d / 30 + d / 60 = (2d / 60) + (d / 60) = 3d / 60 = d / 20 hours. Step 7: Average speed = total distance / total time = (2d) / (d / 20) = 2d * (20 / d) = 40 km/h.

Verification / Alternative check:
Using the harmonic mean formula directly: average speed = (2 * 30 * 60) / (30 + 60) = (3600) / 90 = 40 km/h. This matches the value obtained from the full calculation, confirming the result is correct.

Why Other Options Are Wrong:
A common incorrect guess is 45 km/h, the arithmetic mean of 30 and 60, but this ignores the longer time spent at the lower speed. Options 50, 35 and 36 km/h also do not match the ratio of total distance to total time. Only 40 km/h correctly balances the slower and faster legs of the trip.

Common Pitfalls:
Many students mistakenly average the speeds directly, which is valid only if the times at each speed are equal, not the distances. In this problem, more time is spent travelling at 30 km/h, which pulls the overall average down below 45 km/h. Always base average speed on total distance divided by total time.

Final Answer:
The average speed for the entire journey is 40 km/h.