A boat goes upstream at a speed of 21 km/h against the current and returns downstream over the same distance at a speed of 28 km/h with the current. Considering the complete to and fro journey, what is the average speed of the boat (in km/h) for the total trip?

Introduction / Context:
This question focuses on average speed when an object travels equal distances at different speeds in two directions. This is a classic concept in time and distance where the correct method is to use the harmonic mean rather than the simple arithmetic mean. It is particularly important in river boat and stream related problems in aptitude tests.

Given Data / Assumptions:

Upstream speed of the boat = 21 km/h.
Downstream speed of the boat = 28 km/h.
Distance upstream is equal to the distance downstream.
The current speed is constant and the boat speed relative to water is constant.
We need the average speed over the entire round trip.

Concept / Approach:
When the distances in both directions are equal and the speeds are different, average speed is not the arithmetic mean. Instead, average speed = (2 * u * v) / (u + v), where u and v are speeds in the two directions. This formula is derived from distance = speed * time for both legs of the journey. We apply this formula directly to the upstream and downstream speeds to find the overall average speed.

Step-by-Step Solution:
Step 1: Let the one-way distance be d kilometres. Upstream time = d / 21 hours and downstream time = d / 28 hours. Step 2: Total distance for the round trip = 2d kilometres. Step 3: Total time for the trip = d / 21 + d / 28 hours. Step 4: Average speed = total distance / total time = 2d / (d / 21 + d / 28). Step 5: Factor out d from the denominator to get 2d / [d * (1 / 21 + 1 / 28)] = 2 / (1 / 21 + 1 / 28). Step 6: Compute 1 / 21 + 1 / 28 = (4 + 3) / 84 = 7 / 84 = 1 / 12. Step 7: Hence average speed = 2 / (1 / 12) = 2 * 12 = 24 km/h.

Verification / Alternative check:
Choose a convenient value, for example, d = 84 kilometres, which is a common multiple of 21 and 28. Upstream time = 84 / 21 = 4 hours. Downstream time = 84 / 28 = 3 hours. Total distance = 168 kilometres. Total time = 7 hours. Average speed = 168 / 7 = 24 km/h. This direct calculation confirms that the formula based result is correct.

Why Other Options Are Wrong:
24.5 km/h and 25 km/h are close to the arithmetic mean of 21 and 28 but do not match the correct harmonic mean calculation. 25.4 km/h and 23.5 km/h also do not satisfy the detailed time and distance relationship. Only 24 km/h is consistent with equal distances travelled at 21 km/h and 28 km/h.

Common Pitfalls:
A frequent error is to compute the simple average (21 + 28) / 2 = 24.5 and assume it is the answer. This ignores that time spent at the lower speed is more, which pulls the effective average speed down. Another mistake is to forget that total distance is double the one-way distance when computing the final average speed. Using the harmonic mean formula avoids these errors.

Final Answer:
The average speed of the boat for the entire journey is 24 km/h.