A motor boat takes 12 hours to travel a certain distance downstream and 24 hours to travel the same distance upstream. For that distance, how much time would the boat take to travel in still water?

Introduction / Context:
This is a standard boat and stream question that uses downstream and upstream times for the same distance to infer the time required in still water. The focus is on understanding how speeds relate to times and how the still-water speed is the average of downstream and upstream speeds.

Given Data / Assumptions:

Let the one-way distance be d km.
Time taken downstream = 12 hours.
Time taken upstream = 24 hours.
Downstream speed = d / 12 km/h.
Upstream speed = d / 24 km/h.
Speed of boat in still water = average of downstream and upstream speeds.
We must find the time taken to cover distance d in still water.

Concept / Approach:
In boat-stream problems, the effective downstream speed is (b + c) and upstream speed is (b - c), where b is the boat's speed in still water and c is the current. Therefore, the boat's still-water speed is the average of these two: b = [(b + c) + (b - c)] / 2. Here, we compute these effective speeds from distance and time, then take their average to get b, and finally compute the time in still water as distance divided by b.

Step-by-Step Solution:
Step 1: Find downstream and upstream speeds. Downstream speed v_d = d / 12 km/h. Upstream speed v_u = d / 24 km/h. Step 2: Compute boat speed in still water. b = (v_d + v_u) / 2. b = (d/12 + d/24) / 2. Compute numerator: d/12 + d/24 = (2d + d) / 24 = 3d / 24 = d / 8. So b = (d/8) / 2 = d / 16 km/h. Step 3: Compute time in still water for the same distance d. Time in still water t = distance / speed = d / (d / 16) = 16 hours.

Verification / Alternative check:
We can loosely check consistency: the boat moves fastest downstream (taking 12 h) and slowest upstream (24 h). The still-water time must lie between these, and because the still-water speed is the average of the two speeds, its corresponding time is roughly in the middle of 12 and 24, which is 18, but when computed exactly for the same distance, we get 16 hours due to the harmonic-type relationship between speeds and times. The computed value 16 h satisfies the algebraic derivation and is consistent with the relationship between speeds and times.

Why Other Options Are Wrong:
8 h is too small and would imply a speed faster than the downstream speed, which is impossible in this scenario.
15 h and 20 h do not match the exact algebraic result from combining speeds and would not create the correct 12 h and 24 h times when currents are reintroduced.

Common Pitfalls:
A frequent mistake is to average the times directly (giving 18 h) instead of working with speeds. Remember that the still-water speed is the average of downstream and upstream speeds, not of the times. Another pitfall is forgetting that time is inversely proportional to speed, so you must convert times to speeds before averaging and then convert back to time.

Final Answer:
The boat would take 16 h to travel the same distance in still water.