A motor boat takes 12 hours to travel a certain distance downstream and 24 hours to travel the same distance upstream. How many hours would it take to cover this distance in still water?

Introduction / Context:
This question compares travel times downstream and upstream for the same distance, and asks for the time in still water. It is a standard boats and streams problem where the key is to relate the downstream and upstream speeds to the still-water speed and the stream speed. Once you find the effective distance-speed relationship, you can determine how long the trip would take in still water.

Given Data / Assumptions:

• Time taken downstream for a certain distance D = 12 hours.
• Time taken upstream for the same distance D = 24 hours.
• Let speed of boat in still water be b km/h.
• Let speed of stream be s km/h.
• Downstream speed = b + s.
• Upstream speed = b − s.
• We are asked for the time taken to cover the distance D in still water at speed b.

Concept / Approach:
First we use time = distance / speed to express downstream and upstream speeds in terms of D. Then we relate those speeds back to b and s. This gives a simple algebraic relation between b and s. From that relation, we express D in terms of b and finally compute the time in still water as D / b.

Step-by-Step Solution:
Step 1: Downstream speed = distance / time = D / 12.Step 2: Upstream speed = distance / time = D / 24.Step 3: We know downstream speed = b + s and upstream speed = b − s.Step 4: So b + s = D / 12 and b − s = D / 24.Step 5: Add the two equations: (b + s) + (b − s) = D / 12 + D / 24 ⇒ 2b = D(1/12 + 1/24) = D(3/24) = D/8.Step 6: Therefore b = D / 16.Step 7: Time in still water to cover distance D = D / b = D / (D/16) = 16 hours.

Verification / Alternative check:
We can also express D in terms of b and s. From downstream, D = 12(b + s); from upstream, D = 24(b − s). Equating gives 12(b + s) = 24(b − s) ⇒ b + s = 2(b − s) ⇒ b = 3s. Using this, we find D = 12(b + s) = 12(3s + s) = 48s and b = 3s. Then time in still water D / b = 48s / (3s) = 16 hours, which confirms our previous result.

Why Other Options Are Wrong:
Times like 15 hours or 20 hours do not match the derived relation between downstream and upstream speeds. An answer of 8 hours would be too short, implying a still-water speed greater than either downstream or upstream speed, which is impossible since the stream only adds or subtracts speed. Twelve hours corresponds to the downstream time only, not the balanced still-water time for the same distance.

Common Pitfalls:

• Trying to simply average 12 and 24 to get 18 hours, which is incorrect.
• Assuming that still-water speed is the arithmetic mean of downstream and upstream speeds without relating them to actual distances.
• Not using the fact that the distance is the same in both directions.
• Algebraic slip-ups when adding or subtracting (b + s) and (b − s).

Final Answer:
The boat would take 16 hours to cover the distance in still water.