A boat whose speed in still water is 9 km/h goes 12 km downstream and then returns 12 km upstream to the starting point in a total time of 3 hours. What is the speed of the stream (in km/h)?

Introduction / Context:
In this boats and streams problem, you know the still-water speed of the boat, the distance covered downstream and upstream, and the total time for the round trip. From this, you must determine the speed of the stream. This involves setting up an equation using time = distance / speed for each leg of the journey.

Given Data / Assumptions:

Boat speed in still water, b = 9 km/h.
Let stream speed be c km/h.
Downstream speed = b + c = 9 + c km/h.
Upstream speed = b - c = 9 - c km/h (c < 9 so that upstream is possible).
Downstream distance = 12 km, upstream distance = 12 km.
Total time for downstream plus upstream = 3 hours.

Concept / Approach:
Time downstream is 12 / (9 + c) hours and time upstream is 12 / (9 - c) hours. The sum of these two times equals 3 hours. This gives a rational equation in c, which we solve to find the stream speed. The algebra simplifies nicely because the distances are equal in both directions.

Step-by-Step Solution:
Step 1: Write the time equation. 12 / (9 + c) + 12 / (9 - c) = 3. Step 2: Divide both sides by 3 to simplify. 4 / (9 + c) + 4 / (9 - c) = 1. Step 3: Combine the fractions. [4(9 - c) + 4(9 + c)] / (81 - c^2) = 1. Numerator: 4(9 - c + 9 + c) = 4 * 18 = 72. So 72 / (81 - c^2) = 1. Step 4: Solve for c. 81 - c^2 = 72 ⇒ c^2 = 9 ⇒ c = 3 (taking positive value). Thus the stream speed is 3 km/h.

Verification / Alternative check:
With c = 3 km/h, downstream speed = 9 + 3 = 12 km/h, upstream speed = 9 - 3 = 6 km/h. Downstream time = 12 / 12 = 1 hour. Upstream time = 12 / 6 = 2 hours. Total = 1 + 2 = 3 hours, which matches the given total time, confirming that c = 3 km/h is correct.

Why Other Options Are Wrong:
1 km/h, 4 km/h or 5 km/h, when used as stream speeds with b = 9 km/h, do not produce a total time of 3 hours for the 12 km downstream and 12 km upstream journey.
For example, with c = 1 km/h, total time would be 12/10 + 12/8 = 1.2 + 1.5 = 2.7 hours, not 3 hours.

Common Pitfalls:
Errors often occur in simplifying the rational equation: students may forget to factor the denominator as (9 + c)(9 - c) = 81 - c^2 or may make sign errors when combining terms. Another pitfall is choosing the negative root for c, but since speed cannot be negative, only the positive value is acceptable. Writing each algebraic step clearly helps avoid these issues.

Final Answer:
The speed of the stream is 3 km/h.