A boat travels 72 km downstream in 8 hours and 84 km upstream in 12 hours. Find the speed of the boat in still water and the speed of the stream (in km/h).

Introduction / Context:
This question gives different distances and times for downstream and upstream travel. From these we can calculate the effective upstream and downstream speeds, then use them to determine the boat speed in still water and the stream speed. It is a classic application of boats and streams formulas and reinforces the idea that effective speed depends on whether the boat moves with or against the current.

Given Data / Assumptions:

• Downstream distance = 72 km, time = 8 hours.
• Upstream distance = 84 km, time = 12 hours.
• Let b be the speed of the boat in still water.
• Let c be the speed of the current.
• Speeds are constant in both directions.

Concept / Approach:
We first find effective speeds:

• Downstream speed = distance / time = 72 / 8.
• Upstream speed = distance / time = 84 / 12.

Then we use the relations:

• b + c = downstream speed.
• b - c = upstream speed.

Solving these equations gives b and c. Finally, we match our answer to the option written as boat speed and stream speed.

Step-by-Step Solution:
Downstream speed = 72 / 8 = 9 km/h. Upstream speed = 84 / 12 = 7 km/h. Let b be boat speed, c be stream speed. Then b + c = 9 and b - c = 7. Add the equations: 2b = 16 so b = 8 km/h. Subtract the equations: 2c = 2 so c = 1 km/h. Therefore, boat speed = 8 km/h and stream speed = 1 km/h.

Verification / Alternative check:
Check downstream: effective speed 8 + 1 = 9 km/h; time for 72 km is 72 / 9 = 8 hours, matching the given time. Check upstream: effective speed 8 - 1 = 7 km/h; time for 84 km is 84 / 7 = 12 hours, which matches as well. This confirms that the values b = 8 km/h and c = 1 km/h satisfy all conditions.

Why Other Options Are Wrong:
Option 9 and 3 km/h would imply downstream and upstream speeds that do not produce the given times for 72 km and 84 km. Similarly, 6 and 7 km/h or 7 and 2 km/h do not satisfy both distance time pairs simultaneously. Only the pair 8 and 1 km/h correctly reproduces the 8 hours downstream and 12 hours upstream travel times.

Common Pitfalls:
Some students mistakenly average the distances or times incorrectly or mix up which pair corresponds to boat speed and which to stream speed. Another frequent issue is forgetting that different distances require separate speed calculations before forming equations. Always compute downstream and upstream speeds first and only then set up the equations b + c and b - c to avoid confusion.

Final Answer:
The speed of the boat in still water is 8 km/h and the speed of the stream is 1 km/h.