Ashok can row a boat upstream at 8 km/h and downstream at 12 km/h. What is the speed of the stream (in km/h)?

Introduction / Context:
This is a standard boats-and-streams question. You are given the speed of rowing upstream and downstream and asked to find the speed of the stream. The key idea is that the stream either helps or opposes the boat, changing the effective speed.

Given Data / Assumptions:
• Upstream speed (against current) = 8 km/h.• Downstream speed (with current) = 12 km/h.• Let the speed of the boat in still water be u km/h.• Let the speed of the stream be v km/h.

Concept / Approach:
When the boat moves upstream, the current opposes it, so the effective speed is (u − v). When it moves downstream, the current aids it, so the effective speed is (u + v). These two linear equations can be solved simultaneously to find u and v. The question only asks for v, the speed of the stream.

Step-by-Step Solution:
Step 1: Upstream: u − v = 8.Step 2: Downstream: u + v = 12.Step 3: Add the two equations: (u − v) + (u + v) = 8 + 12.Step 4: This gives 2u = 20 ⇒ u = 10 km/h.Step 5: Substitute u = 10 into u + v = 12 ⇒ 10 + v = 12.Step 6: Therefore, v = 2 km/h.

Verification / Alternative check:
If the boat's own speed is 10 km/h and the stream is 2 km/h, then upstream speed = 10 − 2 = 8 km/h and downstream speed = 10 + 2 = 12 km/h, matching the given values exactly.

Why Other Options Are Wrong:
6 km/hr: Would give impossible values for the boat's own speed when combined with the given upstream speed.3 km/hr or 4.5 km/hr: These do not satisfy both equations simultaneously when checked.

Common Pitfalls:
Students sometimes average 8 and 12 directly and call that the stream speed, forgetting that the average gives the still-water speed, not the stream speed. The stream speed is half the difference of downstream and upstream speeds.

Final Answer:
The speed of the stream is 2 km/h.