A boat covers 1 km downstream in 5 minutes and 1 km upstream in 12 minutes. What is the speed of the current (in km/h)?

Introduction / Context:
River-boat problems compare downstream (with current) and upstream (against current) speeds. The still-water speed and the current's speed can be recovered from these two observed speeds using simple averages and differences.

Given Data / Assumptions:

• Downstream: 1 km in 5 minutes.
• Upstream: 1 km in 12 minutes.
• Speeds are constant; current is uniform.

Concept / Approach:
If v_b is boat speed in still water and v_c is current speed, then downstream speed v_d = v_b + v_c and upstream speed v_u = v_b − v_c. Hence v_b = (v_d + v_u)/2 and v_c = (v_d − v_u)/2.

Step-by-Step Solution:
Convert times to hours: 5 min = 5/60 h; 12 min = 12/60 h.v_d = distance / time = 1 / (5/60) = 12 km/h.v_u = 1 / (12/60) = 5 km/h.Current speed v_c = (v_d − v_u) / 2 = (12 − 5)/2 = 3.5 km/h.

Verification / Alternative check:
Boat's still-water speed v_b = (12 + 5)/2 = 8.5 km/h. Check: 8.5 + 3.5 = 12 (downstream), 8.5 − 3.5 = 5 (upstream), which matches the given data.

Why Other Options Are Wrong:
4.5 km/h or 2.5 km/h assume incorrect averaging; 2 km/h is too small given a large downstream advantage.

Common Pitfalls:
Using arithmetic mean for the current or mixing minutes with hours without converting units consistently.

Final Answer:
3.5 km/h