Finding still water speed from round-trip time: A stream runs at 4 km/h. A boat goes 6 km downstream and returns 6 km upstream to the start in a total of 2 hours. What is the boat's speed in still water?
Aptitude
Boats and Streams
Difficulty: Medium
Choose an option
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A8 km/h
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B7.5 km/h
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C6.8 km/h
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D6 km/h
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E10 km/h
Answer
Correct Answer: 8 km/h
Explanation
Introduction / Context:For a round trip in a current, the total time equals the sum of downstream and upstream times. Using the stream speed, we can set an equation in the unknown still water speed u and solve it.
Given Data / Assumptions:
- Stream speed v = 4 km/h
- Downstream distance = 6 km
- Upstream distance = 6 km
- Total time T = 2 h
Concept / Approach:Time downstream = 6 / (u + v). Time upstream = 6 / (u - v). Their sum equals 2. Solve for u > v.
Step-by-Step Solution:
6/(u + 4) + 6/(u - 4) = 2Multiply through: 6(u - 4) + 6(u + 4) = 2(u^2 - 16)12u = 2u^2 - 322u^2 - 12u - 32 = 0 ⇒ u^2 - 6u - 16 = 0Solve: (u - 8)(u + 2) = 0 ⇒ u = 8 km/h (positive root)Verification / Alternative check:Downstream speed 12 km/h gives 0.5 h; upstream speed 4 km/h gives 1.5 h; total 2 h, consistent.
Why Other Options Are Wrong:
- 7.5, 6.8, and 6 km/h do not satisfy the time equation.
- 10 km/h yields total time less than 2 h.
Common Pitfalls:Adding distances and averaging speeds directly. Always sum times with correct effective speeds and solve the resulting equation carefully.
Final Answer:8 km/h