A motorboat can travel at 10 km/h in still water. It goes 91 km downstream and returns 91 km upstream to the same place in a total of 20 hours. Find the rate of flow of the river (current speed in km/h).

Difficulty: Medium

3 km/h
4 km/h
2 km/h
5 km/h
None of these

Correct Answer: 3 km/h

Explanation:

Introduction / Context:
With still-water speed and total time over symmetric downstream/upstream distances, the current can be solved from a single rational equation.

Given Data / Assumptions:

b = 10 km/h
Distance each way = 91 km
Total time = 20 h
Let current be c

Concept / Approach:
Solve 91/(b + c) + 91/(b − c) = 20 for c (0 < c < b).

Step-by-Step Solution:

91/(10 + c) + 91/(10 − c) = 2091 * ( (10 − c) + (10 + c) ) / (100 − c^2) = 20 ⇒ 91 * 20 / (100 − c^2) = 201820 / (100 − c^2) = 20 ⇒ 100 − c^2 = 91 ⇒ c^2 = 9 ⇒ c = 3 km/h

Verification / Alternative check:
Downstream 91/(13) = 7 h; upstream 91/7 = 13 h; total = 20 h, consistent.

Why Other Options Are Wrong:
2, 4, 5 do not satisfy the sum-of-times equation with b = 10 km/h over 91 km each way.

Common Pitfalls:
Trying to average the two leg speeds directly; always sum times for unequal speeds over fixed distances.

A motorboat can travel at 10 km/h in still water. It goes 91 km downstream and returns 91 km upstream to the same place in a total of 20 hours. Find the rate of flow of the river (current speed in km/h).

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