A rower travels 12 miles downstream in a river in 6 hours less time than it takes to travel the same distance upstream. If the rower doubles their usual rowing speed for the entire 24-mile round trip, the downstream portion would take only 1 hour less than the upstream portion. What is the speed of the river's current in miles per hour?

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Accepted Answer

Let's denote the rower's speed in still water as x mph and the current's speed as y mph.

For a boat traveling in a river:
Speed upstream = (x - y) mph
Speed downstream = (x + y) mph[4]

For the first scenario, using time = distance/speed:
Time upstream = 12/(x - y) hours
Time downstream = 12/(x + y) hours

According to the first condition, the difference between these times is 6 hours:
12/(x - y) - 12/(x + y) = 6[2]

Simplifying this equation:
12(x + y - x + y)/[(x - y)(x + y)] = 6
24y/(x² - y²) = 6
24y = 6(x² - y²)
24y = 6x² - 6y²
6y² + 24y - 6x² = 0...(1)[3]

For the second scenario with doubled rowing rate:
New speed upstream = (2x - y) mph
New speed downstream = (2x + y) mph

Time upstream = 12/(2x - y) hours
Time downstream = 12/(2x + y) hours

From the second condition, the difference is 1 hour:
12/(2x - y) - 12/(2x + y) = 1[3]

Simplifying:
12(2x + y - 2x + y)/[(2x - y)(2x + y)] = 1
24y/(4x² - y²) = 1
24y = 4x² - y²
y² + 24y - 4x² = 0...(2)[3]

From equation (1), dividing by 6:
y² + 4y - x² = 0...(3)

Comparing equations (2) and (3):
Multiplying equation (3) by 4: 4y² + 16y - 4x² = 0
Subtracting from equation (2): y² + 24y - 4x² = 0 - (4y² + 16y - 4x²) = 0
Results in: -3y² + 8y = 0

Factoring: y(-3y + 8) = 0
Therefore: y = 0 or y = 8/3

Since y = 0 would mean no current (which doesn't fit the problem context), the speed of the current is 8/3 mph or 2⅔ mph.

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