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Saw this problem the other day and I have a question about the solution(s):

A river is flowing downstream at a speed of 3 mph. A boat travels up the river 24 miles, turns around and travels down the river back to its starting location. If the total time for the round trip is 6 hours, what is the speed of the boat relative to the river (assumed to be the same for the upstream and downstream trips). So choosing upstream as the positive direction, and using v as the speed (magnitude) of the boat relative to the river:

Time for first leg of trip: 24/(v-3)

Time for second leg of trip: -24/(-v-3) = 24/(v+3)

Therefore, 24/(v-3) + 24/(v+3) = 6

Solving the resultant quadratic equation, we get v = 9 mph and -1 mph. Both of these are numerical solutions to the equation, as can be seen by substitution, but the second one is not physically meaningful: v is defined to be positive and anyhow any v < 3 would mean that the boat could never go upstream.

Why does the -1 solution occur? Is it something about how the problem is formulated?

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