A boat takes 2 hours to travel downstream a river from port A to port B, and 3 hours to return to port A. Another boat takes a total of 6 hours to travel from port B to port A and return to port B. If the speeds of the boats and the river are constant, then the time, in hours, taken by the slower boat to travel from port A to port B is
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3(3 - 5 )
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12( 5 - 2)
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3(3 + 5 )
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3( 5 - 1)
A
Correct answer
Explanation
Let v be boat speed and c be current. Downstream: v+c = distance/2. Upstream: v-c = distance/3. Let distance = 6. v+c = 3, v-c = 2. 2v = 5, v = 2.5, c = 0.5. Second boat: total time 6 hours. Let speed be u. (6/(u+0.5)) + (6/(u-0.5)) = 6. 1/(u+0.5) + 1/(u-0.5) = 1. (u-0.5 + u+0.5) / (u^2 - 0.25) = 1. 2u = u^2 - 0.25. u^2 - 2u - 0.25 = 0. u = (2 + sqrt(4 - 4(1)(-0.25))) / 2 = (2 + sqrt(5)) / 2 = 1 + sqrt(5)/2. Time = 6 / (1 + sqrt(5)/2 + 0.5) = 6 / (1.5 + sqrt(5)/2) = 12 / (3 + sqrt(5)). Rationalizing: 12(3-sqrt(5)) / (9-5) = 3(3-sqrt(5)).