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2011 AMC 12A Problem 12

Problem 12 of 25IntermediateAlgebra

A power boat and a raft both left dock AA on a river and headed downstream. The raft drifted at the speed of the river current. The power boat maintained a constant speed with respect to the river. The power boat reached dock BB downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 99 hours after leaving dock A.A. How many hours did it take the power boat to go from AA to B?B?

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Solution

Measure everything relative to the water. In that frame the raft is stationary at the point where the boat started, and the boat moves at its constant speed vv relative to the water, both downstream and upstream. The boat leaves the raft, travels away for some time, then returns to it at the same relative speed, so it spends equal times going and returning. Hence the outbound leg to BB takes half of 9,9, which is 4.54.5 hours. Thus, the correct answer is D.

More practice

Concepts: relative speed · distance rate and time · symmetry

Problem text and solution from the LIVE past-contest archive. See also the AoPS wiki page for community solutions.