Bunuel wrote:
A power boat and a raft both left dock A 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 B downriver, then immediately turned and traveled back upriver. It eventually met the raft on the river 9 hours after leaving dock A. How many hours did it take the power boat to go from A to B?
A. 3
B. 3.5
C. 4
D. 4.5
E. 5
We can let r = the speed of the current and p = the speed of the power boat in still water. Also, we can let d = the distance between A and B.
Let’s analyze the answer choices. Let’s start with choice E and go backward.
E. 5
If it takes 5 hours for the power boat to go downstream from A to B, then we have (notice that the power boat will then spend 4 hours upstream) :
Downstream: (p + r) x 5 = d
Upstream: (p - r) x 4 + r x 9 = d
Simplifying the two equations, we have 5p + 5r = d for downstream and 4p + 5r = d for upstream. However, the two equations don’t agree (notice that if we subtract the two equations, we will have p = 0, which is impossible).
D. 4.5
If it takes 4.5 hours for the power boat to go downstream from A to B, then we have (notice that the power boat will then spend 4.5 hours upstream then) :
Downstream: (p + r) x 4.5 = d
Upstream: (p - r) x 4.5 + r x 9 = d
Simplifying the two equations, we have 4.5p + 4.5r = d for downstream and 4.5p +4.5r = d for upstream. We see that the two equations agree with each other. So D is the correct answer.
Alternate Solution:
We can let r = the speed of the current and p = the speed of the power boat in still water, d = the distance between A and B and t = the time (in hours) it takes for the powerboat to reach B from A.
Since it takes t hours for the boat to travel from A to B, the distance between A and B is t*(r + p). Therefore, we have d = t*(r + p) or, equivalently, d = tr + tp.
Since the rate of the raft is also r, the raft travels a distance of 9r in 9 hours. In t hours, the boat travels t*(r + p) and reaches B and then, the boat turns back and meets the raft, after traveling a distance of (9 - t)*(p - r) at a rate of p - r (since it is now traveling upstream). In total, they have traveled a distance of 9r + t*(r + p) + (9 - t)*(p - r) = 9r + tr + tp +9p - 9r - tp + tr = 2tr + 9p. Notice that the total distance they traveled is twice the distance between A and B, thus we must have 2tr + 9p = 2d. Dividing each side by 2, we obtain tr + 4.5p = d.
Since we have tr + 4.5p = d and we obtained tr + tp = d earlier, let’s equate the two expressions we have for d:
tr + 4.5p = tr + tp
4.5p = tp
t = 4.5
Since t was the time it took for the boat to get to B from A, the answer is 4.5.
Answer: D
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