I set up the algebra fine but kept second-guessing whether the 23 miles was net displacement or total distance traveled.
First, set up equations for the boat's speed in still water using the first day's trip: upstream time = 4 hours, downstream time = 5 hours, and the distance is the same. Solve for the boat speed, then use the second day's distance (23 miles upstream) to find the time taken, and finally subtract that time from the 6pm arrival to get the departure time.
Pro tip: Always define variables clearly and check units; a common mistake is to mix up the direction of the current. Also, verify your answer by plugging it back into the original scenario.
Let v be the boat's speed in still water (mph). The current is 2 mph. Upstream speed = v - 2, downstream speed = v + 2.
The distance traveled upstream in 4 hours equals the distance traveled downstream in 5 hours: 4(v - 2) = 5(v + 2). Solve for v.
Expand and solve: 4v - 8 = 5v + 10 → v = -18? Wait, that gives negative speed. Re-check: Actually, upstream is against current, so speed upstream = v - 2, downstream = v + 2. But if they paddle upstream for 4 hours then downstream for 5 hours to reach campsite, the campsite is downstream from the turn-around point. The distance from start to campsite is not directly given. The equation 4(v-2) = 5(v+2) would imply the distances are equal, but they are not necessarily equal because the campsite is not the starting point. Actually, the problem says: 'Two friends paddle upstream for 4 hours, then turn around and go downstream for 5 hours to reach their campsite.' So the total distance from start to campsite is: upstream distance (4 hours) minus downstream distance (5 hours)? No: They start at some point, paddle upstream for 4 hours, then turn around and paddle downstream for 5 hours to reach campsite. So the campsite is downstream from the starting point? Let's define: Let starting point be A. They paddle upstream for 4 hours to point B. Then they turn around and paddle downstream for 5 hours to point C (campsite). So distance from A to B = 4(v-2). Distance from B to C = 5(v+2). But C is downstream from A? Actually, if they go upstream then downstream, they might end up downstream of A if the downstream distance is greater. The problem says: 'The next day they paddle back to the original starting point, which is 23 miles upstream, arriving at 6pm.' So from campsite to original starting point is 23 miles upstream. That means the campsite is 23 miles downstream from the starting point. So distance from A to C = 23 miles. But from A to B upstream then B to C downstream: The net distance from A to C is (distance downstream from B to C) - (distance upstream from A to B) = 5(v+2) - 4(v-2) = 5v+10 -4v+8 = v + 18. This net distance should equal 23 miles. So v + 18 = 23 → v = 5 mph. That makes sense. So step 3: Use the fact that the campsite is 23 miles downstream from the start to set up equation: 5(v+2) - 4(v-2) = 23. Solve for v.
On the second day, they paddle from campsite to original starting point, which is 23 miles upstream. Upstream speed = v - 2 = 5 - 2 = 3 mph. Time = distance / speed = 23 / 3 hours ≈ 7.666... hours.
They arrive at 6pm. Subtract the travel time (7 hours 40 minutes) from 6pm to get departure time: 6pm - 7h40m = 10:20am.
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