What is the speed of the river’s current if a boat going 5 mph takes three hours to go upstream from point a to point b and two hours to make the return trip from b to a?
Answer
- What do you know?
- The distance going upstream and downstream is the same
- Time going upstream is 3 hours
- Time going downstream is 2 hours
- Speed of the boat (r) is 5 miles per hour
- Distance (d) = rate (r) X time (t)
- What can you infer?
- It takes more time to go upstream because you are going against the current. So, the speed of the boat relative to the land equals r (speed of the boat relative to the water) – c (speed of the current), [5 – c] or in general terms r – c.
- It takes less time to go downstream because you are going with the current. So, the speed of the boat relative to land equals r (speed of the boat relative to the water) + c (speed of the current [5 + c], or in general terms r + c.
- How do you set up to solve the equation? Write out the two equations in a. and b. below or use the distance, rate, time diagram in part c. below and plug in the rate and the time, and then multiply them to get the distances. Either method works.
- For upstream r = 5 – c, and t = 3 hours, so when you multiply these d = 3(5 – c)
- For downstream r = 5 + c, and t = 2 hours, so when you multiply these d = 2(5 + c)
- Distance, rate, time diagram
|
|
Distance (d) |
Rate (r) |
Time (t) |
|
Upstream |
3(5 – c) |
5 – c |
3 hours |
|
Downstream |
2 (5 + c) |
5 + c |
2 hours |
- For either of the two methods above, the next part of the solution is done the same way.
- Since the distances (d) are the same, set them equal to each other and solve for the speed of the current (c).
3(5 – c) = 2 (5 + c) (distribute)
15 - 3c = 10 + 2c (subtract 10 from both sides)
5 – 3c = 2c (add 3c to both sides)
5 = 5c (divide both sides by 5)
1 mph = c (answer)
- So, the speed of the current is 1 mile per hour