
A canoeist took 2 hours to travel 12 km down a river. The return trip, against the current, took 3 hours. What was the average paddling rate of the canoeist? What was the speed of the current? How is the answer: *average paddling rate is 5 km/h and speed of current* is 1km/h?

To solve this, we can use two variables to represent the unknowns:
- Let \( p \) be the canoeist's paddling rate in still water (in km/h).
- Let \( c \) be the speed of the current (in km/h).

### Step 1: Equation for the downstream trip
When going downstream, the canoeist's effective speed is increased by the current. So the total speed is \( p + c \). 

Since the distance traveled is 12 km and the time taken is 2 hours, we can write:
\[
\frac{12}{p + c} = 2 \quad \text{(time = distance / speed)}
\]
Solving for \( p + c \), we get:
\[
p + c = \frac{12}{2} = 6 \quad \text{km/h}
\]

### Step 2: Equation for the upstream trip
When going upstream, the canoeist's effective speed is decreased by the current. So the total speed is \( p - c \).

Since the distance traveled is again 12 km, but the time taken is 3 hours, we can write:
\[
\frac{12}{p - c} = 3
\]
Solving for \( p - c \), we get:
\[
p - c = \frac{12}{3} = 4 \quad \text{km/h}
\]

### Step 3: Solving the system of equations
Now, we have two equations:
1. \( p + c = 6 \)
2. \( p - c = 4 \)

We can solve this system by adding the two equations together:
\[
(p + c) + (p - c) = 6 + 4
\]
\[
2p = 10 \quad \Rightarrow \quad p = 5
\]
So, the canoeist's paddling rate in still water is 5 km/h.

Now, substitute \( p = 5 \) into one of the original equations to find \( c \). Using \( p + c = 6 \):
\[
5 + c = 6 \quad \Rightarrow \quad c = 1
\]
Thus, the speed of the current is 1 km/h.

### Final Answer:
- The average paddling rate of the canoeist is \( 5 \, \text{km/h} \).
- The speed of the current is \( 1 \, \text{km/h} \).

Would you like more details or have any questions?

Here are 5 related questions you could explore:
1. What if the downstream trip took 1.5 hours instead? How would that affect the calculations?
2. How would the problem change if the distance was different?
3. What if the speed of the current was 2 km/h? How would that impact the paddling rate?
4. How can we calculate the total average speed for the entire trip?
5. How would the scenario change if the canoeist paddled in a lake with no current?

**Tip:** When solving motion problems, always remember the relationship \( \text{time} = \frac{\text{distance}}{\text{speed}} \), and break the problem into parts when dealing with different conditions like with or against a current.

A canoeist took 2 hours to travel 12 km down a river. The return trip, against the current, took 3 hours. What was the average paddling rate of the canoeist? What was the speed of the current? How is the answer: average paddling rate is 5 km/h and speed of current is 1 km/h?

Discover how to solve the canoeist problem, where a canoeist travels downstream and upstream with different speeds due to the current. Learn how to calculate the paddling rate (5 km/h) and the current speed (1 km/h) using systems of equations and the speed formula. Ideal for students in Grades 8-10.

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