
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? Can you solve this in whatever is better? My only two methods are either substitution or elimination.

Let's solve this using a system of equations. We can define two variables:

- Let \( p \) be the paddling rate of the canoeist in still water (in km/h).
- Let \( c \) be the speed of the current (in km/h).

### Step 1: Setting up the equations

1. **Downstream trip**: When the canoeist is traveling downstream (with the current), the total speed is the paddling rate plus the speed of the current, i.e., \( p + c \). The time for this trip is 2 hours, and the distance is 12 km. Using the formula \( \text{speed} = \frac{\text{distance}}{\text{time}} \), we get:

   \[
   p + c = \frac{12}{2} = 6 \, \text{km/h} \tag{Equation 1}
   \]

2. **Upstream trip**: When the canoeist is traveling upstream (against the current), the total speed is the paddling rate minus the speed of the current, i.e., \( p - c \). The time for this trip is 3 hours, and the distance is 12 km. Using the same formula, we get:

   \[
   p - c = \frac{12}{3} = 4 \, \text{km/h} \tag{Equation 2}
   \]

### Step 2: Solving the system of equations

We now have the following system of linear equations:

\[
p + c = 6 \tag{1}
\]
\[
p - c = 4 \tag{2}
\]

#### Method: Elimination

1. **Add** the two equations to eliminate \( c \):

   \[
   (p + c) + (p - c) = 6 + 4
   \]
   \[
   2p = 10
   \]
   \[
   p = 5 \, \text{km/h}
   \]

2. **Substitute** \( p = 5 \) into either equation. Using Equation (1):

   \[
   5 + c = 6
   \]
   \[
   c = 1 \, \text{km/h}
   \]

### Final Answer:
- The average paddling rate of the canoeist is **5 km/h**.
- The speed of the current is **1 km/h**.

This matches the answer you provided.

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Do you have any questions or want further details?

Here are 5 related questions for practice:
1. If the distance traveled downstream was increased to 15 km, how long would it take at the same paddling rate and current speed?
2. If the current speed doubled, what would the new downstream and upstream times be for the same 12 km journey?
3. What would the average paddling rate be if the canoeist took 4 hours upstream and 2.5 hours downstream?
4. How would the times change if the paddling rate decreased by 1 km/h, but the current stayed the same?
5. How can you check the work by calculating the total time for a round trip with the same speeds?

**Tip:** When working with speed and time problems, setting up equations using \( \text{speed} = \frac{\text{distance}}{\text{time}} \) helps ensure consistent and reliable results.

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?

Learn how to determine the average paddling rate of a canoeist and the speed of the river current using algebraic methods. This problem involves setting up and solving a system of linear equations to find the rate of paddling in still water and the speed of the current. Suitable for students in Grades 8-10.

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