Nick and Chloe left their campsite by canoe and paddle downstream at an average speed of 12 km/h. They turned around and paddled back upstream at an average speed of 4 km/h. The total trip took 1 hour. After how much time did the campers turn around downstream
The Correct answer and Explanation is:
Let’s define the problem variables to solve this:
- Let t be the time (in hours) Nick and Chloe spent paddling downstream.
- The time spent paddling upstream is then 1 – t (since the total trip took 1 hour).
Step 1: Express Distance in Terms of Time and Speed
We know the formula:
[
\text{Distance} = \text{Speed} \times \text{Time}
]
- The distance downstream is covered at a speed of 12 km/h in time t. Therefore, the distance downstream is:
[
\text{Distance downstream} = 12 \times t
] - The distance upstream is covered at a speed of 4 km/h in time 1 – t. Therefore, the distance upstream is:
[
\text{Distance upstream} = 4 \times (1 – t)
]
Step 2: Set Up an Equation
Since Nick and Chloe turned around at a point downstream and paddled the same distance upstream, the distance covered downstream is equal to the distance covered upstream:
[
12 \times t = 4 \times (1 – t)
]
Step 3: Solve the Equation
Now, solve for t:
[
12t = 4(1 – t)
]
Distribute the 4 on the right-hand side:
[
12t = 4 – 4t
]
Add 4t to both sides:
[
12t + 4t = 4
]
Simplify:
[
16t = 4
]
Now, divide both sides by 16:
[
t = \frac{4}{16} = \frac{1}{4} \, \text{hours} = 0.25 \, \text{hours}
]
Step 4: Interpret the Result
Nick and Chloe spent 0.25 hours (15 minutes) paddling downstream before turning around.
Step 5: Verify the Result
To verify, check the total time:
- Downstream time: 0.25 hours at 12 km/h, covering 3 km.
- Upstream time: ( 1 – 0.25 = 0.75 ) hours at 4 km/h, covering 3 km.
Since both distances match, the solution is correct.
Explanation Summary
The problem involves two key steps: setting up equations for the distance traveled downstream and upstream, based on the known speeds and total time. By equating these distances and solving for the downstream time, we find that Nick and Chloe paddled downstream for 0.25 hours before turning around. This ensures they traveled equal distances in opposite directions within the 1-hour trip.