Question

Sam and Keith are spending the day at Little Stone Lake. Sam is exploring the lake in his boat, while Keith is enjoying his time on a paddleboard. When it is time to return to their campsite, Sam is 21 kilometers away and Keith is 7 kilometers away. Sam's boat can travel 27 kilometers per hour, and Keith can paddle 6 kilometers per hour.
If they each travel as fast as they can, how long will it take for Sam and Keith to be the same distance from their campsite?
Simplify any fractions.
hours

Answer 1

Let \( t \) be the time in hours after they start returning to the campsite.

For Sam:
- He is initially 21 kilometers away and travels at a speed of 27 kilometers per hour.
- The distance he covers in \( t \) hours is \( 27t \) kilometers.
- Therefore, the distance from the campsite after \( t \) hours is:
\[
21 - 27t
\]

For Keith:
- He is initially 7 kilometers away and paddles at a speed of 6 kilometers per hour.
- The distance he covers in \( t \) hours is \( 6t \) kilometers.
- Therefore, the distance from the campsite after \( t \) hours is:
\[
7 - 6t
\]

We want to find \( t \) such that Sam and Keith are the same distance from the campsite:
\[
21 - 27t = 7 - 6t
\]

Now we solve for \( t \):

1. Add \( 27t \) to both sides:
\[
21 = 7 - 6t + 27t
\]
\[
21 = 7 + 21t
\]

2. Subtract 7 from both sides:
\[
14 = 21t
\]

3. Divide both sides by 21:
\[
t = \frac{14}{21} = \frac{2}{3}
\]

Thus, the time it will take for Sam and Keith to be the same distance from their campsite is \( \frac{2}{3} \) hours.

In decimal form, this is approximately:
\[
\frac{2}{3} \text{ hours} \approx 0.67 \text{ hours}.
\]

So, the final answer is:
\[
\frac{2}{3} \text{ hours}
\]