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

To determine when Sam and Keith are the same distance from their campsite, we need to set up equations for the distances they travel over time.

Let \( t \) be the time in hours that they both travel.

- The distance Sam travels can be expressed as:
\[
d_S = 21 - 27t
\]
Here, 21 kilometers is the initial distance Sam is from the campsite, and he is moving towards it at a speed of 27 kilometers per hour.

- The distance Keith travels can be expressed as:
\[
d_K = 7 - 6t
\]
Again, 7 kilometers is the initial distance Keith is from the campsite, and he is moving towards it at a speed of 6 kilometers per hour.

We want to determine when their distances from the campsite are equal:
\[
21 - 27t = 7 - 6t
\]

Now, let's solve for \( t \):
1. First, simplify the equation:
\[
21 - 27t = 7 - 6t
\]

2. Rearranging gives:
\[
21 - 7 = 27t - 6t
\]
\[
14 = 21t
\]

3. Now, solve for \( t \):
\[
t = \frac{14}{21}
\]
\[
t = \frac{2}{3}
\]

Thus, it will take them \( \frac{2}{3} \) hours to be the same distance from their campsite.

To convert \( \frac{2}{3} \) hours into minutes:
\[
\frac{2}{3} \times 60 = 40 \text{ minutes}
\]

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