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This problem can be solved using the concept of relative motion. We need to determine how long it will take for Sam and Keith to be at the same distance from the campsite.

Let’s denote:

- \( t \) as the time (in hours) after they start moving.
- Sam’s distance from the campsite after \( t \) hours will be \( 21 - 27t \) (since he is 21 km away and traveling towards the campsite at 27 km/h).
- Keith’s distance from the campsite after \( t \) hours will be \( 7 - 6t \) (since he is 7 km away and traveling at 6 km/h).

At the point when they are the same distance from the campsite, these distances will be equal:

\[
21 - 27t = 7 - 6t
\]

Now, let's solve this equation for \( t \):

\[
21 - 27t = 7 - 6t
\]

Subtract 7 from both sides:

\[
14 - 27t = -6t
\]

Add \( 27t \) to both sides:

\[
14 = 21t
\]

Now, divide both sides by 21:

\[
t = \frac{14}{21} = \frac{2}{3}
\]

So, it will take **\( \frac{2}{3} \) hours** (or 40 minutes) for Sam and Keith to be the same distance from the campsite.

Would you like any more details on this solution or have any other questions?

Here are five related questions to deepen your understanding:
1. What is the total distance Sam travels by the time they meet at the same point?
2. How far from the campsite will Sam and Keith be when they are at the same distance?
3. What if Keith could only paddle at 4 km/h? How would this affect the solution?
4. How would the result change if Sam started 30 kilometers away instead of 21?
5. How much time would it take if Keith were 10 kilometers away instead of 7?

**Tip**: When solving relative motion problems, setting up equations for distance is key to finding a common meeting point or time.

Sam and Keith are spending the day at Little Stone Lake. Sam is 21 kilometers away from the campsite and travels at 27 kilometers per hour. Keith is 7 kilometers away from the campsite and travels at 6 kilometers per hour. How long will it take for Sam and Keith to be the same distance from their campsite?

Learn how to solve a relative motion problem involving Sam and Keith traveling towards the same destination at different speeds. The problem requires setting up and solving linear equations to find when they will be the same distance from their campsite. Suitable for students in Grades 6-8.

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