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.

Question

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?

Answer 1

To determine how long it will take for Sam and Keith to be the same distance from their campsite, we can set up equations based on their speeds and distances.

Let's denote:

• \( t \) as the time in hours until they are the same distance from the campsite.

For Sam:

• Initial distance from the campsite: 21 kilometers
• Speed: 27 kilometers per hour

After \( t \) hours, Sam's distance from the campsite will be: \[ \text{Distance}_{Sam} = 21 - 27t \]

For Keith:

• Initial distance from the campsite: 7 kilometers
• Speed: 6 kilometers per hour

After \( t \) hours, Keith's distance from the campsite will be: \[ \text{Distance}_{Keith} = 7 - 6t \]

Set the distances equal:

To find when they are the same distance from the campsite, we set the equations equal to each other: \[ 21 - 27t = 7 - 6t \]

Solve for \( t \):

1. Rearranging the equation: \[ 21 - 7 = 27t - 6t \] simplifies to: \[ 14 = 21t \]
2. Divide both sides by 21: \[ t = \frac{14}{21} = \frac{2}{3} \text{ hours} \]

Conclusion:

It will take \( \frac{2}{3} \) hours, which is 40 minutes, for Sam and Keith to be the same distance from their campsite.