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?
Simplify any fractions.

Answer 1

Let \( t \) be the time in hours that Sam and Keith will be traveling until they are the same distance from their campsite.

For Sam:

• Initial distance from the campsite: 21 km
• Speed of Sam's boat: 27 km/h
• Distance traveled by Sam after \( t \) hours: \( 27t \)
• Distance from the campsite after \( t \) hours: \[ 21 - 27t \]

For Keith:

• Initial distance from the campsite: 7 km
• Speed of Keith's paddleboard: 6 km/h
• Distance traveled by Keith after \( t \) hours: \( 6t \)
• Distance from the campsite after \( t \) hours: \[ 7 - 6t \]

Now, we need to find \( t \) when their distances from the campsite are equal: \[ 21 - 27t = 7 - 6t \]

Next, we'll solve for \( t \):

1. Rearranging the equation: \[ 21 - 7 = 27t - 6t \] simplifies to: \[ 14 = 21t \]

2. Solving for \( t \): \[ t = \frac{14}{21} = \frac{2}{3} \text{ hours} \]

Thus, it will take them \( \frac{2}{3} \) hours or \( 40 \) minutes to be the same distance from their campsite.