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. sams 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

Explanation:

Step1: Set up distance - time equations

Let $t$ be the time in hours. Sam's distance from the campsite after $t$ hours is $d_S=21 - 27t$ (since he is moving towards the campsite). Keith's distance from the campsite after $t$ hours is $d_K = 7-6t$.

Step2: Set the two distances equal

We want to find when $d_S=d_K$, so we set up the equation $21 - 27t=7 - 6t$.

Step3: Solve the equation for $t$

First, add $27t$ to both sides: $21=7 - 6t+27t$.
Simplify the right - hand side: $21=7 + 21t$.
Then subtract 7 from both sides: $21 - 7=21t$, so $14 = 21t$.
Finally, divide both sides by 21: $t=\frac{14}{21}=\frac{2}{3}$.

Answer:

$\frac{2}{3}$