Violet and Owen are both hiking on scout Peak trial. When Violet starts on the trail, Owen has already hiked 1.5 miles. Violet is hiking in a pace of 3.8 mph and Owen is hiking at the pace of 2.6 mph if they both keep hiking the trail out these rates how long will it take for Violet to catch up?

Answer 1

To determine how long it will take Violet to catch up to Owen, we can set up a relationship between the distances each person has traveled over time.

1. Determine their speeds in miles per hour:

   • Violet's speed = 3.8 mph
   • Owen's speed = 2.6 mph

2. Establish the initial distance between them:

   • Owen has already hiked 1.5 miles when Violet starts.

3. Let \( t \) represent the time in hours that Violet hikes.

   • In that time, Violet will travel \( 3.8t \) miles.
   • During the same time, Owen will travel \( 2.6t \) miles.

4. Set up the equation for the distance: Violet catches up to Owen when the distance Violet has traveled is equal to the distance Owen has traveled plus the initial distance he is ahead:

\[ 3.8t = 2.6t + 1.5 \]

5. Solve for \( t \):
   • Subtract \( 2.6t \) from both sides:

\[ 3.8t - 2.6t = 1.5 \]

\[ 1.2t = 1.5 \]

• Now, divide both sides by 1.2:

\[ t = \frac{1.5}{1.2} = 1.25 \text{ hours} \]

6. Convert hours to minutes: To convert 0.25 hours to minutes, multiply by 60:

\[ 0.25 \times 60 = 15 \text{ minutes} \]

Thus, it will take Violet 1 hour and 15 minutes (or 1.25 hours) to catch up to Owen.