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Let's break down the problem and solve it step by step.

### Problem Summary:
- **Naomi** is already **1.3 miles** ahead of the trailhead and is walking at a speed of **2.4 miles per hour**.
- **Shelley** starts at the trailhead and jogs at a speed of **5 miles per hour**.
- We are asked to find how long it takes Shelley to catch up to Naomi.

### Solution:

1. **Distance covered by Naomi:**
   - Naomi is moving at a speed of **2.4 miles per hour**.
   - Let the time it takes Shelley to catch up to Naomi be **t hours**.
   - In that time, Naomi will cover a distance of:
     \[
     \text{Distance by Naomi} = 2.4 \times t \, \text{miles}
     \]

2. **Distance covered by Shelley:**
   - Shelley's speed is **5 miles per hour**.
   - In the same time **t hours**, Shelley will cover a distance of:
     \[
     \text{Distance by Shelley} = 5 \times t \, \text{miles}
     \]

3. **Shelley catching up to Naomi:**
   - Shelley has to cover the initial **1.3 miles** gap that Naomi has already covered, plus the extra distance that Naomi covers during time **t**.
   - So the equation for when Shelley catches up is:
     \[
     5t = 1.3 + 2.4t
     \]

4. **Solve for t:**
   - Subtract **2.4t** from both sides:
     \[
     5t - 2.4t = 1.3
     \]
     \[
     2.6t = 1.3
     \]
   - Divide both sides by **2.6**:
     \[
     t = \frac{1.3}{2.6} = 0.5 \, \text{hours}
     \]

### Answer:
It will take **0.5 hours** (or 30 minutes) for Shelley to catch up to Naomi.

Would you like more details or have any questions?

Here are some related questions:
1. What is the concept of relative speed in this problem?
2. How would the solution change if Naomi's speed increased?
3. Can we solve this using a graphical approach?
4. What is the importance of simplifying fractions in such problems?
5. How does this type of problem relate to real-world scenarios like races?

**Tip:** Always set up an equation based on the distance each person covers when solving catch-up problems.

Naomi and Shelley both hiked the Discovery Point trail. When Shelley started hiking, Naomi was already 1.3 miles from the trailhead, walking at a rate of 2.4 miles per hour. Shelley started at the trailhead and jogged at a rate of 5 miles per hour. How long did it take Shelley to catch up to Naomi?

Learn how to solve a linear motion word problem where Shelley jogs to catch up to Naomi. This problem covers algebraic concepts such as linear equations and relative speed. Suitable for students in Grades 6-8.

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