1. **State the problem:** Shelley starts jogging from the trailhead at 5 miles per hour, while Naomi is already 1.3 miles ahead, walking at 2.4 miles per hour. We need to find how long it takes Shelley to catch up to Naomi. 2. **Set up variables and equations:** Let $t$ be the time in hours after Shelley starts jogging. - Distance Naomi has traveled after time $t$ is $1.3 + 2.4t$ miles. - Distance Shelley has traveled after time $t$ is $5t$ miles. 3. **Set the distances equal to find when Shelley catches Naomi:** $$5t = 1.3 + 2.4t$$ 4. **Solve for $t$:** $$5t - 2.4t = 1.3$$ $$2.6t = 1.3$$ Divide both sides by 2.6: $$t = \frac{1.3}{2.6}$$ 5. **Simplify the fraction:** $$t = \frac{1.3}{2.6} = \frac{13}{26} = \frac{1}{2}$$ 6. **Interpret the result:** It takes Shelley $\frac{1}{2}$ hour, or 0.5 hours, to catch up to Naomi. **Final answer:** $$\boxed{0.5 \text{ hours}}$$