1. **Problem statement:** A person travels at 5 km/hr until the remaining distance equals the time traveled in hours. Then, they increase speed to 8 km/hr and complete the journey in 4 hours 30 minutes. Find the total distance. 2. **Define variables:** Let $t$ be the time traveled at 5 km/hr (in hours). 3. **Distance traveled at 5 km/hr:** $5t$ km. 4. **Remaining distance after $t$ hours:** Since remaining distance equals time traveled, remaining distance = $t$ km. 5. **Total distance:** Sum of distance traveled and remaining distance: $$D = 5t + t = 6t$$ 6. **Time taken for remaining distance at 8 km/hr:** $$\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{t}{8}$$ 7. **Total time given:** 4 hours 30 minutes = 4.5 hours. 8. **Equation for total time:** $$t + \frac{t}{8} = 4.5$$ 9. **Solve for $t$:** $$t \left(1 + \frac{1}{8}\right) = 4.5$$ $$t \times \frac{9}{8} = 4.5$$ $$t = 4.5 \times \frac{8}{9} = 4$$ 10. **Calculate total distance:** $$D = 6t = 6 \times 4 = 24$$ **Final answer:** The total distance covered is 24 km.