1. **State the problem:** Joe's average running speed is 0.5 km/h greater than Bob's. In a 5-km race, Bob finishes 3 minutes (which is 0.05 hours) behind Joe. We need to find their average speeds. 2. **Define variables:** Let Bob's speed be $b$ km/h. Then Joe's speed is $b + 0.5$ km/h. 3. **Use the formula for time:** Time = Distance / Speed. 4. **Write expressions for their times:** - Joe's time: $\frac{5}{b+0.5}$ hours - Bob's time: $\frac{5}{b}$ hours 5. **Set up the equation using the time difference:** $$\frac{5}{b} - \frac{5}{b+0.5} = 0.05$$ 6. **Solve the equation:** Multiply both sides by $b(b+0.5)$ to clear denominators: $$5(b+0.5) - 5b = 0.05 b (b+0.5)$$ Simplify left side: $$5b + 2.5 - 5b = 0.05 b^2 + 0.025 b$$ Which reduces to: $$2.5 = 0.05 b^2 + 0.025 b$$ Multiply both sides by 20 to clear decimals: $$50 = b^2 + 0.5 b$$ Rewrite as quadratic: $$b^2 + 0.5 b - 50 = 0$$ 7. **Use quadratic formula:** $$b = \frac{-0.5 \pm \sqrt{0.5^2 - 4 \times 1 \times (-50)}}{2} = \frac{-0.5 \pm \sqrt{0.25 + 200}}{2} = \frac{-0.5 \pm \sqrt{200.25}}{2}$$ Calculate $\sqrt{200.25} \approx 14.15$: $$b = \frac{-0.5 \pm 14.15}{2}$$ 8. **Choose the positive root (speed must be positive):** $$b = \frac{-0.5 + 14.15}{2} = \frac{13.65}{2} = 6.825$$ 9. **Find Joe's speed:** $$b + 0.5 = 6.825 + 0.5 = 7.325$$ **Final answer:** - Bob's average speed is approximately $6.83$ km/h. - Joe's average speed is approximately $7.33$ km/h.