1. **State the problem:** We have two trains making the same trip. The passenger train takes 3 hours, and the freight train takes 7 hours. The freight train's speed is 40 miles per hour less than the passenger train's speed. We need to find the rate (speed) of each train. 2. **Define variables:** Let $r_p$ be the rate of the passenger train (in miles per hour). Let $r_f$ be the rate of the freight train. 3. **Write the relationships:** Since the freight train's rate is 40 mph less than the passenger train's rate: $$r_f = r_p - 40$$ 4. **Use the formula for distance:** Distance $d = \text{rate} \times \text{time}$. Both trains travel the same distance, so: $$d = r_p \times 3 = r_f \times 7$$ 5. **Substitute $r_f$ in terms of $r_p$:** $$3r_p = 7(r_p - 40)$$ 6. **Solve for $r_p$:** $$3r_p = 7r_p - 280$$ Move terms to one side: $$3r_p - 7r_p = -280$$ $$-4r_p = -280$$ Divide both sides by $-4$: $$r_p = \frac{-280}{-4}$$ $$r_p = 70$$ 7. **Find $r_f$:** $$r_f = r_p - 40 = 70 - 40 = 30$$ 8. **Check the distance:** Passenger train distance: $$d = 70 \times 3 = 210$$ Freight train distance: $$d = 30 \times 7 = 210$$ Both distances are equal, confirming the solution. **Final answer:** - Passenger train rate: $70$ mph - Freight train rate: $30$ mph