Train Speeds 366880

1. **State the problem:** We need to find the speeds of two trains: a passenger train and a freight train. The freight train's speed is 8 mph less than the passenger train's speed. The passenger train travels 400 miles, and the freight train travels 360 miles, both in the same amount of time. 2. **Define variables:** Let $p$ be the speed of the passenger train in mph. Then the speed of the freight train is $p - 8$ mph. 3. **Use the formula for time:** Time = Distance \div Speed. Since both trains travel for the same time, we set their times equal: $$\frac{400}{p} = \frac{360}{p - 8}$$ 4. **Solve the equation:** Cross-multiply: $$400(p - 8) = 360p$$ Distribute: $$400p - 3200 = 360p$$ Subtract $360p$ from both sides: $$400p - 360p - 3200 = 0$$ Simplify: $$40p - 3200 = 0$$ Add $3200$ to both sides: $$40p = 3200$$ Divide both sides by 40: $$p = \frac{3200}{40}$$ Show cancellation: $$p = \frac{\cancel{3200}^{80} }{\cancel{40}^{1}} = 80$$ 5. **Find the freight train speed:** $$p - 8 = 80 - 8 = 72$$ 6. **Answer:** The speed of the passenger train is $80$ mph. The speed of the freight train is $72$ mph.