A train travels at an average of 50 miles per hour for $2\frac{1}{2}$ hours and then travels at a speed of 70 miles per hour for $1\frac{1}{2}$ hours. How far did the train travel in the entire 4 hours?

### Concept & Formula Total distance traveled in a journey with multiple legs is simply the sum of the distances traveled in each leg. $$Distance = Speed \times Time$$ $$Total Distance = D_1 + D_2$$ ### Step-by-Step Solution 1. **Calculate Distance for the First Leg ($D_1$):** * Speed = $50$ miles per hour. * Time = $2\frac{1}{2}$ hours = $2.5$ hours. * $D_1 = 50 \times 2.5 = 125$ miles. 2. **Calculate Distance for the Second Leg ($D_2$):** * Speed = $70$ miles per hour. * Time = $1\frac{1}{2}$ hours = $1.5$ hours. * $D_2 = 70 \times 1.5 = 105$ miles. 3. **Calculate Total Distance:** * Total Distance = $D_1 + D_2$ * Total Distance = $125 + 105 = 230$ miles. ### Exam Strategy & Shortcut You can mentally calculate the parts: First leg: $50 \times 2 = 100$, plus half of $50$ ($25$) = $125$. Second leg: $70 \times 1 = 70$, plus half of $70$ ($35$) = $105$. $125 + 105 = 230$. This skips writing down decimal conversions. ### Common Pitfall A common mistake is trying to average the speeds first ($50$ and $70$) and then multiplying by total time. Since the times driven at these speeds are not equal, a simple average of the speeds will yield an incorrect total distance. ### Final Answer Therefore, the correct answer is **230 miles**.