A freight train and a passenger train start toward each other at the same time from two towns that are 500 miles apart. After 3 hours, the trains are still 80 miles apart. If the average rate of speed of the passenger train in 20 miles per hour faster than the average rate of speed of the freight train, what is the average rate of speed, in miles per hour, of the freight train?

Options

A. `40`
B. `45`
C. `50`
D. `60`

Answer

To solve the problem step by step, we will define the variables, set up the equations based on the information provided, and then solve for the speed of the freight train.

### Step-by-Step Solution:

1. **Define Variables:**
Let the speed of the freight train be \( x \) miles per hour. Then, the speed of the passenger train will be \( x + 20 \) miles per hour since it is 20 miles per hour faster than the freight train.

2. **Calculate Distance Covered in 3 Hours:**
- Distance covered by the freight train in 3 hours:
\[ \text{Distance}_{\text{freight}} = \text{Speed} \times \text{Time} = x \times 3 = 3x \text{ miles} \]
- Distance covered by the passenger train in 3 hours:
\[ \text{Distance}_{\text{passenger}} = (x + 20) \times 3 = 3x + 60 \text{ miles} \]

3. **Set Up the Equation:**
After 3 hours, the distance between the two trains is 80 miles. The total distance covered by both trains plus the distance remaining between them equals the initial distance of 500 miles:
\[ \text{Distance}_{\text{freight}} + \text{Distance}_{\text{passenger}} + \text{Distance}_{\text{remaining}} = 500 \]
Substituting the distances we calculated:
\[ 3x + (3x + 60) + 80 = 500 \]

4. **Simplify the Equation:**
Combine like terms:
\[ 3x + 3x + 60 + 80 = 500 \]
\[ 6x + 140 = 500 \]

5. **Solve for \( x \):**
Subtract 140 from both sides:
\[ 6x = 500 - 140 \]
\[ 6x = 360 \]
Divide by 6:
\[ x = \frac{360}{6} = 60 \]

6. **Conclusion:**
The average rate of speed of the freight train is \( 60 \) miles per hour.

### Final Answer:
The average rate of speed of the freight train is **60 miles per hour**.

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