Two trains leave the station at the same time, one heading west and the other east. The westbound train travels 12 miles per hour slower than the eastbound train. If the two trains are 750 miles apart after 5 hours, what is the rate of the west bound train?

To find the rate of the westbound train, we can use the following steps:

### Step 1: Define the variables
Let's define the speed of the eastbound train as \( x \) miles per hour. Therefore, the speed of the westbound train is \( x - 12 \) miles per hour, since it is 12 miles per hour slower.

### Step 2: Use the formula for distance
The distance each train travels can be expressed as:

\[
\text{Distance} = \text{Speed} \times \text{Time}
\]

Since both trains have been traveling for 5 hours:

- The distance traveled by the eastbound train: \( 5x \)
- The distance traveled by the westbound train: \( 5(x - 12) \)

### Step 3: Set up the equation
The total distance between the two trains after 5 hours is given as 750 miles:

\[
5x + 5(x - 12) = 750
\]

### Step 4: Simplify and solve the equation
First, distribute and combine like terms:

\[
5x + 5x - 60 = 750
\]
\[
10x - 60 = 750
\]

Next, add 60 to both sides:

\[
10x = 810
\]

Finally, divide both sides by 10 to solve for \( x \):

\[
x = 81 \text{ miles per hour}
\]

### Step 5: Find the westbound train's speed
The speed of the westbound train is \( x - 12 \):

\[
\text{Westbound train's speed} = 81 - 12 = 69 \text{ miles per hour}
\]

### Final Answer
The rate of the westbound train is **69 miles per hour**.

Would you like more details, or do you have any questions?

Here are five related questions:
1. What is the speed of the eastbound train?
2. How far did each train travel in the 5 hours?
3. What if the westbound train was 15 miles per hour slower instead of 12 miles per hour?
4. How would the scenario change if they were 900 miles apart after 5 hours?
5. What would be the meeting time if both trains traveled in the same direction?

**Tip:** When two objects are moving away from each other, their distances add up to the total separation.

Learn how to solve the problem where two trains leave a station simultaneously, traveling in opposite directions. After 5 hours, they are 750 miles apart. Find the speed of the westbound train, given that it travels 12 miles per hour slower than the eastbound train. Includes step-by-step solution and relevant algebraic formulas.

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