The problem states that the distance $y$ is proportional to time $x$, and the relationship is given by:

$y = kx$

where $k$ is the constant of proportionality.

We are told that in 4 hours, the train travels 380 miles. Therefore, we can substitute $y = 380$ and $x = 4$ into the equation:

$380 = k \times 4$

Now, solve for $k$:

$k = \frac{380}{4} = 95$

Thus, the value of $k$ is 95.

Would you like more details or clarification?

Here are five related questions for further practice:

1. If the train continues for 6 hours, how far will it travel?
2. What would the distance be after 10 hours, using the value of $k$?
3. If the train's speed doubles, how would the constant $k$ change?
4. What is the time it takes for the train to travel 570 miles?
5. How would the equation change if the relationship were not proportional?

Tip: Always ensure that the units match when working with proportional relationships, like miles and hours in this case, to avoid mistakes in the final answer.