The distance between two towns is 458 km. Two trains start from these two stations at the same time and travel towards each other at 45 km/hr. and 60 km/hr. respectively. In how many hours will they be 38 km apart from each other?

Hint: We are given the total distance between the two towns. To find the time we need to find the total relative velocity by adding both the given velocities. Here, we will use the distance-time relationship to find the value of how much time is taken. The time will be in hours as the velocity given is in km/hour.

Complete step-by-step solution:
Consider the quantities given in the question,
Total distance between two towns is 458 km, velocity of train A is 45 km/hr. and velocity of train B is 60 km/hr.
We have to find out in how many hours the trains will be 38 km apart from each other. Since, the trains are travelling towards each other from two different stations it shows that the trains are moving in opposite directions.
Thus, to calculate the relative velocity we need to add up both the velocities given in the question.
Here, we get that,
\[
  {\text{v}} = 60 + 45 \\
   = 105 \\
\]
Thus, we get the total velocity as 105 km/hr.
Next, we have to determine the time as the trains are 38 km apart from each other
So, we will find the relative distance travelled by both the trains by subtracting 38 km from the total distance given.
That is,
\[
  {\text{d}} = 458 - 38 \\
   = 420 \\
 \]
Hence, the total distance is 420 km.
Now, we need to determine the time as to how many hours it will take to meet up.
So, Apply the distance-time relationship,
We know, \[{\text{Time = }}\dfrac{{{\text{distance}}}}{{{\text{velocity}}}}\]
We will substitute the values in the above relationship and find the time taken by both the trains to meet up.
\[
  {\text{Time}} = \dfrac{{420}}{{105}} \\
   = 4 \\
 \]
Thus, the time taken by both the trains when they will be 38 km apart from each other is 4 hr.

Note: Relative velocity is obtained by adding up both the velocities when the trains are travelling in opposite directions. Do not subtract the velocities from each other as subtraction is done when trains are travelling in the same direction. Use the time-distance relationship accurately. There is no need to change the units as the units are in standard form already for all the data given.