Understanding the Train Meeting Problem

This problem involves two trains starting at the same time from different stations and traveling towards each other. We are given their speeds and the difference in the distance they cover when they meet. Our goal is to find the total distance between the two stations.

Given Information:

• Speed of the first train (from station P): $S_P = 75$ km/h
• Speed of the second train (from station Q): $S_Q = 100$ km/h
• Difference in distance traveled when they meet: 50 km
• The trains start at the same time.

Step-by-Step Solution

Let's break down the problem to find the solution.

1. Define Time and Distance Variables:

   Let the time (in hours) taken for the two trains to meet be $t$. Since they start at the same time, they travel for the same duration until they meet.

   The distance traveled by the first train (from P) is $D_P$.

   The distance traveled by the second train (from Q) is $D_Q$.

2. Express Distances in Terms of Time:

   Using the formula distance = speed × time:

   • Distance traveled by the first train: $D_P = S_P \times t = 75t$ km
   • Distance traveled by the second train: $D_Q = S_Q \times t = 100t$ km

3. Use the Distance Difference Condition:

   We are told that when they meet, one train has traveled 50 km more than the other. Since the second train is faster (100 km/h vs 75 km/h), it will cover more distance.

   Therefore, the difference in distance is: $D_Q - D_P = 50$ km

4. Solve for Time ($t$):

   Substitute the expressions for $D_Q$ and $D_P$ into the difference equation:

   $100t - 75t = 50$

   Simplify the equation:

   $25t = 50$

   Solve for $t$:

   $t = \frac{50}{25} = 2$ hours

   So, the trains meet after 2 hours.

5. Calculate the Total Distance:

   The total distance between stations P and Q is the sum of the distances traveled by both trains until they meet.

   Total Distance = $D_P + D_Q$

   We can calculate this in two ways:

   • Method 1: Calculate individual distances and add them.
     • $D_P = 75 \times t = 75 \times 2 = 150$ km
     • $D_Q = 100 \times t = 100 \times 2 = 200$ km
     • Total Distance = $150 + 200 = 350$ km
   • Method 2: Use the combined speed. The relative speed of the trains approaching each other is $S_P + S_Q = 75 + 100 = 175$ km/h. The total distance covered is this relative speed multiplied by the time taken to meet.
     • Total Distance = $(S_P + S_Q) \times t = (75 + 100) \times 2 = 175 \times 2 = 350$ km

   Both methods confirm the total distance.

Final Answer

The distance between the two stations P and Q is 350 km.