Train X departs from station A at 10 a.m. for station B, which is 360 km so far. Train Y departs from station B at 10 a.m. for station A Train X travels at an average speed of 90 km/hr and does not stop anywhere until it arrives at station B Train Y travels at an average speed of 60 km/hr, but has to stop for 20 min at station C, which is 60 km away from station B enroots to station A Ignoring the lengths of the trains, what is the distance, to the nearest kilometer, from station A to the point where the trains cross each other?



To find the distance from Station A to the point where Trains X and Y cross each other, we need to determine the time at which they meet and use that time to calculate the distance traveled by Train X.

### Step 1: Calculate the time it takes for Train Y to reach Station C and stop there

- **Distance from B to C**: 60 km
- **Speed of Train Y**: 60 km/h
- **Time to reach Station C**: 

\[
\text{Time} = \frac{60 \text{ km}}{60 \text{ km/h}} = 1 \text{ hour}
\]

- Train Y stops at Station C for 20 minutes (which is \(\frac{20}{60} = \frac{1}{3}\) hours).

### Step 2: Calculate the distance covered by Train X in the first 1 hour and 20 minutes

- Train X's speed = 90 km/h
- Time taken by Train Y to cover 60 km and stop for 20 minutes = 1 hour + 20 minutes = \(1 + \frac{1}{3}\) hours = \(\frac{4}{3}\) hours.
- Distance covered by Train X in \(\frac{4}{3}\) hours:

\[
\text{Distance} = 90 \times \frac{4}{3} = 120 \text{ km}
\]

### Step 3: Determine the effective speed and time after Train Y resumes

- After Train Y resumes its journey, the remaining distance between the two trains is \(360 - 120 - 60 = 180\) km.
- Combined speed of both trains when they are moving towards each other:

\[
\text{Combined speed} = 90 \text{ km/h} + 60 \text{ km/h} = 150 \text{ km/h}
\]

- Time to meet after Train Y resumes its journey:

\[
\text{Time} = \frac{180 \text{ km}}{150 \text{ km/h}} = 1.2 \text{ hours}
\]

### Step 4: Calculate the total distance covered by Train X

- In \(1.2\) hours, Train X will cover:

\[
\text{Distance} = 90 \times 1.2 = 108 \text{ km}
\]

- Total distance from Station A to the meeting point:

\[
\text{Total distance} = 120 \text{ km} + 108 \text{ km} = 228 \text{ km}
\]

So, the distance from Station A to the point where the trains cross each other is **228 km**.

Would you like more details on any part of this solution or have any questions?

Here are some related questions you might consider:
1. What if Train Y had to stop at Station C for 30 minutes instead of 20 minutes?
2. How would the result change if Train X had a delay and departed 10 minutes late?
3. If Train Y had a higher speed of 80 km/h, how would that affect the meeting point?
4. What if the distance between Station A and Station B was 500 km instead of 360 km?
5. How long would it take for Train Y to reach Station A after passing Train X?

**Tip:** When solving problems involving relative motion, always break down the journey into segments where different conditions apply (like stops or changes in speed) and use the concept of relative speed when the objects are moving towards each other.

Learn how to calculate the distance from Station A to the point where Trains X and Y cross each other. Detailed step-by-step solution involving relative motion, speed, distance, and time calculations. Suitable for high school students studying motion problems.

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