These questions are based on the information given below.
Train P has a length of 300 m and a speed of 72 kmph. Train has a length of 600 m and a speed of 90 kmph. Both enter the tunnel from opposite ends simultaneously. The length of the tunnel is 600 m.
Find the distance between the point where the rear ends of the trains cross each other and the point of entry of the slower train (in m).

Options

A. `398(1)/(3)`
B. `366(2)/(3)`
C. `216(2)/(3)`
D. `233(1)/(3)`

Answer

To solve the problem, we need to determine the distance between the point where the rear ends of the trains cross each other and the point of entry of the slower train (Train P).

### Step-by-Step Solution:

1. **Convert Speeds from km/h to m/s:**
- Speed of Train P = 72 km/h = \( 72 \times \frac{5}{18} = 20 \) m/s
- Speed of Train Q = 90 km/h = \( 90 \times \frac{5}{18} = 25 \) m/s

**Hint:** Remember that to convert km/h to m/s, multiply by \( \frac{5}{18} \).

2. **Calculate the Total Distance to be Covered:**
- The total distance when two trains cross each other is the sum of their lengths and the length of the tunnel.
- Total Distance = Length of Train P + Length of Train Q + Length of Tunnel
- Total Distance = 300 m + 600 m + 600 m = 1500 m

**Hint:** When two objects move towards each other, the total distance is the sum of their lengths and any additional distance they need to cover.

3. **Calculate the Relative Speed:**
- Since the trains are moving towards each other, their speeds add up.
- Relative Speed = Speed of Train P + Speed of Train Q = 20 m/s + 25 m/s = 45 m/s

**Hint:** When two objects move towards each other, their speeds combine to give you the relative speed.

4. **Calculate the Time Taken to Cross Each Other:**
- Time = Total Distance / Relative Speed
- Time = \( \frac{1500 \text{ m}}{45 \text{ m/s}} = \frac{100}{3} \) seconds

**Hint:** Use the formula \( \text{Time} = \frac{\text{Distance}}{\text{Speed}} \) to find the time taken.

5. **Calculate the Distance Covered by Train P in that Time:**
- Distance covered by Train P = Speed of Train P × Time
- Distance covered by Train P = \( 20 \text{ m/s} \times \frac{100}{3} \text{ s} = \frac{2000}{3} \text{ m} \)

**Hint:** To find how far an object has traveled, multiply its speed by the time it has been traveling.

6. **Determine the Position of the Rear End of Train P:**
- The rear end of Train P will be at the distance covered by Train P from its entry point.
- Since Train P has a length of 300 m, the position of the rear end of Train P when it crosses Train Q will be:
- Position of rear end = Distance covered by Train P - Length of Train P
- Position of rear end = \( \frac{2000}{3} \text{ m} - 300 \text{ m} \)
- Convert 300 m to a fraction: \( 300 = \frac{900}{3} \)
- Position of rear end = \( \frac{2000}{3} - \frac{900}{3} = \frac{1100}{3} \text{ m} \)

**Hint:** To find the position of the rear end, subtract the length of the train from the distance it has traveled.

7. **Calculate the Distance from the Entry Point of Train P:**
- The distance from the entry point of Train P to the point where the rear ends cross each other is:
- Distance = Position of rear end of Train P
- Distance = \( \frac{1100}{3} \text{ m} \)

**Hint:** The distance from the entry point to the crossing point is simply the position of the rear end of the train.

### Final Answer:
The distance between the point where the rear ends of the trains cross each other and the point of entry of the slower train (Train P) is \( \frac{1100}{3} \) m, which is approximately 366.67 m.