**(1) Two trains of equal length are running on parallel lines in the same direction at 56 km/hr and 40 km/hr. The faster train passes the slower train in 45 seconds. The length of each train is?**

**Convert the speed of the trains from km/hr to m/s.**

**Answer :** 25.59

**Explanation**

56 km/hr = 56000 m / 3600 s = 15 m/s

40 km/hr = 40000 m / 3600 s = 11.11 m/s

Find the relative speed of the trains.

Relative speed = Speed of faster train – Speed of slower train

Relative speed = 15 m/s – 11.11 m/s = 3.89 m/s

Find the total distance that the trains need to travel.

The total distance is equal to the length of the faster train plus the length of the slower train.

Let the length of each train be x meters.

Total distance = 2x meters

Divide the total distance by the relative speed of the trains to find the time it takes the trains to pass each other.

Time = Distance / Speed

Time = 2x / 3.89 m/s

Time = 51.19 seconds

Therefore, it takes the trains 51.19 seconds to pass each other.

Since the trains are of equal length, each train is 51.19 / 2 = 25.59 meters long.

**(2) Two trains of equal length are running on parallel lines in the opposite direction at 50 km/hr and 60 km/hr. The trains pass each other in 20 seconds. The length of each train is?**

**Answer** : 305.6 meters.

**Explanation**

Let’s denote the length of each train as L (in km). Since the trains are moving in opposite directions,

their relative speed is the sum of their individual speeds, which is 50 + 60 = 110 km/hr.

To convert the relative speed from km/hr to meters per second (m/s), we can use the following conversion factor:

1 km/hr = 1000/3600 m/s

Therefore, the relative speed of the two trains in m/s is:

110 km/hr × (1000/3600) m/s = 110 × (10/36) m/s = 30.56 m/s (approximately)

Now, we know that the trains pass each other in 20 seconds. We can use the formula:

Distance = Speed × Time

Since the trains are of equal length, the total distance they cover while passing each other is the sum of their lengths, which is 2L.

So, 2L = 30.56 m/s × 20 s

Solving for L:

2L = 611.2 m L = 305.6 m

Thus, the length of each train is approximately 305.6 meters.

**(3) Two trains of lengths 300 meters and 400 meters are running on parallel lines in the same direction at 60 km/hr and 80 km/hr. In how many seconds will the faster train pass the slower train?**

**Answer : 126 seconds**

**Explanation**

To find the time taken by the faster train to pass the slower train, we need to calculate the relative speed between the two trains. Since the two trains are moving in the same direction, the relative speed between them will be the difference between their speeds.

The speed of the slower train = 60 km/hr = 60 x (5/18) m/s = 50/3 m/s

The speed of the faster train = 80 km/hr = 80 x (5/18) m/s = 200/9 m/s

Relative speed = (200/9) – (50/3) = 50/9 m/s

Now, we need to calculate the distance that the faster train needs to cover to pass the slower train. The distance will be equal to the sum of the lengths of the two trains.

Total distance = 300 + 400 = 700 meters

We can use the formula:

time = distance / speed

To find the time taken by the faster train to cover the distance of 700 meters at a relative speed of 50/9 m/s.

time = 700 / (50/9) = 126 seconds

Therefore, the faster train will pass the slower train in 126 seconds.

**(4) A train of length 400 meters is running at a speed of 72 km/hr. How long does it take to cross a 200-meter-long bridge?**

**Answer : 30 seconds**

**Explanation**

To cross the bridge, the train will need to cover a total distance equal to the length of the train plus the length of the bridge. Therefore, the total distance to be covered by the train is:

Total distance = length of train + length of bridge = 400 + 200 = 600 meters

We know that the speed of the train is 72 km/hr. To calculate the time taken by the train to cross the bridge, we need to convert the speed from km/hr to m/s.

Speed of train = 72 km/hr = 72 x (5/18) m/s = 20 m/s

Now, we can use the formula:

time = distance / speed

To find the time taken by the train to cross the bridge.

time = 600 / 20 = 30 seconds

Therefore, it will take 30 seconds for the train to cross the 200-meter-long bridge.

**(5) A train of length 500 meters is running at a speed of 60 km/hr. How long does it take to cross a 300-meter-long tunnel?**

**Answer : 48 seconds**

**Explanation**

To cross the tunnel, the train will need to cover a distance equal to the length of the tunnel plus the length of the train. Therefore, the total distance to be covered by the train is:

Total distance = length of train + length of tunnel = 500 + 300 = 800 meters

We know that the speed of the train is 60 km/hr. To calculate the time taken by the train to cross the tunnel, we need to convert the speed from km/hr to m/s.

Speed of train = 60 km/hr = 60 x (5/18) m/s = 50/3 m/s

Now, we can use the formula:

time = distance / speed

To find the time taken by the train to cross the tunnel.

time = 800 / (50/3) = 48 seconds

Therefore, it will take 48 seconds for the train to cross the 300-meter-long tunnel.