Problems On Trains

Answer: Option D
Solution:
To find the length of the train, we use the formula for distance:

\[
\text{Distance} = \text{Speed} \times \text{Time}
\]

Given:
– Speed of the train = 60 km/h
– Time to cross the pole = 9 seconds

Step 1: Convert speed from km/h to m/s.
To convert km/h to m/s, we multiply by \(\frac{5}{18}\):

\[
\text{Speed in m/s} = 60 \times \frac{5}{18} = \frac{300}{18} = 16.67 \, \text{m/s}
\]

Step 2: Use the formula to find the distance (length of the train).
Since the train crosses a pole, the distance covered by the train is equal to its length. Now we can calculate the length using the formula:

\[
\text{Length of the train} = \text{Speed} \times \text{Time}
\] \[
\text{Length of the train} = 16.67 \times 9 = 150.03 \, \text{m}
\]

Conclusion:
The length of the train is approximately 150 meters.

Answer: Option B
Solution:
To solve this, we will first find the relative speed of the train and the man, and then use it to calculate the speed of the train.

Given:
– Length of the train = 125 meters
– Speed of the man = 5 km/h
– Time taken by the train to pass the man = 10 seconds

Step 1: Convert the speed of the man from km/h to m/s.
To convert km/h to m/s, multiply by \(\frac{5}{18}\):

\[
\text{Speed of the man in m/s} = 5 \times \frac{5}{18} = \frac{25}{18} \approx 1.39 \, \text{m/s}
\]

Step 2: Calculate the relative speed between the train and the man.
Since the train and the man are moving in the same direction, the relative speed is the difference between the speed of the train and the speed of the man.

Let the speed of the train be \(V_{\text{train}}\) in m/s. The relative speed is:

\[
\text{Relative speed} = V_{\text{train}} – 1.39 \, \text{m/s}
\]

Step 3: Use the formula for speed to find the speed of the train.
The train covers a distance of 125 meters to pass the man in 10 seconds. So, the relative speed can be calculated as:

\[
\text{Relative speed} = \frac{\text{Distance}}{\text{Time}} = \frac{125}{10} = 12.5 \, \text{m/s}
\]

Step 4: Set up the equation and solve for \(V_{\text{train}}\).
\[
V_{\text{train}} – 1.39 = 12.5
\] \[
V_{\text{train}} = 12.5 + 1.39 = 13.89 \, \text{m/s}
\]

Step 5: Convert the speed of the train from m/s to km/h.
To convert m/s to km/h, multiply by \(\frac{18}{5}\):

\[
V_{\text{train}} = 13.89 \times \frac{18}{5} = 50 \, \text{km/h}
\]

Conclusion:
The speed of the train is 50 km/h.

Answer: Option C
Solution:
To find the length of the bridge, we need to calculate the total distance the train covers in 30 seconds. The train travels this distance by crossing both the bridge and its own length.

Given:
– Length of the train = 130 meters
– Speed of the train = 45 km/h
– Time taken to cross the bridge = 30 seconds

Step 1: Convert the speed of the train from km/h to m/s.
To convert from km/h to m/s, multiply by \(\frac{5}{18}\):

\[
\text{Speed of the train in m/s} = 45 \times \frac{5}{18} = \frac{225}{18} = 12.5 \, \text{m/s}
\]

Step 2: Calculate the distance the train travels in 30 seconds.
The total distance covered by the train in 30 seconds is the speed of the train multiplied by the time:

\[
\text{Distance covered in 30 seconds} = \text{Speed} \times \text{Time} = 12.5 \times 30 = 375 \, \text{meters}
\]

Step 3: Calculate the length of the bridge.
The total distance covered is the sum of the length of the train and the length of the bridge. So, the length of the bridge is:

\[
\text{Length of the bridge} = \text{Total distance covered} – \text{Length of the train}
\] \[
\text{Length of the bridge} = 375 – 130 = 245 \, \text{meters}
\]

Conclusion:
The length of the bridge is 245 meters.

Answer: Option B
Solution:
Let’s define the following:

– Let the length of the train be \( L \) meters.
– Let the length of the platform be \( P \) meters.
– The speed of the train is \( 54 \, \text{km/h} \), which we need to convert to meters per second.

Step 1: Convert the speed from km/h to m/s

The formula for converting speed from km/h to m/s is:

\[
\text{Speed in m/s} = \text{Speed in km/h} \times \frac{1000}{3600}
\]

So the speed of the train in meters per second is:

\[
54 \times \frac{1000}{3600} = 15 \, \text{m/s}
\]

Step 2: Use the time taken to pass the man standing on the platform

When the train passes the man, it covers a distance equal to the length of the train \( L \). The time taken to pass the man is 20 seconds, so:

\[
L = \text{Speed} \times \text{Time}
\] \[
L = 15 \, \text{m/s} \times 20 \, \text{seconds} = 300 \, \text{meters}
\]

So the length of the train is \( L = 300 \, \text{meters} \).

Step 3: Use the time taken to pass the platform

When the train passes the platform, it covers a distance equal to the sum of the length of the train \( L \) and the length of the platform \( P \). The time taken to pass the platform is 36 seconds, so:

\[
L + P = \text{Speed} \times \text{Time}
\] \[
300 + P = 15 \times 36
\] \[
300 + P = 540
\] \[
P = 540 – 300 = 240 \, \text{meters}
\]

Final Answer:
The length of the platform is 240 meters.

Answer: Option B
Solution:
Let’s break down the problem step by step.

Given:
– The length of the train = 240 meters
– Time taken to pass a pole = 24 seconds
– The length of the platform = 650 meters

Step 1: Calculate the speed of the train

When the train passes a pole, it covers a distance equal to its own length, which is 240 meters. The time taken to pass the pole is 24 seconds.

The speed of the train can be calculated as:

\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}} = \frac{240}{24} = 10 \, \text{m/s}
\]

Step 2: Time taken to pass the platform

When the train passes the platform, it covers a distance equal to the sum of its own length (240 meters) and the length of the platform (650 meters), which is:

\[
\text{Total distance} = 240 + 650 = 890 \, \text{meters}
\]

Now, we can calculate the time taken to pass the platform using the formula:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{890}{10} = 89 \, \text{seconds}
\]

Final Answer:
The train will take 89 seconds to pass the platform.

Answer: Option A
Solution:
Let’s define the following:

– Let the length of each train be \( L \) meters.
– The speed of the faster train is 46 km/hr.
– The speed of the slower train is 36 km/hr.
– The time taken by the faster train to pass the slower train is 36 seconds.

Step 1: Convert the speeds to meters per second

We need to convert the speeds from km/h to m/s. The formula for converting km/h to m/s is:

\[
\text{Speed in m/s} = \text{Speed in km/h} \times \frac{1000}{3600}
\]

So the speed of the faster train in meters per second is:

\[
46 \times \frac{1000}{3600} = \frac{46000}{3600} \approx 12.78 \, \text{m/s}
\]

And the speed of the slower train in meters per second is:

\[
36 \times \frac{1000}{3600} = \frac{36000}{3600} = 10 \, \text{m/s}
\]

Step 2: Calculate the relative speed

When two trains are moving in the same direction, the relative speed is the difference in their speeds. Therefore, the relative speed is:

\[
\text{Relative speed} = 12.78 \, \text{m/s} – 10 \, \text{m/s} = 2.78 \, \text{m/s}
\]

Step 3: Calculate the total distance covered

When the faster train passes the slower train, it covers a distance equal to the combined length of the two trains, which is \( 2L \).

The formula for distance is:

\[
\text{Distance} = \text{Relative speed} \times \text{Time}
\]

Substituting the known values:

\[
2L = 2.78 \times 36
\] \[
2L = 100.08
\] \[
L = \frac{100.08}{2} = 50.04 \, \text{meters}
\]

Final Answer:
The length of each train is approximately 50 meters.

Answer: Option A
Solution:
Let’s break down the problem step by step.

Given:
– Length of the train = 360 meters
– Speed of the train = 45 km/hr
– Length of the bridge = 140 meters

Step 1: Convert the speed from km/hr to m/s

To convert the speed from km/h to m/s, we use the conversion factor \( \frac{1000}{3600} \). So, the speed in m/s is:

\[
\text{Speed in m/s} = 45 \times \frac{1000}{3600} = 12.5 \, \text{m/s}
\]

Step 2: Total distance to be covered

When the train passes the bridge, it covers a distance equal to the sum of the length of the train and the length of the bridge:

\[
\text{Total distance} = 360 \, \text{m} + 140 \, \text{m} = 500 \, \text{m}
\]

Step 3: Calculate the time taken to pass the bridge

The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

Substituting the known values:

\[
\text{Time} = \frac{500 \, \text{m}}{12.5 \, \text{m/s}} = 40 \, \text{seconds}
\]

Final Answer:
The train will take 40 seconds to pass the bridge.

Answer: Option C
Solution:
Let’s solve the problem step by step.

Given:
– Speed of the first train = 60 km/hr
– Speed of the second train = 90 km/hr
– Length of the first train = 1.10 km
– Length of the second train = 0.9 km

We need to find the time taken by the slower train (speed = 60 km/hr) to cross the faster train (speed = 90 km/hr).

Step 1: Convert the speeds to meters per second

To convert km/hr to m/s, we multiply by \( \frac{1000}{3600} \).

– Speed of the first train in m/s:
\[
60 \times \frac{1000}{3600} = 16.67 \, \text{m/s}
\]

– Speed of the second train in m/s:
\[
90 \times \frac{1000}{3600} = 25 \, \text{m/s}
\]

Step 2: Find the relative speed

When two trains are moving in opposite directions, the relative speed is the sum of their speeds. Thus:

\[
\text{Relative speed} = 16.67 \, \text{m/s} + 25 \, \text{m/s} = 41.67 \, \text{m/s}
\]

Step 3: Find the total distance to be covered

When the slower train crosses the faster train, it covers a distance equal to the sum of the lengths of both trains:

\[
\text{Total distance} = 1.10 \, \text{km} + 0.9 \, \text{km} = 2.00 \, \text{km} = 2000 \, \text{m}
\]

Step 4: Calculate the time taken to cross

The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

Substituting the values:

\[
\text{Time} = \frac{2000 \, \text{m}}{41.67 \, \text{m/s}} \approx 48 \, \text{seconds}
\]

Final Answer:
The time taken by the slower train to cross the faster train is approximately 48 seconds.

Answer: Option C
Solution:
Let’s break down the problem step by step:

Given:
– Speed of the jogger = 9 km/h
– Speed of the train = 45 km/h
– Length of the train = 120 meters
– Initial distance between the jogger and the engine of the train = 240 meters

Step 1: Convert the speeds from km/h to m/s

To convert km/h to m/s, we multiply by \( \frac{1000}{3600} \).

– Speed of the jogger in m/s:
\[
9 \times \frac{1000}{3600} = 2.5 \, \text{m/s}
\]

– Speed of the train in m/s:
\[
45 \times \frac{1000}{3600} = 12.5 \, \text{m/s}
\]

Step 2: Find the relative speed

Since the jogger and the train are running in the same direction, the relative speed is the difference between their speeds:

\[
\text{Relative speed} = 12.5 \, \text{m/s} – 2.5 \, \text{m/s} = 10 \, \text{m/s}
\]

Step 3: Find the total distance to be covered

The train has to cover the combined distance of the jogger’s initial distance (240 meters) and the length of the train (120 meters), so the total distance to be covered is:

\[
\text{Total distance} = 240 \, \text{m} + 120 \, \text{m} = 360 \, \text{m}
\]

Step 4: Calculate the time taken to pass the jogger

The time taken to pass the jogger is given by:

\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}}
\]

Substituting the known values:

\[
\text{Time} = \frac{360 \, \text{m}}{10 \, \text{m/s}} = 36 \, \text{seconds}
\]

Final Answer:
The train will take 36 seconds to pass the jogger.

Answer: Option A
Solution:
Let’s break down the problem step by step:

Given:
– Speed of the first train = 120 km/h
– Speed of the second train = 80 km/h
– Length of the first train = 270 meters
– Time taken for the two trains to cross each other = 9 seconds

Step 1: Convert the speeds to meters per second

To convert km/h to m/s, we multiply by \( \frac{1000}{3600} \).

– Speed of the first train in m/s:
\[
120 \times \frac{1000}{3600} = 33.33 \, \text{m/s}
\]

– Speed of the second train in m/s:
\[
80 \times \frac{1000}{3600} = 22.22 \, \text{m/s}
\]

Step 2: Find the relative speed

Since the two trains are moving in opposite directions, the relative speed is the sum of their speeds:

\[
\text{Relative speed} = 33.33 \, \text{m/s} + 22.22 \, \text{m/s} = 55.55 \, \text{m/s}
\]

Step 3: Find the total distance covered

When the two trains cross each other, the total distance covered is the sum of their lengths. Let the length of the second train be \( L \). The total distance covered is:

\[
\text{Total distance} = 270 \, \text{m} + L
\]

Step 4: Use the time taken to cross each other

The formula for distance is:

\[
\text{Distance} = \text{Relative speed} \times \text{Time}
\]

Substitute the known values:

\[
270 + L = 55.55 \times 9
\] \[
270 + L = 500
\] \[
L = 500 – 270 = 230 \, \text{meters}
\]

Final Answer:
The length of the second train is 230 meters.

Answer: Option D
Solution:
Let’s break down the problem step by step:

Given:
– Speed of the goods train = 72 km/h
– Length of the platform = 250 meters
– Time taken to cross the platform = 26 seconds

Step 1: Convert the speed from km/h to m/s

To convert the speed from km/h to m/s, we use the conversion factor \( \frac{1000}{3600} \). So, the speed of the train in m/s is:

\[
\text{Speed in m/s} = 72 \times \frac{1000}{3600} = 20 \, \text{m/s}
\]

Step 2: Calculate the total distance covered

When the goods train crosses the platform, it covers a distance equal to the sum of the length of the train and the length of the platform. Let the length of the train be \( L \) meters. The total distance covered is:

\[
\text{Total distance} = L + 250 \, \text{meters}
\]

Step 3: Use the formula for time

The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

Substitute the known values into this formula:

\[
26 = \frac{L + 250}{20}
\]

Step 4: Solve for the length of the train

Multiply both sides of the equation by 20:

\[
26 \times 20 = L + 250
\] \[
520 = L + 250
\] \[
L = 520 – 250 = 270 \, \text{meters}
\]

Final Answer:
The length of the goods train is 270 meters.

Answer: Option C
Solution:
Let’s break down the problem step by step:

Given:
– Length of each train = 100 meters
– Time taken for the two trains to cross each other = 8 seconds
– One train is moving twice as fast as the other

### Step 1: Define variables for the speeds

Let:
– \( v \) be the speed of the slower train (in meters per second).
– \( 2v \) be the speed of the faster train (since it is moving twice as fast as the slower one).

Step 2: Calculate the total distance covered

When the two trains cross each other, the total distance covered is the sum of their lengths, which is:

\[
\text{Total distance} = 100 \, \text{m} + 100 \, \text{m} = 200 \, \text{m}
\]

Step 3: Use the formula for relative speed

When the two trains are moving in opposite directions, their relative speed is the sum of their individual speeds:

\[
\text{Relative speed} = v + 2v = 3v
\]

Step 4: Use the formula for time

The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

Substitute the known values into the formula:

\[
8 = \frac{200}{3v}
\]

Step 5: Solve for \( v \)

Multiply both sides of the equation by \( 3v \):

\[
8 \times 3v = 200
\] \[
24v = 200
\] \[
v = \frac{200}{24} = \frac{25}{3} \approx 8.33 \, \text{m/s}
\]

Step 6: Calculate the speed of the faster train

The speed of the faster train is \( 2v \):

\[
\text{Speed of the faster train} = 2 \times \frac{25}{3} = \frac{50}{3} \approx 16.67 \, \text{m/s}
\]

Final Answer:
The speed of the faster train is approximately 16.67 m/s.

To convert this to km/h:

\[
16.67 \, \text{m/s} \times \frac{3600}{1000} = 60 \, \text{km/h}
\]

Thus, the speed of the faster train is 60 km/h.

Answer: Option D
Solution:
Let’s solve the problem step by step.

Given:
– Length of the first train = 140 meters
– Length of the second train = 160 meters
– Speed of the first train = 60 km/h
– Speed of the second train = 40 km/h
– The trains are moving in opposite directions.

Step 1: Convert speeds from km/h to m/s

To convert the speeds from km/h to m/s, we use the conversion factor \( \frac{1000}{3600} \):

– Speed of the first train in m/s:
\[
60 \times \frac{1000}{3600} = 16.67 \, \text{m/s}
\]

– Speed of the second train in m/s:
\[
40 \times \frac{1000}{3600} = 11.11 \, \text{m/s}
\]

Step 2: Calculate the relative speed

Since the trains are moving in opposite directions, the relative speed is the sum of their individual speeds:

\[
\text{Relative speed} = 16.67 \, \text{m/s} + 11.11 \, \text{m/s} = 27.78 \, \text{m/s}
\]

Step 3: Calculate the total distance to be covered

When the two trains cross each other, the total distance covered is the sum of the lengths of both trains:

\[
\text{Total distance} = 140 \, \text{m} + 160 \, \text{m} = 300 \, \text{m}
\]

Step 4: Use the formula for time

The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

Substitute the known values into the formula:

\[
\text{Time} = \frac{300 \, \text{m}}{27.78 \, \text{m/s}} \approx 10.8 \, \text{seconds}
\]

Final Answer:
The time taken by the two trains to cross each other is approximately 10.8 seconds.

Answer: Option B
Solution:
Let’s break down the problem step by step.

Given:
– Length of the train = 110 meters
– Speed of the train = 60 km/h
– Speed of the man = 6 km/h
– The man is running in the opposite direction to the train.

Step 1: Convert the speeds to meters per second

To convert km/h to m/s, we use the conversion factor \( \frac{1000}{3600} \).

– Speed of the train in m/s:
\[
60 \times \frac{1000}{3600} = 16.67 \, \text{m/s}
\]

– Speed of the man in m/s:
\[
6 \times \frac{1000}{3600} = 1.67 \, \text{m/s}
\]

Step 2: Calculate the relative speed

Since the man and the train are running in opposite directions, the relative speed is the sum of their speeds:

\[
\text{Relative speed} = 16.67 \, \text{m/s} + 1.67 \, \text{m/s} = 18.34 \, \text{m/s}
\]

Step 3: Use the formula for time

The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

In this case, the distance is the length of the train (since the train must pass the man), which is 110 meters. So,

\[
\text{Time} = \frac{110 \, \text{m}}{18.34 \, \text{m/s}} \approx 6.0 \, \text{seconds}
\]

Final Answer:
The train will take approximately 6 seconds to pass the man.

Answer: Option B
Solution:
Let’s break down the problem and calculate the time it takes for the train to pass through the tunnel.

Given:
– Speed of the train = 75 mph
– Length of the tunnel = \( \frac{3}{4} \) miles
– Length of the train = \( \frac{1}{2} \) miles

Step 1: Calculate the total distance to be covered

When the train enters the tunnel, it must cover the entire length of the tunnel plus the length of the train itself before the rear of the train emerges from the tunnel. Therefore, the total distance to be covered is:

\[
\text{Total distance} = \text{Length of the train} + \text{Length of the tunnel}
\] \[
\text{Total distance} = \frac{1}{2} \, \text{mile} + \frac{3}{4} \, \text{mile} = \frac{5}{4} \, \text{miles}
\]

Step 2: Convert the speed to miles per minute

Since the speed of the train is given in miles per hour, we need to convert this to miles per minute by dividing by 60:

\[
\text{Speed in miles per minute} = \frac{75}{60} = 1.25 \, \text{miles per minute}
\]

Step 3: Calculate the time taken to cover the total distance

Now, we can use the formula for time:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

Substitute the values into the formula:

\[
\text{Time} = \frac{\frac{5}{4}}{1.25}
\]

Simplifying:

\[
\text{Time} = \frac{5}{4} \times \frac{1}{1.25} = \frac{5}{4} \times \frac{4}{5} = 1 \, \text{minute}
\]

Final Answer:
The train will take 1 minute to pass through the tunnel from the moment the front enters to the moment the rear emerges.

Answer: Option C
Solution:
Let’s solve the problem step by step:

Given:
– Length of the train = 800 meters
– Speed of the train = 78 km/h
– Time taken to cross the tunnel = 1 minute

Step 1: Convert the speed from km/h to m/s

To convert km/h to m/s, we use the conversion factor \( \frac{1000}{3600} \):

\[
\text{Speed in m/s} = 78 \times \frac{1000}{3600} = 21.67 \, \text{m/s}
\]

Step 2: Calculate the total distance covered

When the train crosses the tunnel, it covers a distance equal to the sum of the length of the train and the length of the tunnel. Let the length of the tunnel be \( L \) meters. The total distance covered is:

\[
\text{Total distance} = 800 + L \, \text{meters}
\]

Step 3: Use the formula for time

The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

We are given that the time taken to cross the tunnel is 1 minute, which is 60 seconds. Substituting the known values into the formula:

\[
60 = \frac{800 + L}{21.67}
\]

Step 4: Solve for the length of the tunnel

Multiply both sides of the equation by 21.67:

\[
60 \times 21.67 = 800 + L
\] \[
1300.2 = 800 + L
\] \[
L = 1300.2 – 800 = 500.2 \, \text{meters}
\]

Final Answer:
The length of the tunnel is approximately 500 meters.

Answer: Option B
Solution:
Let’s solve the problem step by step.

Given:
– Length of the train = 300 meters
– Time taken to cross the signal pole = 18 seconds
– Time taken to cross the platform = 39 seconds

Step 1: Calculate the speed of the train

When the train crosses the signal pole, the distance covered is equal to the length of the train. The formula for speed is:

\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Substitute the known values:

\[
\text{Speed} = \frac{300 \, \text{m}}{18 \, \text{s}} = 16.67 \, \text{m/s}
\]

Step 2: Calculate the distance covered when crossing the platform

When the train crosses the platform, the distance covered is the sum of the length of the train and the length of the platform. The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

We are given that the time taken to cross the platform is 39 seconds, so:

\[
39 = \frac{300 + \text{Length of the platform}}{16.67}
\]

Step 3: Solve for the length of the platform

Multiply both sides of the equation by 16.67:

\[
39 \times 16.67 = 300 + \text{Length of the platform}
\] \[
650.13 = 300 + \text{Length of the platform}
\] \[
\text{Length of the platform} = 650.13 – 300 = 350.13 \, \text{meters}
\]

Final Answer:
The length of the platform is approximately 350 meters.

Answer: Option B
Solution:
Let’s solve the problem step by step:

Given:
– Time taken to pass a pole = 15 seconds
– Time taken to pass a platform 100 meters long = 25 seconds
– Length of the platform = 100 meters

Step 1: Let the length of the train be \( L \) meters.
– When the train passes a pole, it covers a distance equal to its own length, which is \( L \).
– When the train passes a platform, the total distance covered is the sum of the length of the train and the length of the platform, i.e., \( L + 100 \) meters.

Step 2: Calculate the speed of the train
The speed of the train is the same in both cases. We can calculate the speed using the time taken to pass the pole. The formula for speed is:

\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

The speed when passing the pole is:

\[
\text{Speed} = \frac{L}{15} \, \text{m/s}
\]

Step 3: Calculate the speed using the time to pass the platform
The total distance when the train passes the platform is \( L + 100 \). The time taken is 25 seconds, so the speed when passing the platform is:

\[
\text{Speed} = \frac{L + 100}{25} \, \text{m/s}
\]

Step 4: Set the speeds equal to each other
Since the train’s speed is the same in both cases, we can set the two speed equations equal to each other:

\[
\frac{L}{15} = \frac{L + 100}{25}
\]

Step 5: Solve for \( L \)
To solve for \( L \), cross-multiply:

\[
25L = 15(L + 100)
\]

Expand the equation:

\[
25L = 15L + 1500
\]

Now, solve for \( L \):

\[
25L – 15L = 1500
\] \[
10L = 1500
\] \[
L = \frac{1500}{10} = 150 \, \text{meters}
\]

Final Answer:
The length of the train is 150 meters.

Answer: Option D
Solution:
Let’s break down the problem step by step.

Given:
– Time taken to pass the telegraph post = 8 seconds
– Time taken to pass the bridge = 20 seconds
– Length of the bridge = 264 meters

Step 1: Let the length of the train be \( L \) meters.
– When the train passes the telegraph post, the distance covered is equal to the length of the train, which is \( L \) meters.
– When the train passes the bridge, the distance covered is the sum of the length of the train and the length of the bridge, i.e., \( L + 264 \) meters.

Step 2: Speed of the train
We can use the formula for speed:

\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Speed when passing the telegraph post:
\[
\text{Speed} = \frac{L}{8} \, \text{m/s}
\]

Speed when passing the bridge:
\[
\text{Speed} = \frac{L + 264}{20} \, \text{m/s}
\]

Step 3: Set the two speeds equal
Since the train moves at the same speed in both cases, we can set the two speed equations equal to each other:

\[
\frac{L}{8} = \frac{L + 264}{20}
\]

Step 4: Solve for \( L \)
Now, cross-multiply to solve for \( L \):

\[
20L = 8(L + 264)
\]

Expand both sides:

\[
20L = 8L + 2112
\]

Now, solve for \( L \):

\[
20L – 8L = 2112
\] \[
12L = 2112
\] \[
L = \frac{2112}{12} = 176 \, \text{meters}
\]

Step 5: Find the speed of the train
Now that we know the length of the train is 176 meters, we can find the speed using the formula:

\[
\text{Speed} = \frac{L}{8} = \frac{176}{8} = 22 \, \text{m/s}
\]

Step 6: Convert the speed to km/h
To convert the speed from meters per second to kilometers per hour, multiply by \( \frac{3600}{1000} \):

\[
\text{Speed in km/h} = 22 \times \frac{3600}{1000} = 79.2 \, \text{km/h}
\]

Final Answer:
The speed of the train is 79.2 km/h.

Answer: Option B
Solution:
Let’s break down the problem step by step.

Given:
– Length of the train = 500 meters
– Speed of the train = 63 km/h
– Speed of the man = 3 km/h
– The man is walking in the same direction as the train.

Step 1: Convert the speeds to meters per second
To convert km/h to m/s, use the conversion factor \( \frac{1000}{3600} \).

– Speed of the train in m/s:
\[
\text{Speed of the train} = 63 \times \frac{1000}{3600} = 17.5 \, \text{m/s}
\]

– Speed of the man in m/s:
\[
\text{Speed of the man} = 3 \times \frac{1000}{3600} = 0.8333 \, \text{m/s}
\]

Step 2: Calculate the relative speed
Since the man is walking in the same direction as the train, the relative speed between the train and the man is the difference of their speeds:

\[
\text{Relative speed} = 17.5 \, \text{m/s} – 0.8333 \, \text{m/s} = 16.6667 \, \text{m/s}
\]

Step 3: Calculate the time taken for the train to pass the man
The time taken for the train to pass the man is the time required to cover the length of the train with the relative speed. The formula for time is:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

Substitute the values:

\[
\text{Time} = \frac{500 \, \text{m}}{16.6667 \, \text{m/s}} = 30 \, \text{seconds}
\]

Final Answer:
The train will take 30 seconds to cross the man.

Answer: Option D
Solution:
To solve this, let’s break it down:

Let:
– \( S_1 \) be the speed of the train from Howrah to Patna.
– \( S_2 \) be the speed of the train from Patna to Howrah.
– The total distance between Howrah and Patna be \( D \).

Now, when the trains meet:
– The first train (from Howrah to Patna) has covered some distance, and after meeting, it takes 9 more hours to reach Patna.
– The second train (from Patna to Howrah) has covered some distance as well, and after meeting, it takes 16 more hours to reach Howrah.

Key Idea:
– The total time taken by both trains after meeting is proportional to the distances they need to cover.
– Since both trains meet at some point, the remaining distance for each train after meeting is proportional to their speeds.

Let’s denote the time taken by the first train to meet as \( t \) and the second train to meet as \( t’ \).
– After the meeting, the first train covers the remaining distance in 9 hours, and the second train covers the remaining distance in 16 hours.

Using the relationship between speed, distance, and time:

For the first train:
\[
\text{Distance covered after meeting} = S_1 \times 9
\]

For the second train:
\[
\text{Distance covered after meeting} = S_2 \times 16
\]

Since the total distance is the same for both trains, the remaining distances for both trains after meeting must be equal:
\[
S_1 \times 9 = S_2 \times 16
\]

Solving for the ratio of their speeds:
\[
\frac{S_1}{S_2} = \frac{16}{9}
\]

Thus, the ratio of their speeds is 16:9.

Answer: Option C
Solution:
To find the time it takes for the 100 m long train to cross a 140 m long railway bridge, we first need to calculate the total distance the train needs to cover.

Total distance to be covered:
– Length of the train = 100 meters
– Length of the bridge = 140 meters
– Total distance = 100 + 140 = **240 meters**

Now, let’s convert the speed of the train into meters per second:
– Speed of the train = 60 km/hr

To convert this into meters per second, use the conversion factor:
\[
1 \, \text{km/hr} = \frac{1000}{3600} \, \text{m/s}
\]

So,
\[
60 \, \text{km/hr} = 60 \times \frac{1000}{3600} = 16.67 \, \text{m/s}
\]

Time taken to cross the bridge:
The formula for time is:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\] Substituting the values:
\[
\text{Time} = \frac{240 \, \text{m}}{16.67 \, \text{m/s}} \approx 14.4 \, \text{seconds}
\]

Therefore, it will take approximately 14.4 seconds for the train to cross the 140 m long railway bridge.

Answer: Option C
Solution:
To find the speed of the train, we can use the relationship between speed, distance, and time:

\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Given:
– Length of the train = 132 meters
– Time taken to pass the telegraph pole = 6 seconds

Speed of the train:
\[
\text{Speed} = \frac{132 \, \text{m}}{6 \, \text{seconds}} = 22 \, \text{m/s}
\]

Now, to convert this speed into kilometers per hour (km/h), we use the conversion factor:
\[
1 \, \text{m/s} = \frac{18}{5} \, \text{km/h}
\]

So:
\[
22 \, \text{m/s} = 22 \times \frac{18}{5} = 79.2 \, \text{km/h}
\]

Therefore, the speed of the train is 79.2 km/h.

Answer: Option C
Solution:
We are given the following:

– The speed of the train = 60 km/h
– The length of the platform = 200 meters
– Time taken to cross the platform = 27 seconds

Step 1: Convert speed from km/h to m/s
To convert speed from km/h to m/s, we use the conversion factor:
\[
1 \, \text{km/h} = \frac{1000}{3600} \, \text{m/s} = \frac{5}{18} \, \text{m/s}
\] So, the speed of the train in m/s is:
\[
60 \, \text{km/h} = 60 \times \frac{5}{18} = 16.67 \, \text{m/s}
\]

Step 2: Use the formula for distance
The total distance covered by the train while crossing the platform is the sum of the length of the train and the length of the platform.

Let the length of the train be \( L \) meters.

The formula for distance is:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\] The total distance covered by the train is the length of the train plus the length of the platform:
\[
L + 200 = 16.67 \times 27
\]

Now calculate the total distance:
\[
16.67 \times 27 = 450.09 \, \text{meters}
\]

So:
\[
L + 200 = 450.09
\]

Step 3: Solve for \( L \)
\[
L = 450.09 – 200 = 250.09 \, \text{meters}
\]

Therefore, the length of the train is approximately 250 meters.

Answer: Option E
Solution:
Let’s solve this step by step:

Given:
– Speed of the train = 90 km/h
– Time taken to cross a platform double its length = 36 seconds

Let the length of the train be \( L \) meters.
Thus, the length of the platform will be \( 2L \) meters (since the platform is double the length of the train).

Step 1: Convert the speed from km/h to m/s
To convert speed from km/h to m/s, use the conversion factor:
\[
1 \, \text{km/h} = \frac{1000}{3600} \, \text{m/s} = \frac{5}{18} \, \text{m/s}
\] So:
\[
90 \, \text{km/h} = 90 \times \frac{5}{18} = 25 \, \text{m/s}
\]

Step 2: Use the formula for distance
The total distance covered by the train while crossing the platform is the sum of the length of the train and the length of the platform.

The formula for distance is:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]

The total distance covered by the train is:
\[
L + 2L = 3L
\]

Now, using the given time (36 seconds) and the speed (25 m/s):
\[
3L = 25 \times 36
\]

Calculating the right-hand side:
\[
3L = 900
\]

Step 3: Solve for \( L \)
\[
L = \frac{900}{3} = 300 \, \text{meters}
\]

Step 4: Find the length of the platform
Since the platform’s length is double the length of the train:
\[
\text{Length of the platform} = 2L = 2 \times 300 = 600 \, \text{meters}
\]

Therefore, the length of the platform is 600 meters.

Answer: Option C
Solution:
To find the speed of the train, we need to calculate the total distance the train travels while crossing the tunnel, and then use the formula for speed.

Given:
– Length of the train = 150 meters
– Length of the tunnel = 300 meters
– Time taken to cross the tunnel = 40.5 seconds

Step 1: Calculate the total distance traveled by the train
When the train crosses the tunnel, it covers the length of the train plus the length of the tunnel:
\[
\text{Total distance} = 150 \, \text{meters} + 300 \, \text{meters} = 450 \, \text{meters}
\]

Step 2: Use the formula for speed
The formula for speed is:
\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Substituting the given values:
\[
\text{Speed} = \frac{450 \, \text{meters}}{40.5 \, \text{seconds}} = 11.11 \, \text{m/s}
\]

Step 3: Convert speed from m/s to km/h
To convert speed from meters per second (m/s) to kilometers per hour (km/h), use the conversion factor:
\[
1 \, \text{m/s} = \frac{18}{5} \, \text{km/h}
\]

Thus:
\[
\text{Speed in km/h} = 11.11 \times \frac{18}{5} = 40 \, \text{km/h}
\]

Therefore, the speed of the train is 40 km/h.

Answer: Option C
Solution:
To find the speed of the train, let’s go through the steps.

Given:
– Length of the train = 280 meters
– Length of the platform = \( 3 \times \text{Length of the train} = 3 \times 280 = 840 \) meters
– Time taken to cross the platform = 50 seconds

Step 1: Calculate the total distance traveled by the train
When the train crosses the platform, the total distance traveled is the sum of the length of the train and the length of the platform:
\[
\text{Total distance} = \text{Length of the train} + \text{Length of the platform} = 280 \, \text{m} + 840 \, \text{m} = 1120 \, \text{m}
\]

Step 2: Use the formula for speed
The formula for speed is:
\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Substitute the values:
\[
\text{Speed} = \frac{1120 \, \text{meters}}{50 \, \text{seconds}} = 22.4 \, \text{m/s}
\]

Step 3: Convert speed from m/s to km/h
To convert speed from meters per second (m/s) to kilometers per hour (km/h), use the conversion factor:
\[
1 \, \text{m/s} = \frac{18}{5} \, \text{km/h}
\]

Thus:
\[
\text{Speed in km/h} = 22.4 \times \frac{18}{5} = 80.64 \, \text{km/h}
\]

Therefore, the speed of the train is approximately 80.64 km/h.

Answer: Option B
Solution:
To find the time taken by the train to pass a man running in the opposite direction, let’s break it down step by step.

Given:
– Length of the train = 110 meters
– Speed of the train = 60 km/h
– Speed of the man = 6 km/h
– The man and the train are moving in opposite directions.

Step 1: Find the relative speed of the train and the man
Since the train and the man are moving in opposite directions, their relative speed is the sum of their individual speeds:

\[
\text{Relative speed} = 60 \, \text{km/h} + 6 \, \text{km/h} = 66 \, \text{km/h}
\]

Step 2: Convert the relative speed to meters per second
To convert the relative speed from km/h to m/s, we use the conversion factor:

\[
1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}
\]

So:

\[
66 \, \text{km/h} = 66 \times \frac{5}{18} = 18.33 \, \text{m/s}
\]

Step 3: Calculate the time to pass the man
Now, the train needs to cover the length of the train (110 meters) at the relative speed of 18.33 m/s.

Using the formula for time:

\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}}
\]

Substituting the values:

\[
\text{Time} = \frac{110 \, \text{meters}}{18.33 \, \text{m/s}} \approx 6 \, \text{seconds}
\]

Final Answer:
The train will take approximately 6 seconds to pass the man.

Answer: Option D
Solution:
Let’s break this down step by step:

Given:
– Speed of Train A = 60 km/h
– Speed of Train B = 72 km/h
– Length of each train = 240 meters

Since both trains are running in the same direction, the relative speed between them is the difference of their speeds.

Step 1: Calculate the relative speed
The relative speed of Train B with respect to Train A (since they are moving in the same direction) is:
\[
\text{Relative speed} = 72 \, \text{km/h} – 60 \, \text{km/h} = 12 \, \text{km/h}
\]

Step 2: Convert the relative speed into meters per second
To convert the relative speed from km/h to m/s, we use the conversion factor:
\[
1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}
\] So:
\[
12 \, \text{km/h} = 12 \times \frac{5}{18} = \frac{60}{18} = 3.33 \, \text{m/s}
\]

Step 3: Calculate the distance that Train B needs to cover
Train B needs to cover the entire length of Train A to pass it completely. The distance to be covered by Train B is the length of Train A, which is 240 meters.

Step 4: Calculate the time taken for Train B to cross Train A
Using the formula for time:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}}
\]

Substituting the values:
\[
\text{Time} = \frac{240 \, \text{meters}}{3.33 \, \text{m/s}} \approx 72 \, \text{seconds}
\]

Final Answer:
It will take approximately 72 seconds for Train B to completely cross Train A.

Answer: Option C
Solution:
To calculate the time taken for the slower train to cross the faster train, let’s break it down step by step.

Given:
– Speed of the slower train = 60 km/h
– Speed of the faster train = 90 km/h
– Length of the slower train = 1.10 km = 1100 meters
– Length of the faster train = 0.9 km = 900 meters

Since the trains are moving in opposite directions, their relative speed is the sum of their individual speeds.

Step 1: Calculate the relative speed
The relative speed of the two trains (since they are moving in opposite directions) is:
\[
\text{Relative speed} = 60 \, \text{km/h} + 90 \, \text{km/h} = 150 \, \text{km/h}
\]

Step 2: Convert the relative speed to meters per second
To convert the relative speed from km/h to m/s, use the conversion factor:
\[
1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}
\]

Thus:
\[
150 \, \text{km/h} = 150 \times \frac{5}{18} = 41.67 \, \text{m/s}
\]

Step 3: Calculate the total distance to be covered
The total distance to be covered is the sum of the lengths of both trains:
\[
\text{Total distance} = 1100 \, \text{meters} + 900 \, \text{meters} = 2000 \, \text{meters}
\]

Step 4: Calculate the time taken for the slower train to cross the faster train
The time taken to cross the faster train is given by the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}}
\]

Substituting the values:
\[
\text{Time} = \frac{2000 \, \text{meters}}{41.67 \, \text{m/s}} \approx 48 \, \text{seconds}
\]

Final Answer:
The time taken by the slower train to cross the faster train is approximately 48 seconds.

Answer: Option C
Solution:
Let’s solve this step by step:

Given:
– Length of Train 1 = 120 meters
– Length of Train 2 = 90 meters
– Speed of Train 1 = 80 km/h
– Speed of Train 2 = 55 km/h
– Distance between the two trains = 90 meters

Since the trains are moving towards each other, their relative speed will be the sum of their individual speeds.

Step 1: Calculate the relative speed
The relative speed is the sum of the speeds of the two trains because they are moving towards each other:
\[
\text{Relative speed} = 80 \, \text{km/h} + 55 \, \text{km/h} = 135 \, \text{km/h}
\]

Step 2: Convert the relative speed from km/h to m/s
To convert the relative speed into meters per second, we use the conversion factor:
\[
1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}
\]

So:
\[
135 \, \text{km/h} = 135 \times \frac{5}{18} = 37.5 \, \text{m/s}
\]

Step 3: Calculate the total distance to be covered
The total distance to be covered is the sum of the lengths of the two trains plus the distance between them:
\[
\text{Total distance} = 120 \, \text{m} + 90 \, \text{m} + 90 \, \text{m} = 300 \, \text{m}
\]

Step 4: Calculate the time taken to cross each other
The formula for time is:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}}
\]

Substituting the values:
\[
\text{Time} = \frac{300 \, \text{m}}{37.5 \, \text{m/s}} = 8 \, \text{seconds}
\]

Final Answer:
It will take 8 seconds for the two trains to cross each other.

Answer: Option C
Solution:
To find the speed of the train, we will use the following information:

Given:
– Length of the train = 240 meters
– Speed of the man = 3 km/h (walking in the opposite direction)
– Time taken by the train to cross the man = 10 seconds

Since the man is walking in the opposite direction to the train, the relative speed of the train with respect to the man is the sum of their speeds.

Step 1: Convert the speed of the man from km/h to m/s
To convert the speed from km/h to m/s, we use the conversion factor:
\[
1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}
\]

So, the speed of the man in m/s is:
\[
\text{Speed of the man} = 3 \, \text{km/h} \times \frac{5}{18} = \frac{15}{18} = 0.833 \, \text{m/s}
\]

Step 2: Calculate the relative speed
Let the speed of the train be \( v \) m/s. The relative speed of the train with respect to the man is:
\[
\text{Relative speed} = v + 0.833 \, \text{m/s}
\]

Step 3: Use the formula for time to cross the man
The time taken by the train to cross the man is given by:
\[
\text{Time} = \frac{\text{Length of the train}}{\text{Relative speed}}
\]

We are given that the time is 10 seconds, so:
\[
10 = \frac{240}{v + 0.833}
\]

Step 4: Solve for \( v \)
Multiply both sides by \( v + 0.833 \):
\[
10(v + 0.833) = 240
\] \[
10v + 8.33 = 240
\] \[
10v = 240 – 8.33
\] \[
10v = 231.67
\] \[
v = \frac{231.67}{10} = 23.167 \, \text{m/s}
\]

Step 5: Convert the speed from m/s to km/h
To convert \( v \) from m/s to km/h, multiply by \( \frac{18}{5} \):
\[
v = 23.167 \times \frac{18}{5} = 83.8 \, \text{km/h}
\]

Final Answer:
The speed of the train is approximately 83.8 km/h.

Answer: Option A
Solution:
To solve this, we need to calculate the length of each train.

Given:
– Speed of the faster train = 46 km/h
– Speed of the slower train = 36 km/h
– Time taken for the faster train to pass the slower train = 36 seconds
– Both trains have equal lengths (let the length of each train be \( L \) meters)

Step 1: Calculate the relative speed
Since the trains are moving in the same direction, their relative speed is the difference between their speeds:
\[
\text{Relative speed} = 46 \, \text{km/h} – 36 \, \text{km/h} = 10 \, \text{km/h}
\]

Step 2: Convert the relative speed into meters per second
To convert the relative speed from km/h to m/s, use the conversion factor:
\[
1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}
\] Thus:
\[
10 \, \text{km/h} = 10 \times \frac{5}{18} = \frac{50}{18} = 2.78 \, \text{m/s}
\]

Step 3: Use the formula for time
The time taken for the faster train to pass the slower train is given as 36 seconds. The total distance traveled during this time is the sum of the lengths of both trains, which is \( 2L \).

Using the formula for time:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}}
\] Substitute the given values:
\[
36 = \frac{2L}{2.78}
\]

Step 4: Solve for \( L \)
Multiply both sides by 2.78:
\[
36 \times 2.78 = 2L
\] \[
100.08 = 2L
\] \[
L = \frac{100.08}{2} = 50.04 \, \text{meters}
\]

Final Answer:
The length of each train is approximately 50 meters.

Answer: Option B
Solution:
Let’s break this down step by step.

Given:
– Length of each train = 120 meters
– Time taken by the first train to cross the telegraph post = 10 seconds
– Time taken by the second train to cross the telegraph post = 15 seconds

We are asked to find the time it will take for the two trains to cross each other when traveling in opposite directions.

Step 1: Find the speed of each train
The speed of a train can be calculated using the formula:
\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Speed of the first train:
\[
\text{Speed of the first train} = \frac{120 \, \text{meters}}{10 \, \text{seconds}} = 12 \, \text{m/s}
\]

Speed of the second train:
\[
\text{Speed of the second train} = \frac{120 \, \text{meters}}{15 \, \text{seconds}} = 8 \, \text{m/s}
\]

Step 2: Find the relative speed
Since the two trains are traveling in opposite directions, their relative speed will be the sum of their individual speeds:
\[
\text{Relative speed} = 12 \, \text{m/s} + 8 \, \text{m/s} = 20 \, \text{m/s}
\]

Step 3: Find the total distance to be covered
When the two trains cross each other, the total distance covered is the sum of the lengths of both trains:
\[
\text{Total distance} = 120 \, \text{meters} + 120 \, \text{meters} = 240 \, \text{meters}
\]

Step 4: Calculate the time to cross each other
The time taken to cross each other is given by the formula:
\[
\text{Time} = \frac{\text{Total distance}}{\text{Relative speed}}
\]

Substitute the values:
\[
\text{Time} = \frac{240 \, \text{meters}}{20 \, \text{m/s}} = 12 \, \text{seconds}
\]

Final Answer:
It will take 12 seconds for the two trains to cross each other when traveling in opposite directions.

Answer: Option A
Solution:
Let’s solve this problem step by step.

Given:
– Speed of Train B = 120 km/h
– Length of Train B = 100 meters
– Length of Train C = 200 meters
– Time taken to cross Train C = 2 minutes

We need to find the speed of Train C.

Step 1: Convert the time to seconds
Since the time is given in minutes, we need to convert it to seconds:
\[
\text{Time} = 2 \, \text{minutes} \times 60 \, \text{seconds/minute} = 120 \, \text{seconds}
\]

Step 2: Find the relative speed
When the two trains are moving in the same direction, the relative speed is the difference between their speeds. Let’s denote the speed of Train C as \( x \) km/h. The relative speed of Train B with respect to Train C will be:
\[
\text{Relative speed} = 120 \, \text{km/h} – x \, \text{km/h}
\]

Step 3: Convert the relative speed to m/s
To use the lengths in meters and time in seconds, we need to convert the relative speed from km/h to m/s. The conversion factor is:
\[
1 \, \text{km/h} = \frac{5}{18} \, \text{m/s}
\] Thus, the relative speed in m/s is:
\[
\text{Relative speed in m/s} = (120 – x) \times \frac{5}{18}
\]

Step 4: Use the formula for time
The total distance traveled by Train B to cross Train C is the sum of their lengths:
\[
\text{Total distance} = 100 \, \text{m} + 200 \, \text{m} = 300 \, \text{m}
\]

The formula for time is:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}}
\] Substitute the values:
\[
120 = \frac{300}{(120 – x) \times \frac{5}{18}}
\]

Step 5: Solve for \( x \)
First, simplify the equation:
\[
120 = \frac{300 \times 18}{(120 – x) \times 5}
\] \[
120 = \frac{5400}{5 \times (120 – x)}
\] \[
120 \times 5 \times (120 – x) = 5400
\] \[
600 \times (120 – x) = 5400
\] \[
120 – x = \frac{5400}{600} = 9
\] \[
120 – x = 9
\] \[
x = 120 – 9 = 111 \, \text{km/h}
\]

Final Answer:
The speed of Train C is 111 km/h.

Answer: Option D
Solution:
Let’s solve this step by step.

Given:
– Length of the platform = 50 meters
– Time taken by the train to cross the platform = 14 seconds
– Time taken by the train to pass a man = 10 seconds

We are asked to find the speed of the train.

Step 1: Calculate the speed of the train
Let the length of the train be \( L \) meters, and the speed of the train be \( S \) meters per second.

When the train crosses the man, the distance covered by the train is the length of the train \( L \). The time taken to pass the man is 10 seconds. So, the formula for speed is:
\[
S = \frac{L}{10} \quad \text{(Equation 1)}
\]

When the train crosses the platform, the total distance covered is the sum of the length of the train and the length of the platform:
\[
\text{Total distance} = L + 50 \, \text{meters}
\] The time taken to pass the platform is 14 seconds. So, the formula for speed when crossing the platform is:
\[
S = \frac{L + 50}{14} \quad \text{(Equation 2)}
\]

Step 2: Set up the equations
From Equation 1 and Equation 2, we have:
\[
\frac{L}{10} = \frac{L + 50}{14}
\]

Step 3: Solve for \( L \)
Cross-multiply to solve for \( L \):
\[
L \times 14 = (L + 50) \times 10
\] \[
14L = 10L + 500
\] \[
14L – 10L = 500
\] \[
4L = 500
\] \[
L = \frac{500}{4} = 125 \, \text{meters}
\]

Step 4: Calculate the speed of the train
Now that we know the length of the train is 125 meters, substitute \( L = 125 \) into Equation 1:
\[
S = \frac{125}{10} = 12.5 \, \text{m/s}
\]

Step 5: Convert the speed to km/h
To convert the speed from meters per second to kilometers per hour, multiply by \( \frac{18}{5} \):
\[
S = 12.5 \times \frac{18}{5} = 45 \, \text{km/h}
\]

Final Answer:
The speed of the train is 45 km/h.

Answer: Option B
Solution:
To solve this, let’s break it down step by step.

Given:
– Speed of the train = 132 km/h
– Length of the train = 110 meters
– Length of the railway platform = 165 meters

Step 1: Convert the speed of the train to meters per second
Since the lengths are given in meters and the time is in seconds, we first need to convert the speed from km/h to m/s. To convert from km/h to m/s, multiply by \( \frac{5}{18} \):
\[
132 \, \text{km/h} = 132 \times \frac{5}{18} = \frac{660}{18} = 36.67 \, \text{m/s}
\]

Step 2: Calculate the total distance to be covered
The total distance to be covered is the sum of the length of the train and the length of the platform:
\[
\text{Total distance} = 110 \, \text{meters} + 165 \, \text{meters} = 275 \, \text{meters}
\]

Step 3: Use the formula for time
The time taken to cross the platform is given by:
\[
\text{Time} = \frac{\text{Total distance}}{\text{Speed}}
\] Substitute the values:
\[
\text{Time} = \frac{275 \, \text{meters}}{36.67 \, \text{m/s}} \approx 7.5 \, \text{seconds}
\]

Final Answer:
It will take approximately 7.5 seconds for the train to cross the platform.

Answer: Option B
Solution:
To calculate the speed of the train, we can break it down step by step.

Given:
– Length of the train = 500 feet
– Length of the platform = 700 feet
– Time taken to cross the platform = 10 seconds

Step 1: Calculate the total distance covered
When the train crosses the platform, the total distance covered is the sum of the lengths of the train and the platform:
\[
\text{Total distance} = 500 \, \text{feet} + 700 \, \text{feet} = 1200 \, \text{feet}
\]

Step 2: Use the formula for speed
Speed is given by the formula:
\[
\text{Speed} = \frac{\text{Distance}}{\text{Time}}
\]

Substitute the values:
\[
\text{Speed} = \frac{1200 \, \text{feet}}{10 \, \text{seconds}} = 120 \, \text{feet/second}
\]

Step 3: Convert the speed from feet per second to miles per hour (mph)
To convert from feet per second to miles per hour, use the conversion factors:
– \( 1 \, \text{mile} = 5280 \, \text{feet} \)
– \( 1 \, \text{hour} = 3600 \, \text{seconds} \)

\[
\text{Speed in mph} = 120 \, \text{feet/second} \times \frac{3600 \, \text{seconds/hour}}{5280 \, \text{feet/mile}} = 120 \times \frac{3600}{5280} \approx 81.82 \, \text{mph}
\]

Final Answer:
The speed of the train is approximately 81.82 mph.

Answer: Option B
Solution:
Let’s break this down step by step.

Given:
– Speed of the Ghaziabad – Hapur – Meerut EMU = 16 km/h
– Speed of the Meerut – Hapur – Ghaziabad EMU = 21 km/h
– The difference in the distance traveled by the two trains when they meet = 60 km
– The two trains meet after traveling towards each other.

Step 1: Set up variables
Let the distance between the two stations (Ghaziabad and Meerut) be \( D \) kilometers.

When the two trains meet, the total distance \( D \) is covered by both trains together. Since one train travels 60 km more than the other, let’s assume:
– The first train (Ghaziabad – Hapur – Meerut EMU) travels \( x \) kilometers.
– The second train (Meerut – Hapur – Ghaziabad EMU) travels \( x + 60 \) kilometers.

Thus, the total distance traveled by both trains when they meet is:
\[
x + (x + 60) = D
\] \[
2x + 60 = D
\]

Step 2: Relate the distances with time
Since both trains start at the same time and travel towards each other, they meet after traveling for the same amount of time. The time taken by each train is:
\[
\text{Time taken by first train} = \frac{x}{16} \, \text{hours}
\] \[
\text{Time taken by second train} = \frac{x + 60}{21} \, \text{hours}
\]

Since the time taken by both trains is the same, we can set up the equation:
\[
\frac{x}{16} = \frac{x + 60}{21}
\]

Step 3: Solve for \( x \)
Cross-multiply to solve for \( x \):
\[
21x = 16(x + 60)
\] \[
21x = 16x + 960
\] \[
21x – 16x = 960
\] \[
5x = 960
\] \[
x = \frac{960}{5} = 192
\]

Step 4: Calculate the total distance \( D \)
Now that we know \( x = 192 \), we can calculate the total distance \( D \):
\[
D = 2x + 60 = 2(192) + 60 = 384 + 60 = 444 \, \text{km}
\]

Final Answer:
The distance between Ghaziabad and Meerut is 444 km.

Answer: Option D
Solution:
Let’s break this down step by step to find the rate of the slower train.

Given:
– The two stations are 270 km apart.
– The trains meet at some point between the two stations.
– After meeting, the first train takes \( 6\frac{1}{4} \) hours (or 6.25 hours) to reach the other station.
– After meeting, the second train takes 4 hours to reach the other station.

We are tasked with finding the rate of the slower train.

Step 1: Let the speed of the two trains be \( v_1 \) and \( v_2 \) (where \( v_1 \) is the speed of the first train and \( v_2 \) is the speed of the second train).

Step 2: Define the distances each train travels before they meet
Let the distance traveled by the first train before they meet be \( x \) kilometers, and the distance traveled by the second train before they meet be \( 270 – x \) kilometers (since the total distance between the stations is 270 km).

The total time taken by each train to meet is the same, and it is the same for both trains. So, the time taken by the first train to travel distance \( x \) is:
\[
\text{Time for the first train} = \frac{x}{v_1}
\] And the time taken by the second train to travel distance \( 270 – x \) is:
\[
\text{Time for the second train} = \frac{270 – x}{v_2}
\]

Step 3: Use the time ratio after meeting
After meeting, the first train takes 6.25 hours to cover the remaining distance (which is \( 270 – x \) kilometers), and the second train takes 4 hours to cover the remaining distance (which is \( x \) kilometers).

We know the total time taken by each train to reach its destination is proportional to their speeds. Therefore, the ratio of the speeds of the two trains is the inverse of the ratio of the times taken by each train to reach its destination after they meet.

The time taken for the first train to cover the remaining distance is 6.25 hours, and for the second train, it is 4 hours. Hence, the ratio of their speeds is the inverse of the ratio of their times:
\[
\frac{v_1}{v_2} = \frac{4}{6.25} = \frac{4}{\frac{25}{4}} = \frac{16}{25}
\]

So, the ratio of the speeds of the two trains is \( \frac{16}{25} \).

Step 4: Solve for the speed of the slower train
Now, we know the ratio of the speeds of the two trains is \( \frac{16}{25} \), so let the speed of the slower train be \( v_2 \) and the speed of the faster train be \( v_1 = \frac{16}{25} v_2 \).

We can use this ratio to find \( v_2 \).

The remaining distance each train travels after meeting is proportional to their speeds. The first train travels \( 270 – x \) kilometers in 6.25 hours, so:
\[
v_1 = \frac{270 – x}{6.25}
\] Similarly, the second train travels \( x \) kilometers in 4 hours, so:
\[
v_2 = \frac{x}{4}
\]

Using the ratio \( \frac{v_1}{v_2} = \frac{16}{25} \), we have:
\[
\frac{\frac{270 – x}{6.25}}{\frac{x}{4}} = \frac{16}{25}
\] Simplifying this equation:
\[
\frac{(270 – x) \times 4}{6.25 \times x} = \frac{16}{25}
\] \[
\frac{4(270 – x)}{6.25x} = \frac{16}{25}
\] Now, cross-multiply to solve for \( x \):
\[
4(270 – x) \times 25 = 16 \times 6.25x
\] \[
100(270 – x) = 100x
\] \[
27000 – 100x = 100x
\] \[
27000 = 200x
\] \[
x = \frac{27000}{200} = 135 \, \text{km}
\]

Step 5: Find the speed of the slower train
Now that we know \( x = 135 \, \text{km} \), we can calculate the speed of the slower train.

The time taken by the slower train to cover 135 km is 4 hours, so the speed of the slower train is:
\[
v_2 = \frac{x}{4} = \frac{135}{4} = 33.75 \, \text{km/h}
\]

Final Answer:
The speed of the slower train is 33.75 km/h.

Answer: Option B
Solution:
To solve this problem, let’s break it down step by step.

Given:
– Train A starts from station X and travels towards station Y.
– Train B starts from station Y and travels towards station X.
– The time taken by Train A after meeting is 4 hours 48 minutes (which is 4.8 hours).
– The time taken by Train B after meeting is 3 hours 20 minutes (which is 3.33 hours).
– The speed of Train A is 45 km/h.
– We need to find the speed of Train B.

Step 1: Let the speeds of the two trains be:
– Speed of Train A = 45 km/h (Given)
– Speed of Train B = \( v_B \) km/h (To be determined)

Step 2: Use the formula for distance
The distance traveled by each train after they meet is equal to the product of their speed and the time they travel.

– Distance traveled by Train A after meeting = \( 45 \times 4.8 = 216 \) km.
– Distance traveled by Train B after meeting = \( v_B \times 3.33 \) km.

Since the total distance between the two stations (X and Y) is the sum of the distances covered by both trains before and after they meet, we can relate the total distance.

Step 3: Total distance covered
The total distance between the stations is the same for both trains, and it is equal to the sum of the distances they cover before and after the meeting point.

Let the distance between the stations (X and Y) be \( D \).

The distance covered by Train A before meeting is \( D_1 \), and the distance covered by Train B before meeting is \( D_2 \). After meeting:
– Train A covers 216 km in 4.8 hours.
– Train B covers \( v_B \times 3.33 \) km in 3.33 hours.

Thus, the total distance between the stations is:
\[
D = D_1 + 216 = D_2 + (v_B \times 3.33)
\]

Since the times taken to cover their respective distances are proportional to their speeds, we can use the following relation based on the inverse of their speeds:

\[
\frac{\text{Time taken by Train A after meeting}}{\text{Time taken by Train B after meeting}} = \frac{\text{Speed of Train B}}{\text{Speed of Train A}}
\]

Substituting the known values:
\[
\frac{4.8}{3.33} = \frac{v_B}{45}
\]

Step 4: Solve for \( v_B \)
Now, solving for \( v_B \):
\[
\frac{4.8}{3.33} = \frac{v_B}{45}
\] \[
v_B = 45 \times \frac{4.8}{3.33}
\] \[
v_B \approx 45 \times 1.44 = 64.8 \, \text{km/h}
\]

Final Answer:
The speed of Train B is approximately 64.8 km/h.

Answer: Option B
Solution:
Let’s break down the problem step by step to find the length of the train.

Given:
– The train passes a lamp post in 7 seconds.
– The train passes the entire platform (length = 390 meters) in 28 seconds.

Step 1: Calculate the speed of the train
When the train passes the lamp post, the distance covered is the length of the train itself, and this is done in 7 seconds. Let the length of the train be \( L \) meters.

The speed of the train is the distance traveled divided by the time taken:
\[
\text{Speed of train} = \frac{L}{7} \, \text{m/s}
\]

Step 2: Use the time taken to pass the platform
When the train passes the entire platform, the distance covered is the length of the train plus the length of the platform (390 meters). This is done in 28 seconds.

The speed of the train, as calculated before, is also the distance covered in this case divided by the time taken. So we have:
\[
\text{Speed of train} = \frac{L + 390}{28} \, \text{m/s}
\]

Step 3: Set up the equation
Since the speed of the train is the same in both cases, we can equate the two expressions for speed:
\[
\frac{L}{7} = \frac{L + 390}{28}
\]

Step 4: Solve for \( L \)
Cross-multiply to solve for \( L \):
\[
28L = 7(L + 390)
\] \[
28L = 7L + 2730
\] \[
28L – 7L = 2730
\] \[
21L = 2730
\] \[
L = \frac{2730}{21} = 130 \, \text{meters}
\]

Final Answer:
The length of the train is 130 meters.

Answer: Option A
Solution:
Let’s solve this step by step:

Given:
– Distance between stations A and B = 220 km
– Speed of train A = 50 km/h
– Speed of train B = 60 km/h

Step 1: Relative speed
When two trains are moving towards each other, their relative speed is the sum of their individual speeds.

\[
\text{Relative speed} = 50 \, \text{km/h} + 60 \, \text{km/h} = 110 \, \text{km/h}
\]

Step 2: Time taken to meet
The time taken for the two trains to meet is given by the formula:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}}
\] Substituting the given values:
\[
\text{Time} = \frac{220 \, \text{km}}{110 \, \text{km/h}} = 2 \, \text{hours}
\]

Final Answer:
The two trains will meet after 2 hours.

Answer: Option A
Solution:
To find the speed of the train, let’s break down the information step by step.

Given:
– Length of the train = 100 meters
– Speed of the man = 5 km/h
– Time taken to pass the man = 7 seconds

Step 1: Convert the man’s speed into meters per second
First, we need to convert the man’s speed from km/h to meters per second:
\[
\text{Speed of man} = 5 \, \text{km/h} = \frac{5 \times 1000}{3600} = \frac{5000}{3600} \approx 1.39 \, \text{m/s}
\]

Step 2: Relative speed of the train and the man
Since the train and the man are moving in opposite directions, the relative speed is the sum of their speeds.

Let the speed of the train be \( v \) km/h. Convert it to meters per second:
\[
\text{Speed of train in m/s} = \frac{v \times 1000}{3600} = \frac{v}{3.6}
\]

Thus, the relative speed between the train and the man is:
\[
\text{Relative speed} = \frac{v}{3.6} + 1.39 \, \text{m/s}
\]

Step 3: Use the formula for time to pass the man
The time taken to pass the man is given by the formula:
\[
\text{Time} = \frac{\text{Length of the train}}{\text{Relative speed}}
\]

Substitute the known values:
\[
7 = \frac{100}{\frac{v}{3.6} + 1.39}
\]

Step 4: Solve for \( v \)
Multiply both sides of the equation by the relative speed:
\[
7 \left( \frac{v}{3.6} + 1.39 \right) = 100
\]

Distribute 7:
\[
7 \times \frac{v}{3.6} + 7 \times 1.39 = 100
\] \[
\frac{7v}{3.6} + 9.73 = 100
\]

Subtract 9.73 from both sides:
\[
\frac{7v}{3.6} = 90.27
\]

Now, multiply both sides by 3.6:
\[
7v = 90.27 \times 3.6 = 325.0
\]

Finally, solve for \( v \):
\[
v = \frac{325.0}{7} \approx 46.43 \, \text{km/h}
\]

Final Answer:
The speed of the train is approximately 46.43 km/h.

Answer: Option B
Solution:
To find the length of the train, let’s break it down:

Given:
– Time taken to cross the pole = 9 seconds
– Speed of the train = 48 km/h

Step 1: Convert the speed into meters per second
We need to convert the speed from km/h to meters per second (m/s), because the time is given in seconds.

\[
\text{Speed in m/s} = \frac{\text{Speed in km/h} \times 1000}{3600}
\] \[
\text{Speed in m/s} = \frac{48 \times 1000}{3600} = \frac{48000}{3600} = 13.33 \, \text{m/s}
\]

Step 2: Use the formula for distance
The distance traveled by the train is equal to the length of the train (since the train passes the pole completely). We can use the formula:
\[
\text{Distance} = \text{Speed} \times \text{Time}
\]

Substituting the values:
\[
\text{Length of the train} = 13.33 \, \text{m/s} \times 9 \, \text{seconds}
\] \[
\text{Length of the train} = 120 \, \text{meters}
\]

Final Answer:
The length of the train is 120 meters.

Answer: Option A
Solution:
Let’s break down the problem step by step.

Given:
– Speed of train A = 75 km/h
– Speed of train B = 50 km/h
– When the trains meet, train A has traveled 175 km more than train B.

Step 1: Let the distance traveled by train B before they meet be \( x \) km.
– The distance traveled by train A before they meet is \( x + 175 \) km (since train A has traveled 175 km more than train B).

Step 2: Use the concept of time
The trains start at the same time and travel toward each other. Since they meet at the same time, the time taken by both trains to meet is the same.

– Time taken by train A to meet = \( \frac{x + 175}{75} \) hours
– Time taken by train B to meet = \( \frac{x}{50} \) hours

Since the time taken by both trains is the same:
\[
\frac{x + 175}{75} = \frac{x}{50}
\]

Step 3: Solve for \( x \)
Cross-multiply to solve for \( x \):
\[
50(x + 175) = 75x
\] \[
50x + 8750 = 75x
\] \[
8750 = 75x – 50x
\] \[
8750 = 25x
\] \[
x = \frac{8750}{25} = 350 \, \text{km}
\]

Step 4: Find the total distance
The total distance between stations A and B is the sum of the distances traveled by both trains:
\[
\text{Total distance} = x + (x + 175) = 350 + 350 + 175 = 875 \, \text{km}
\]

Final Answer:
The distance between stations A and B is 875 km.

Answer: Option B
Solution:
To find the time taken for the two trains to cross each other after they meet, we need to first determine the relative speed between the two trains and then use it to calculate the time.

Given:
– Length of train 1 = 180 meters
– Length of train 2 = 120 meters
– Speed of train 1 = 65 km/h
– Speed of train 2 = 55 km/h

Step 1: Convert the speeds into meters per second
We need to convert the speeds from km/h to meters per second (m/s) because the lengths are given in meters and the time we need to find will be in seconds.

\[
\text{Speed of train 1 in m/s} = \frac{65 \times 1000}{3600} = \frac{65000}{3600} \approx 18.06 \, \text{m/s}
\] \[
\text{Speed of train 2 in m/s} = \frac{55 \times 1000}{3600} = \frac{55000}{3600} \approx 15.28 \, \text{m/s}
\]

Step 2: Calculate the relative speed
Since the trains are moving towards each other, their relative speed is the sum of their individual speeds:
\[
\text{Relative speed} = 18.06 \, \text{m/s} + 15.28 \, \text{m/s} = 33.34 \, \text{m/s}
\]

Step 3: Total distance to be covered
The total distance to be covered for the trains to completely pass each other is the sum of their lengths:
\[
\text{Total distance} = 180 \, \text{m} + 120 \, \text{m} = 300 \, \text{m}
\]

Step 4: Calculate the time to cross each other
Now, we can calculate the time it will take for the two trains to cross each other using the formula:
\[
\text{Time} = \frac{\text{Total distance}}{\text{Relative speed}}
\] \[
\text{Time} = \frac{300 \, \text{m}}{33.34 \, \text{m/s}} \approx 8.99 \, \text{seconds}
\]

Final Answer:
The trains will cross each other in approximately 9 seconds.

Answer: Option B
Solution:
To solve this problem, we need to calculate the speeds of the two trains based on the information provided.

Given:
– Length of train 1 = 100 meters
– Length of train 2 = 95 meters
– Time taken to pass each other when running in the same direction = 27 seconds
– Time taken to pass each other when running in opposite directions = 9 seconds

Let the speed of train 1 be \( v_1 \) meters per second, and the speed of train 2 be \( v_2 \) meters per second.

Step 1: Total distance covered when the trains are moving in the same direction
When the trains are running in the same direction, the distance covered when they pass each other is the sum of their lengths:
\[
\text{Total distance} = 100 + 95 = 195 \, \text{meters}
\] The relative speed when they run in the same direction is \( v_1 – v_2 \) (since the trains are moving in the same direction). Using the formula for distance:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{195}{v_1 – v_2}
\] We are given that this time is 27 seconds:
\[
\frac{195}{v_1 – v_2} = 27
\] So,
\[
v_1 – v_2 = \frac{195}{27} = 7.22 \, \text{m/s}
\]

Step 2: Total distance covered when the trains are moving in opposite directions
When the trains are running in opposite directions, the total distance covered when they pass each other is also the sum of their lengths:
\[
\text{Total distance} = 100 + 95 = 195 \, \text{meters}
\] The relative speed when they run in opposite directions is \( v_1 + v_2 \). Using the formula for distance:
\[
\text{Time} = \frac{\text{Distance}}{\text{Relative speed}} = \frac{195}{v_1 + v_2}
\] We are given that this time is 9 seconds:
\[
\frac{195}{v_1 + v_2} = 9
\] So,
\[
v_1 + v_2 = \frac{195}{9} = 21.67 \, \text{m/s}
\]

Step 3: Solve the system of equations
We now have the following system of equations:
1. \( v_1 – v_2 = 7.22 \)
2. \( v_1 + v_2 = 21.67 \)

To solve for \( v_1 \) and \( v_2 \), add the two equations:
\[
(v_1 – v_2) + (v_1 + v_2) = 7.22 + 21.67
\] \[
2v_1 = 28.89
\] \[
v_1 = \frac{28.89}{2} = 14.445 \, \text{m/s}
\]

Now, substitute \( v_1 = 14.445 \) into \( v_1 + v_2 = 21.67 \):
\[
14.445 + v_2 = 21.67
\] \[
v_2 = 21.67 – 14.445 = 7.225 \, \text{m/s}
\]

Step 4: Convert the speeds to km/h
To convert the speeds from meters per second to kilometers per hour, multiply by 3.6:
\[
v_1 = 14.445 \times 3.6 = 51.89 \, \text{km/h}
\] \[
v_2 = 7.225 \times 3.6 = 26.02 \, \text{km/h}
\]

Final Answer:
The speed of the first train is approximately 51.89 km/h, and the speed of the second train is approximately 26.02 km/h.

Answer: Option C
Solution:
To find the length of the train, let’s break down the information step by step.

Given:
– Speed of the train = 84 km/h
– Speed of the man = 6 km/h
– Time taken to pass the man = 4 seconds

Step 1: Convert the speeds into meters per second
First, we need to convert the speeds from km/h to m/s because the time is given in seconds.

\[
\text{Speed of the train in m/s} = \frac{84 \times 1000}{3600} = \frac{84000}{3600} = 23.33 \, \text{m/s}
\]

\[
\text{Speed of the man in m/s} = \frac{6 \times 1000}{3600} = \frac{6000}{3600} = 1.67 \, \text{m/s}
\]

Step 2: Calculate the relative speed
Since the train and the man are moving in opposite directions, their relative speed is the sum of their speeds:

\[
\text{Relative speed} = 23.33 \, \text{m/s} + 1.67 \, \text{m/s} = 25 \, \text{m/s}
\]

Step 3: Use the formula for distance
The distance traveled by the train while passing the man is the length of the train. We can use the formula for distance:

\[
\text{Distance} = \text{Relative speed} \times \text{Time}
\]

Substituting the values:

\[
\text{Length of the train} = 25 \, \text{m/s} \times 4 \, \text{seconds} = 100 \, \text{meters}
\]

Final Answer:
The length of the train is 100 meters.

Answer: Option B
Solution:
Let’s solve this problem step by step.

Given:
– Length of the first bridge = 500 meters
– Length of the second bridge = 250 meters
– Time taken to pass the first bridge = 100 seconds
– Time taken to pass the second bridge = 60 seconds

Let the length of the train be \( L \) meters, and let the speed of the train be \( v \) meters per second.

Step 1: Use the first bridge to find the speed of the train

When the train passes the first bridge, the total distance traveled by the train is the sum of the length of the train and the length of the first bridge:
\[
\text{Distance} = L + 500 \, \text{meters}
\] The time taken to cover this distance is 100 seconds. The speed of the train is:
\[
v = \frac{\text{Distance}}{\text{Time}} = \frac{L + 500}{100} \, \text{m/s}
\]

Step 2: Use the second bridge to find the speed of the train

Similarly, when the train passes the second bridge, the total distance traveled is the sum of the length of the train and the length of the second bridge:
\[
\text{Distance} = L + 250 \, \text{meters}
\] The time taken to cover this distance is 60 seconds. The speed of the train is:
\[
v = \frac{L + 250}{60} \, \text{m/s}
\]

Step 3: Set up the equation to solve for \( L \)

Now that we have two expressions for the speed \( v \), we can set them equal to each other:
\[
\frac{L + 500}{100} = \frac{L + 250}{60}
\]

Step 4: Solve the equation for \( L \)

Cross-multiply to solve for \( L \):
\[
60(L + 500) = 100(L + 250)
\] Expand both sides:
\[
60L + 30000 = 100L + 25000
\] Simplify the equation:
\[
30000 – 25000 = 100L – 60L
\] \[
5000 = 40L
\] Solve for \( L \):
\[
L = \frac{5000}{40} = 125 \, \text{meters}
\]

Final Answer:
The length of the train is 125 meters.

Answer: Option D
Solution:
Let’s solve this step by step.

Given:
– Time taken to pass a lamp post = 3 seconds
– Time taken to pass a 900-meter-long platform = 30 seconds
– Length of the platform for the next scenario = 800 meters

Let the length of the train be \( L \) meters and its speed be \( v \) meters per second.

Step 1: Calculate the speed of the train
The time taken to pass the lamp post is 3 seconds. When the train passes a lamp post, it travels a distance equal to its own length. Therefore, the speed of the train is:
\[
v = \frac{L}{3} \, \text{m/s}
\]

Step 2: Use the information about the 900-meter-long platform
When the train passes the 900-meter-long platform, the total distance traveled is the length of the train plus the length of the platform:
\[
\text{Total distance} = L + 900 \, \text{meters}
\] The time taken to cover this distance is 30 seconds, so:
\[
v = \frac{L + 900}{30} \, \text{m/s}
\]

Step 3: Set up the equation for speed
We now have two expressions for the speed \( v \):
1. \( v = \frac{L}{3} \)
2. \( v = \frac{L + 900}{30} \)

Set these two expressions equal to each other:
\[
\frac{L}{3} = \frac{L + 900}{30}
\]

Step 4: Solve for \( L \)
Cross-multiply to solve for \( L \):
\[
30L = 3(L + 900)
\] Expand both sides:
\[
30L = 3L + 2700
\] Simplify:
\[
30L – 3L = 2700
\] \[
27L = 2700
\] \[
L = \frac{2700}{27} = 100 \, \text{meters}
\]

Step 5: Calculate the time taken to cross the 800-meter-long platform
Now that we know the length of the train is 100 meters, we can calculate the time it will take for the train to cross the 800-meter-long platform.

The total distance to be covered is the length of the train plus the length of the new platform:
\[
\text{Total distance} = 100 + 800 = 900 \, \text{meters}
\] The speed of the train is \( v = \frac{L}{3} = \frac{100}{3} \, \text{m/s} \).

Now, use the formula for time:
\[
\text{Time} = \frac{\text{Distance}}{\text{Speed}} = \frac{900}{\frac{100}{3}} = 900 \times \frac{3}{100} = 27 \, \text{seconds}
\]

Final Answer:
The train will take 27 seconds to cross a platform that is 800 meters long.