Problems On Trains All Competence Aptitude

Let's denote the length of each train as \( L \) meters.

When the faster train overtakes the slower train, the relative speed between them is the difference in their speeds.

Relative speed \( S_r = (46 - 36) \) km/hr = \( 10 \) km/hr.

Now, we convert this relative speed to meters per second (\( m/s \)):\[ 10 \text{ km/hr} = \dfrac{10 \times 1000}{3600} \text{ m/s} \]\[ = \dfrac{10000}{3600} \text{ m/s} \]\[ = \dfrac{25}{9} \text{ m/s} \]

Now, using the formula: Time = Distance / Speed, we can find the length of each train.

Since the faster train overtakes the slower train in 36 seconds, the distance it travels in this time is the length of the slower train plus the length of the faster train.

\[ L_{\text{slower train}} + L_{\text{faster train}} = \text{Distance covered by faster train} \]

\[ L + L = \left( \dfrac{25}{9} \times 36 \right) \]

\[ 2L = \dfrac{900}{9} \]

\[ 2L = 100 \]

\[ L = \dfrac{100}{2} = 50 \text{ meters} \]

So, the length of each train is \( 50 \) meters. Therefore, the answer is \( 50 \) meters.

Let's denote the length of each train as \( L \) meters.

When the faster train overtakes the slower train, the relative speed between them is the difference in their speeds.

Relative speed \( S_r = (46 - 36) \) km/hr = \( 10 \) km/hr.

Now, we convert this relative speed to meters per second (\( m/s \)):\[ 10 \text{ km/hr} = \dfrac{10 \times 1000}{3600} \text{ m/s} \]\[ = \dfrac{10000}{3600} \text{ m/s} \]\[ = \dfrac{25}{9} \text{ m/s} \]

Now, using the formula: Time = Distance / Speed, we can find the length of each train.

Since the faster train overtakes the slower train in 36 seconds, the distance it travels in this time is the length of the slower train plus the length of the faster train.

\[ L_{\text{slower train}} + L_{\text{faster train}} = \text{Distance covered by faster train} \]

\[ L + L = \left( \dfrac{25}{9} \times 36 \right) \]

\[ 2L = \dfrac{900}{9} \]

\[ 2L = 100 \]

\[ L = \dfrac{100}{2} = 50 \text{ meters} \]

So, the length of each train is \( 50 \) meters. Therefore, the answer is \( 50 \) meters.

Let's denote the length of each train as \( L \) meters.

When the faster train completely crosses the slower train, the relative speed between them is the difference in their speeds.

Relative speed \( S_r = (90 - 60) \) km/hr = \( 30 \) km/hr.

Now, we convert this relative speed to meters per second (\( m/s \)):\[ 30 \text{ km/hr} = \dfrac{30 \times 1000}{3600} \text{ m/s} \]\[ = \dfrac{30000}{3600} \text{ m/s} \]\[ = \dfrac{25}{3} \text{ m/s} \]

Now, using the formula: Time = Distance / Speed, we can find the length of each train.

Since the faster train completely crosses the slower train in 30 seconds, the distance it travels in this time is the length of the slower train plus the length of the faster train.

\[ L_{\text{slower train}} + L_{\text{faster train}} = \text{Distance covered by faster train} \]

\[ L + L = \left( \dfrac{25}{3} \times 30 \right) \]

\[ 2L = \dfrac{750}{3} \]

\[ 2L = 250 \]

\[ L = \dfrac{250}{2} = 125 \text{ meters} \]

So, the length of each train is \( 125 \) meters. Therefore, the answer is \( 125 \) meters.

First, let's find the speeds of the policeman and the thief.

The policeman covers 1 kilometer in 8 minutes, so his speed is:\[ \text{Speed of policeman} = \dfrac{1 \text{ km}}{8 \text{ min}} = \dfrac{1}{8} \text{ km/min} \]

Similarly, the thief covers 1 kilometer in 10 minutes, so his speed is:\[ \text{Speed of thief} = \dfrac{1 \text{ km}}{10 \text{ min}} = \dfrac{1}{10} \text{ km/min} \]

Now, let's denote the distance covered by the thief before he is overpowered as \( x \) meters. Since the policeman runs 100 meters faster per minute than the thief, the time it takes for the policeman to cover the same distance as the thief plus the 100-meter headstart is the same as the time it takes for the thief to cover the distance.

So, the equation becomes:\[ \text{Time taken by policeman to cover } x + 100 \text{ meters} = \text{Time taken by thief to cover } x \text{ meters} \]

Time is given by the formula: Distance/Speed.

So, we have:\[ \dfrac{x + 100}{\dfrac{1}{8}} = \dfrac{x}{\dfrac{1}{10}} \]

Now, let's solve for \( x \):

\[ \dfrac{x + 100}{\dfrac{1}{8}} = \dfrac{x}{\dfrac{1}{10}} \]\[ 8(x + 100) = 10x \]\[ 8x + 800 = 10x \]\[ 800 = 2x \]\[ x = 400 \]

So, the distance covered by the thief before he is overpowered is \( 400 \) meters. Therefore, the answer is \( 400 \) meters.

Let's denote the length of each train as \( L \) meters.

When the faster train overtakes the slower train, the relative speed between them is the difference in their speeds.

Relative speed \( S_r = (46 - 36) \) km/hr = \( 10 \) km/hr.

Now, we convert this relative speed to meters per second (\( m/s \)):\[ 10 \text{ km/hr} = \dfrac{10 \times 1000}{3600} \text{ m/s} \]\[ = \dfrac{10000}{3600} \text{ m/s} \]\[ = \dfrac{25}{9} \text{ m/s} \]

Now, using the formula: Time = Distance / Speed, we can find the length of each train.

Since the faster train overtakes the slower train in 36 seconds, the distance it travels in this time is the length of the slower train plus the length of the faster train.

\[ L_{\text{slower train}} + L_{\text{faster train}} = \text{Distance covered by faster train} \]

\[ L + L = \left( \dfrac{25}{9} \times 36 \right) \]

\[ 2L = \dfrac{900}{9} \]

\[ 2L = 100 \]

\[ L = \dfrac{100}{2} = 50 \text{ meters} \]

So, the length of each train is \( 50 \) meters. Therefore, the answer is \( 50 \) meters.

When one train passes another train moving in the same direction, the relative speed between them is the difference in their speeds.

Let's denote:
- Length of the first train \( L_1 = 50 \) meters
- Length of the second train \( L_2 = 75 \) meters
- Speed of the first train \( S_1 = 68 \) km/h
- Speed of the second train \( S_2 = 50 \) km/h

Relative speed \( S_r = S_1 - S_2 \) km/h.

Converting the relative speed to meters per second (\( m/s \)):\[ S_r = (68 - 50) \times \dfrac{5}{18} \]\[ S_r = 18 \times \dfrac{5}{18} \]\[ S_r = 5 \, \text{m/s} \]

Now, when one train passes another train, the distance to be covered is the sum of the lengths of the two trains.

So, the distance to be covered = \( L_1 + L_2 = 50 + 75 = 125 \) meters.

Now, using the formula: Time = Distance / Speed, we can find the time taken to pass:

\[ \text{Time} = \dfrac{\text{Distance}}{\text{Relative speed}} \]\[ \text{Time} = \dfrac{125}{5} \]\[ \text{Time} = 25 \text{ seconds} \]

So, it takes 25 seconds for the first train to pass the second train. Therefore, the answer is 25 seconds.

Let's denote the length of the train as \( L \) meters. 

When the train crosses the platform, it essentially covers the sum of its own length and the length of the platform.

Given that the speed of the train is \( 90 \) km/hr and it crosses the platform in \( 1 \) minute, which is \( \dfrac{1}{60} \) hours, we can use the formula:

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

The total distance covered is the sum of the lengths of the train and the platform.

So, we have:

\[ L + L = 90 \times \dfrac{1}{60} \]

\[ 2L = \dfrac{90}{60} \]

\[ 2L = \dfrac{3}{2} \]

\[ L = \dfrac{3}{4} \times 1000 \]

\[ L = 750 \text{ meters} \]

So, the length of the train is \( 750 \) meters. Therefore, the answer is \( 750 \) meters.

To find the time it takes for the train to cross the bridge, we need to consider the total distance the train needs to cover, which is the sum of the length of the train and the length of the bridge.

Given:
- Length of the train = 200 meters
- Speed of the train = 72 km/hr

Converting the speed from km/hr to m/s:\[ 72 \text{ km/hr} = \dfrac{72 \times 1000}{3600} \text{ m/s} = 20 \text{ m/s} \]

Now, the total distance to be covered is:\[ \text{Total distance} = \text{Length of train} + \text{Length of bridge} = 200 + 800 = 1000 \text{ meters} \]

Using the formula: Time = Distance / Speed, we can find the time taken to cross:\[ \text{Time} = \dfrac{\text{Total distance}}{\text{Speed}} = \dfrac{1000}{20} = 50 \text{ seconds} \]

So, it will take 50 seconds for the train to cross the bridge. Therefore, the answer is 50 seconds.

To find the speed of the train, we first need to calculate the total distance covered by the train while crossing both its own length and the length of the platform.

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

Total distance covered = Length of train + Length of platform = 500 + 700 = 1200 feet

Now, using the formula: Speed = Distance / Time, we can find the speed of the train:\[ \text{Speed} = \dfrac{\text{Total distance}}{\text{Time}} = \dfrac{1200 \text{ feet}}{10 \text{ seconds}} = 120 \text{ feet/second} \]

So, the speed of the train is 120 feet/second. Therefore, the answer is 120 feet/second.

To find the time it takes for the train to cross the platform, we need to consider the total distance the train needs to cover, which is the sum of the length of the train and the length of the platform.

Given:
- Speed of the train = 132 km/hr
- Length of the train = 110 meters
- Length of the platform = 165 meters

First, we need to convert the speed of the train from km/hr to m/s:\[ 132 \text{ km/hr} = \dfrac{132 \times 1000}{3600} \text{ m/s} = 36.67 \text{ m/s} \]

Now, the total distance to be covered is:\[ \text{Total distance} = \text{Length of train} + \text{Length of platform} = 110 + 165 = 275 \text{ meters} \]

Using the formula: Time = Distance / Speed, we can find the time taken to cross:\[ \text{Time} = \dfrac{\text{Total distance}}{\text{Speed}} = \dfrac{275 \text{ meters}}{36.67 \text{ m/s}} \]

Now, calculate the time:\[ \text{Time} ≈ \dfrac{275}{36.67} \approx 7.5 \text{ seconds} \]

So, it will take approximately 7.5 seconds for the train to cross the platform. Therefore, the answer is 7.5 seconds.