Problems On Trains All Competence Aptitude

To solve this problem, we'll use the concept of relative speed.

When the train crosses the bridge, it covers its own length as well as the length of the bridge. So, the total distance covered is the sum of the length of the train and the length of the bridge.

Given:
- Length of the train = 150 m
- Length of the bridge = 500 m

Total distance covered when crossing the bridge = Length of train + Length of bridge\[= 150 \text{ m} + 500 \text{ m} = 650 \text{ m}\]

The time taken to cross the bridge is 30 seconds.

Now, let's find the speed of the train.\[ \text{Speed} = \dfrac{\text{Distance}}{\text{Time}} = \dfrac{650 \text{ m}}{30 \text{ s}} = \dfrac{65}{3} \text{ m/s}\]

Now, the train needs to cross a platform 370 m long. When crossing the platform, it covers its own length, the length of the platform, and the distance it travels relative to the platform. The relative distance covered is the length of the train plus the length of the platform.\[ \text{Relative distance} = \text{Length of train} + \text{Length of platform} = 150 \text{ m} + 370 \text{ m} = 520 \text{ m}\]

Now, let's find the time taken to cross the platform.\[ \text{Time} = \dfrac{\text{Relative distance}}{\text{Speed}} = \dfrac{520 \text{ m}}{\dfrac{65}{3} \text{ m/s}} = \dfrac{520 \times 3}{65} \text{ s} \]

\[ \text{Time} = \dfrac{1560}{65} \text{ s} = 24 \text{ s}\]

Therefore, the train will take 24 seconds to cross the platform.

So, the correct answer is 24 seconds.

To solve this problem, we first need to convert the speed of the train from kilometers per hour (km/h) to meters per second (m/s) since the length of the bridge is given in meters.

Given:
- Length of the train = 200 m
- Speed of the train = 36 km/h

First, let's convert the speed from km/h to m/s.\[ \text{Speed} = 36 \text{ km/h} \times \dfrac{1000 \text{ m}}{3600 \text{ s}} = 10 \text{ m/s} \]

Now, the train needs to cover its own length as well as the length of the bridge during the time it takes to cross the bridge.

Given:
- Time taken to cross the bridge = 55 seconds

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

The total distance covered when crossing the bridge is the sum of the length of the train and the length of the bridge.\[ \text{Total distance} = \text{Length of train} + \text{Length of bridge} \]\[ = 200 \text{ m} + L \text{ m} \]

We know that:\[ \text{Speed} = \dfrac{\text{Distance}}{\text{Time}} \]\[ 10 \text{ m/s} = \dfrac{200 \text{ m} + L \text{ m}}{55 \text{ s}} \]

Now, let's solve for \( L \):\[ L = 10 \text{ m/s} \times 55 \text{ s} - 200 \text{ m} \]\[ L = 550 \text{ m} - 200 \text{ m} \]\[ L = 350 \text{ m} \]

Therefore, the length of the bridge is 350 meters.

So, the correct answer is 350 m.

To find the length of the bridge, we first need to convert the speed of the man from kilometers per hour (km/h) to meters per minute since the time taken to cross the bridge is given in minutes.

Given:
- Speed of the man = 15 km/h
- Time taken to cross the bridge = 5 minutes

First, let's convert the speed from km/h to meters per minute.\[ \text{Speed} = 15 \text{ km/h} \times \dfrac{1000 \text{ m}}{60 \text{ minutes}} = 250 \text{ m/minute} \]

Now, the man needs to cover his own distance as well as the length of the bridge during the time it takes to cross the bridge.

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

The total distance covered when crossing the bridge is the sum of the speed of the man and the length of the bridge.\[ \text{Total distance} = \text{Speed of man} \times \text{Time taken} \]\[ = 250 \text{ m/minute} \times 5 \text{ minutes} \]\[ = 1250 \text{ meters} \]

Now, we know that the total distance covered is equal to the length of the bridge.\[ L = 1250 \text{ meters} \]

Therefore, the length of the bridge is 1250 meters.

So, the correct answer is 1250 meters.

To find the time taken by the train to cross the bridge, we need to calculate the total distance that the train needs to cover and then divide it by the speed of the train.

Given:
- Length of the train = 200 meters
- Speed of the train = 54 km/hr
- Length of the bridge = 175 meters

First, let's convert the speed of the train from kilometers per hour (km/h) to meters per second (m/s) since the lengths are given in meters.\[ \text{Speed} = 54 \text{ km/hr} \times \dfrac{1000 \text{ m}}{3600 \text{ s}} = 15 \text{ m/s} \]

Now, let's calculate the total distance that the train needs to cover to cross both the train and the bridge.
Total distance = Length of train + Length of bridge\[ = 200 \text{ m} + 175 \text{ m} = 375 \text{ m} \]

Now, let's calculate the time taken by the train to cover this distance.\[ \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} = \dfrac{375 \text{ m}}{15 \text{ m/s}} = 25 \text{ seconds} \]

Therefore, the time taken by the train to cross the 175-meter long bridge is 25 seconds.

So, the correct answer is 25 seconds.

To find the time taken by the train to cross a man standing on the platform, we need to calculate the time it takes for the front of the train to travel a distance equal to its length past the man.

Given:
- Length of the train = 75 meters
- Speed of the train = 20 km/h

First, let's convert the speed of the train from kilometers per hour (km/h) to meters per second (m/s) since the lengths are given in meters and we need to keep units consistent.\[ \text{Speed} = 20 \text{ km/h} \times \dfrac{1000 \text{ m}}{3600 \text{ s}} = \dfrac{50}{9} \text{ m/s} \]

Now, let's calculate the time it takes for the train to cross the man.\[ \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} = \dfrac{75 \text{ m}}{\dfrac{50}{9} \text{ m/s}} = \dfrac{75 \times 9}{50} \text{ seconds} = \dfrac{675}{50} \text{ seconds} = 13.5 \text{ seconds} \]

Therefore, the time taken by the train to cross a man standing on the platform is 13.5 seconds.

So, the correct answer is 13.5 seconds.

Let's first convert the speed of the man from kilometers per hour to meters per second to ensure units are consistent:

\[ 3 \text{ km/hr} = \dfrac{3 \times 1000}{3600} \text{ m/s} = \dfrac{5}{6} \text{ m/s} \]

Now, let's denote the speed of the train as \( v \) m/s. 

When the train passes the man, both the train and the man cover the length of the train. So, the relative speed between them is the sum of their speeds.

Relative speed \( = \text{Speed of train} - \text{Speed of man} \)\[ = v - \dfrac{5}{6} \text{ m/s} \]

The time taken for the train to pass the man is given as 33 seconds.

\[ \text{Distance} = \text{Speed} \times \text{Time} \]\[ 300 = \left( v - \dfrac{5}{6} \right) \times 33 \]

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

\[ 300 = 33v - \dfrac{165}{6} \]\[ 300 + \dfrac{165}{6} = 33v \]\[ 33v = 300 + \dfrac{165}{6} \]\[ 33v = \dfrac{1800 + 165}{6} \]\[ 33v = \dfrac{1965}{6} \]\[ v = \dfrac{1965}{6 \times 33} \]\[ v = \dfrac{1965}{198} \]\[ v \approx 9.924 \text{ m/s} \]

Finally, let's convert the speed of the train from meters per second to kilometers per hour:

\[ \text{Speed of train} = 9.924 \times \dfrac{3600}{1000} \]\[ \text{Speed of train} \approx 35.728 \text{ km/h} \]

Therefore, the speed of the train is approximately \( 35.73 \text{ km/h} \), which when simplified is \( \dfrac{360}{11} \) km/h or \( 32 \dfrac{8}{11} \) km/h.

So, the correct answer is \( 32 \dfrac{8}{11} \) km/h.

To find the speed of the train, we'll use the formula:

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

Given:
- Length of the train (\( L \)) = 120 m
- Time taken to pass the telegraph post (\( T \)) = 6 seconds

Substituting the given values into the formula:

\[ \text{Speed} = \dfrac{120 \text{ m}}{6 \text{ s}} = 20 \text{ m/s} \]

Now, let's convert the speed from meters per second to kilometers per hour since it's commonly expressed in that unit:

\[ \text{Speed} = 20 \text{ m/s} \times \dfrac{3600 \text{ s}}{1000 \text{ m}} = 72 \text{ km/hr} \]

Therefore, the speed of the train is 72 km/hr.

So, the correct answer is 72 km/hr.

To find the speed of the train, we'll use the formula:

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

Given:
- Length of the train (\( L \)) = 250 m
- Time taken to cross the tree (\( T \)) = 50 seconds

Substituting the given values into the formula:

\[ \text{Speed} = \dfrac{250 \text{ m}}{50 \text{ s}} = 5 \text{ m/s} \]

Now, let's convert the speed from meters per second to kilometers per hour since it's commonly expressed in that unit:

\[ \text{Speed} = 5 \text{ m/s} \times \dfrac{3600 \text{ s}}{1000 \text{ m}} = 18 \text{ km/hr} \]

Therefore, the speed of the train is 18 km/hr.

So, the correct answer is 18 km/hr.

To find the time it takes for the train to pass the telegraph post, we'll first convert the speed of the train from kilometers per hour to meters per second. Then, we'll use the formula for time:

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

Given:
- Length of the train (\( L \)) = 60 m
- Speed of the train = 36 km/hr

First, let's convert the speed from kilometers per hour to meters per second:\[ \text{Speed} = 36 \text{ km/hr} \times \dfrac{1000 \text{ m}}{3600 \text{ s}} = 10 \text{ m/s} \]

Now, let's use the formula to find the time:\[ \text{Time} = \dfrac{60 \text{ m}}{10 \text{ m/s}} = 6 \text{ seconds} \]

Therefore, the train will pass the telegraph post in 6 seconds.

So, the correct answer is 6 seconds.

When two trains are moving in opposite directions, the distance they cover relative to each other is the sum of their lengths.

Given:
- Length of the first train (\(L_1\)) = 140 m
- Length of the second train (\(L_2\)) = 160 m
- Speed of the first train (\(S_1\)) = 60 km/hr
- Speed of the second train (\(S_2\)) = 40 km/hr

First, let's convert the speeds from kilometers per hour to meters per second, as we need to keep the units consistent:\[ S_1 = 60 \times \dfrac{1000}{3600} = \dfrac{50}{3} \text{ m/s} \]\[ S_2 = 40 \times \dfrac{1000}{3600} = \dfrac{100}{9} \text{ m/s} \]

Now, let's find the relative speed of the two trains:\[ \text{Relative speed} = S_1 + S_2 = \dfrac{50}{3} + \dfrac{100}{9} \]

To add these fractions, we need to find a common denominator:\[ \text{Relative speed} = \dfrac{150}{9} + \dfrac{100}{9} = \dfrac{250}{9} \text{ m/s} \]

Now, let's find the time taken for the trains to cross each other:\[ \text{Time} = \dfrac{\text{Distance}}{\text{Relative speed}} = \dfrac{140 + 160}{\dfrac{250}{9}} = \dfrac{300 \times 9}{250} \]\[ \text{Time} = \dfrac{2700}{250} = 10.8 \text{ seconds} \]

Therefore, the time taken for the trains to cross each other is 10.8 seconds.

So, the correct answer is 10.8 seconds.