Problems On Trains All Competence Aptitude

To find the time taken by the train to cross the electric pole, we first need to convert its speed from kilometers per hour to meters per second, as the length of the train is given in meters.

Given:
- Length of the train = 160 meters
- Speed of the train = 72 km/hr

First, let's convert the speed of the train from kilometers per hour to meters per second:\[ \text{Speed} = 72 \times \dfrac{1000}{3600} = 20 \, \text{m/s} \]

Now, using the formula:\[ \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} \]

We can find the time taken by the train to cross the electric pole:\[ \text{Time} = \dfrac{160 \, \text{m}}{20 \, \text{m/s}} \]\[ \text{Time} = 8 \, \text{seconds} \]

So, the time taken by the train to cross the electric pole is \( \textbf{8 seconds} \).

Therefore, the correct answer is \( \textbf{8 seconds} \).

To find the time taken by the train to pass a person standing near the line, we need to consider that the length of the train and the speed of the train are given.

Given:
- Length of the train = 100 meters
- Speed of the train = 120 km/hr

First, let's convert the speed of the train from kilometers per hour to meters per second, as the length of the train is given in meters.\[ \text{Speed} = 120 \times \dfrac{1000}{3600} = 33.33 \, \text{m/s} \]

Now, using the formula:\[ \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} \]

We can find the time taken by the train to pass the person standing near the line:\[ \text{Time} = \dfrac{100 \, \text{m}}{33.33 \, \text{m/s}} \]\[ \text{Time} \approx 3 \, \text{seconds} \]

So, the time taken by the train to pass a person standing near the line is approximately \( \textbf{3 seconds} \).

Therefore, the correct answer is \( \textbf{3 seconds} \).

To find the time taken by the train to cross a pillar, we need to consider that the length of the train and the speed of the train are given.

Given:
- Length of the train = 100 meters
- Speed of the train = 50 km/hr

First, let's convert the speed of the train from kilometers per hour to meters per second, as the length of the train is given in meters.\[ \text{Speed} = 50 \times \dfrac{1000}{3600} = \dfrac{125}{9} \, \text{m/s} \]

Now, using the formula:\[ \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} \]

We can find the time taken by the train to cross the pillar:\[ \text{Time} = \dfrac{100 \, \text{m}}{\dfrac{125}{9} \, \text{m/s}} \]\[ \text{Time} = \dfrac{100 \times 9}{125} \, \text{seconds} \]\[ \text{Time} = \dfrac{900}{125} \, \text{seconds} \]\[ \text{Time} = 7.2 \, \text{seconds} \]

So, the time taken by the train to cross a pillar is \( \textbf{7.2 seconds} \).

Therefore, the correct answer is \( \textbf{7.2 seconds} \).

To find the time taken by the train to pass a post, we need to consider that the length of the train and the speed of the train are given.

Given:
- Length of the train = 180 meters
- Speed of the train = 90 km/hr

First, let's convert the speed of the train from kilometers per hour to meters per second, as the length of the train is given in meters.\[ \text{Speed} = 90 \times \dfrac{1000}{3600} = 25 \, \text{m/s} \]

Now, using the formula:\[ \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} \]

We can find the time taken by the train to pass the post:\[ \text{Time} = \dfrac{180 \, \text{m}}{25 \, \text{m/s}} \]\[ \text{Time} = 7.2 \, \text{seconds} \]

So, the time taken by the train to pass a post is \( \textbf{7.2 seconds} \).

Therefore, the correct answer is \( \textbf{7.2 seconds} \).

To find the length of the train, we need to use the formula for speed:

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

Given:
- Speed of the train = 90 km/hr
- Time taken to cross the signal = 10 seconds

First, let's convert the speed of the train from kilometers per hour to meters per second:\[ \text{Speed} = 90 \times \dfrac{1000}{3600} = 25 \, \text{m/s} \]

Now, using the formula, we can rearrange it to solve for distance (length of the train):\[ \text{Distance} = \text{Speed} \times \text{Time} \]

\[ \text{Distance} = 25 \times 10 = 250 \, \text{meters} \]

So, the length of the train is \( \textbf{250 meters} \).

Therefore, the correct answer is \( \textbf{250 meters} \).

Let's denote the speed of the train as \( v \) km/h.

The relative speed of the train with respect to the first person is \( v - 3 \) km/h, and with respect to the second person is \( v - 5 \) km/h.

Given:
- Time taken to pass the first person = 10 seconds
- Time taken to pass the second person = 11 seconds

We know that Distance = Speed × Time. 

For the first person:\[ (v - 3) \times \dfrac{10}{3600} = \dfrac{v - 3}{360} \]

For the second person:\[ (v - 5) \times \dfrac{11}{3600} = \dfrac{v - 5}{327.27} \]

Since the distances covered by the train to pass both persons are the same, we equate the two distances:

\[ \dfrac{v - 3}{360} = \dfrac{v - 5}{327.27} \]

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

\[ 327.27(v - 3) = 360(v - 5) \]\[ 327.27v - 981.81 = 360v - 1800 \]\[ 360v - 327.27v = 1800 - 981.81 \]\[ 32.73v = 818.19 \]\[ v = \dfrac{818.19}{32.73} \]\[ v ≈ 24.99 \]

So, the speed of the train is approximately \( \textbf{24.99 km/h} \).

Therefore, the correct answer is \( \textbf{25 km/h} \) (rounded to the nearest integer).

To find the time it takes for the trains to cross each other, we can use the concept of relative motion.

When two objects are moving towards each other, their relative speed is the sum of their individual speeds.

Given:
- Length of the first train = 105 meters
- Speed of the first train = 45 km/h
- Length of the second train = 90 meters
- Speed of the second train = 72 km/h

First, let's convert the speeds from kilometers per hour to meters per second, as the lengths of the trains are given in meters and we want the answer in seconds.

For the first train:\[ \text{Speed} = 45 \times \dfrac{1000}{3600} = 12.5 \, \text{m/s} \]

For the second train:\[ \text{Speed} = 72 \times \dfrac{1000}{3600} = 20 \, \text{m/s} \]

Now, let's find the relative speed of the trains:\[ \text{Relative speed} = \text{Speed of first train} + \text{Speed of second train} \]\[ \text{Relative speed} = 12.5 \, \text{m/s} + 20 \, \text{m/s} = 32.5 \, \text{m/s} \]

Now, we can use the formula:\[ \text{Time} = \dfrac{\text{Total distance}}{\text{Relative speed}} \]

The total distance is the sum of the lengths of the trains:\[ \text{Total distance} = 105 \, \text{m} + 90 \, \text{m} = 195 \, \text{m} \]

\[ \text{Time} = \dfrac{195 \, \text{m}}{32.5 \, \text{m/s}} \]\[ \text{Time} = 6 \, \text{seconds} \]

So, the time it takes for the trains to cross each other is \( \textbf{6 seconds} \).

Therefore, the correct answer is \( \textbf{6 seconds} \).

Let's denote the speed of the first train as \( v_1 \) km/hr.

Given:
- Length of the first train = 150 meters
- Length of the second train = 120 meters
- Speed of the second train = 43.2 km/hr
- Time taken for the trains to cross each other = 10 seconds

To find the speed of the first train, we first need to find the relative speed of the two trains when they are moving in opposite directions.

Relative speed of the trains = Sum of their speeds

For the second train:\[ \text{Speed} = 43.2 \, \text{km/hr} = \dfrac{43.2 \times 1000}{3600} = 12 \, \text{m/s} \]

Now, let's use the formula for relative speed:\[ \text{Relative speed} = v_1 + 12 \, \text{m/s} \]

Given that the total distance covered when the two trains pass each other is the sum of their lengths:\[ 150 + 120 = 270 \, \text{meters} \]

Using the formula for speed:\[ \text{Speed} = \dfrac{\text{Distance}}{\text{Time}} \]\[ \text{Relative speed} = \dfrac{270}{10} = 27 \, \text{m/s} \]

Now, we have:\[ v_1 + 12 = 27 \]\[ v_1 = 27 - 12 \]\[ v_1 = 15 \, \text{m/s} \]

Now, let's convert this speed back to km/hr:\[ v_1 = 15 \times \dfrac{3600}{1000} = 54 \, \text{km/hr} \]

So, the speed of the first train is \( \textbf{54 km/hr} \).

Therefore, the correct answer is \( \textbf{54 km/hr} \).

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

Given:
- Speed of the train = 84 km/hr
- Speed of the man = 6 km/hr
- Time taken for the train to pass the man = 4 seconds

First, let's convert the speeds from kilometers per hour to meters per second, as the time is given in seconds and we want the length of the train in meters.

For the train:\[ \text{Speed} = 84 \times \dfrac{1000}{3600} = 23.33 \, \text{m/s} \]

For the man:\[ \text{Speed} = 6 \times \dfrac{1000}{3600} = 1.67 \, \text{m/s} \]

When the train passes the man, the relative speed between them is the sum of their speeds, as they are moving in opposite directions:\[ \text{Relative speed} = 23.33 \, \text{m/s} + 1.67 \, \text{m/s} = 25 \, \text{m/s} \]

Now, using the formula:\[ \text{Distance} = \text{Speed} \times \text{Time} \]

We can find the length of the train:\[ L = 25 \times 4 = 100 \, \text{meters} \]

So, the length of the train is \( \textbf{100 meters} \).

Therefore, the correct answer is \( \textbf{100 meters} \).

Let's denote the speed of the second train as \( v \) m/s.

Given:
- Length of the train = 150 meters
- Time taken for the first train to pass the pole = 15 seconds
- Time taken for the second train (of the same length) to pass the first train in the opposite direction = 12 seconds

First, let's find the speed of the first train when it passes the pole:\[ \text{Speed} = \dfrac{\text{Distance}}{\text{Time}} = \dfrac{150 \, \text{m}}{15 \, \text{s}} = 10 \, \text{m/s} \]

Now, let's consider the relative speed when the two trains are moving towards each other. The relative speed is the sum of their individual speeds since they are moving in opposite directions.

For the second train:\[ \text{Relative speed} = v + 10 \, \text{m/s} \]

The distance covered by the second train in 12 seconds is the sum of its length and the length of the first train:\[ 150 \, \text{m} + 150 \, \text{m} = 300 \, \text{m} \]

Using the formula for speed:\[ \text{Speed} = \dfrac{\text{Distance}}{\text{Time}} = \dfrac{300 \, \text{m}}{12 \, \text{s}} = 25 \, \text{m/s} \]

Now, we have:\[ v + 10 = 25 \]\[ v = 25 - 10 = 15 \, \text{m/s} \]

Now, let's convert this speed from meters per second to kilometers per hour:\[ v = 15 \times \dfrac{3600}{1000} = 54 \, \text{km/hr} \]

So, the speed of the second train is \( \textbf{54 km/hr} \).

Therefore, the correct answer is \( \textbf{54 km/hr} \).