Problems On Trains All Competence Aptitude

Let's denote:
- Distance between stations A and B as \( D \) km.
- Time taken for the trains to meet as \( t \) hours.

Given:
- Speed of the first train = 50 kmph
- Speed of the second train = 60 kmph
- Distance traveled by the second train when they meet = 120 km more than the distance traveled by the first train.

Since they are traveling towards each other, their relative speed is the sum of their speeds:

\[ \text{Relative speed} = 50 \text{ kmph} + 60 \text{ kmph} = 110 \text{ kmph} \]

Now, using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), we can find the distances traveled by each train:

- Distance traveled by the first train = \( 50 \times t \) km
- Distance traveled by the second train = \( 60 \times t \) km

Given that the second train has traveled 120 km more than the first when they meet:

\[ 60 \times t = 50 \times t + 120 \]

\[ 60t - 50t = 120 \]

\[ 10t = 120 \]

\[ t = 12 \text{ hours} \]

Now, let's find the distance between stations A and B using the time taken by the trains to meet:

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

\[ D = 50 \times 12 \]

\[ D = 600 \text{ km} \]

So, the distance between stations A and B is 600 km. However, this is the distance traveled by both trains combined. Since they are traveling towards each other, the actual distance between stations A and B would be the sum of the distances traveled by both trains. 

\[ \text{Actual distance between A and B} = 50 \times 12 + 60 \times 12 \]

\[ \text{Actual distance between A and B} = 600 + 720 \]

\[ \text{Actual distance between A and B} = 1320 \text{ km} \]

Therefore, the correct answer is 1320 km.

To find the time it takes for the two trains to pass each other, we first determine the total distance to be covered and the relative speed at which they approach one another.

Step 1: Calculate the Total Distance
When two trains pass each other, the total distance covered is the sum of their lengths:

\[ \text{Total Distance} = 180 \text{ m} + 120 \text{ m} = 300 \text{ m} \]

Step 2: Calculate the Relative Speed
Since the trains are moving toward each other, their relative speed is the sum of their individual speeds. First, let's find the combined speed in km/h:

\[ \text{Relative Speed (km/h)} = 65 \text{ km/h} + 55 \text{ km/h} = 120 \text{ km/h} \]

Now, we convert this to m/s using the conversion factor \( \displaystyle \frac{5}{18} \):

\[ \displaystyle \text{Relative Speed (m/s)} = 120 \times \frac{5}{18} = \frac{600}{18} = \frac{100}{3} \text{ m/s} \]

Step 3: Calculate the Time Taken
Using the formula \( \displaystyle \text{Time} = \frac{\text{Distance}}{\text{Speed}} \):

\[ \displaystyle \text{Time} = \frac{300}{\left(\frac{100}{3}\right)} \] \[ \displaystyle \text{Time} = 300 \times \frac{3}{100} = 3 \times 3 = 9 \text{ seconds} \]

The two trains will pass each other in 9 seconds.

Let's first find out how far each train travels in 45 minutes:

For the first train with a speed of 45 km/hr:\[ \text{Distance} = \text{Speed} \times \text{Time} \]\[ \text{Distance} = 45 \times \dfrac{45}{60} \]\[ \text{Distance} = 33.75 \text{ km} \]

For the second train with a speed of 40 km/hr:\[ \text{Distance} = \text{Speed} \times \text{Time} \]\[ \text{Distance} = 40 \times \dfrac{45}{60} \]\[ \text{Distance} = 30 \text{ km} \]

Now, let's find the distance between the two trains after 45 minutes:

The distance between the two trains is the difference in their distances traveled in 45 minutes:

\[ \text{Distance between the two trains} = 33.75 \text{ km} - 30 \text{ km} \]\[ \text{Distance between the two trains} = 3.75 \text{ km} \]

We need to convert this into meters to match the format of the answer choices. Since 1 kilometer is equal to 1000 meters:

\[ 3.75 \text{ km} = 3 \text{ km} + 0.75 \times 1000 \text{ meters} \]\[ 3.75 \text{ km} = 3 \text{ km} + 750 \text{ meters} \]

Therefore, the distance between the two trains after 45 minutes is 3 km 750 m. Thus, the answer is 3 km 750 m.

Let's denote:
- Distance between stations A and B as \( D \) km.
- Time taken for the trains to meet as \( t \) hours.

Given:
- Speed of the first train = 75 km/hr
- Speed of the second train = 50 km/hr
- One train travels 175 km more than the other when they meet.

Since they are traveling towards each other, their relative speed is the sum of their speeds:

\[ \text{Relative speed} = 75 \text{ km/hr} + 50 \text{ km/hr} = 125 \text{ km/hr} \]

Now, using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), we can find the distances traveled by each train:

- Distance traveled by the first train = \( 75 \times t \) km
- Distance traveled by the second train = \( 50 \times t \) km

Given that one train travels 175 km more than the other when they meet:

\[ 75 \times t - 50 \times t = 175 \]

\[ 25t = 175 \]

\[ t = 7 \text{ hours} \]

Now, let's find the distance between stations A and B using the time taken by the trains to meet:

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

\[ D = 75 \times 7 \]

\[ D = 525 \text{ km} \]

So, the distance between stations A and B is 525 km. However, this is the distance traveled by both trains combined. Since they are traveling towards each other, the actual distance between stations A and B would be the sum of the distances traveled by both trains. 

\[ \text{Actual distance between A and B} = 75 \times 7 + 50 \times 7 \]

\[ \text{Actual distance between A and B} = 525 + 350 \]

\[ \text{Actual distance between A and B} = 875 \text{ km} \]

Therefore, the correct answer is 875 km.

When two trains are moving towards each other and they cross each other, they cover the sum of their lengths.

Given:
- Length of the first train = 137 meters
- Length of the second train = 163 meters
- Speed of the first train = 42 km/hr
- Speed of the second train = 48 km/hr

First, let's convert the speeds from km/h to m/s, because the lengths are given in meters and we want to find the time in seconds.

Speed of the first train = \( \dfrac{42 \times 1000}{3600} \) m/s ≈ \( \dfrac{42000}{3600} \) m/s ≈ \( \dfrac{35}{3} \) m/s

Speed of the second train = \( \dfrac{48 \times 1000}{3600} \) m/s ≈ \( \dfrac{48000}{3600} \) m/s ≈ \( \dfrac{40}{3} \) m/s

The relative speed of the two trains is the sum of their speeds, as they are moving towards each other:

Relative speed = Speed of the first train + Speed of the second train

Relative speed = \( \dfrac{35}{3} + \dfrac{40}{3} \) m/s = \( \dfrac{75}{3} \) m/s = \( 25 \) m/s

Now, using the formula \( \text{Time} = \dfrac{\text{Distance}}{\text{Speed}} \), we can find the time taken for them to cross each other, which is the time taken to cover the sum of their lengths.

Total distance = Length of first train + Length of second train
Total distance = 137 meters + 163 meters = 300 meters

Time = \( \dfrac{300}{25} \) seconds
Time = \( 12 \) seconds

Therefore, they will cross each other in \( 12 \) seconds. So, the correct answer is 12 seconds.

Let's denote:
- Distance between the two men initially = 1200 meters
- Rate at which the first man walks = 5 m/minute
- Rate at which the second man walks = 10 m/minute
- Time taken for them to meet = \( t \) minutes

As they are walking towards each other, their relative speed is the sum of their individual speeds.

Relative speed = Speed of the first man + Speed of the second man
Relative speed = \( 5 + 10 = 15 \) m/minute

Now, using the formula \( \text{Distance} = \text{Speed} \times \text{Time} \), we can find the time taken for them to meet, which is the time taken to cover the distance between them.

\[ \text{Distance} = \text{Speed} \times \text{Time} \]\[ 1200 = 15 \times t \]

Solving for \( t \):\[ t = \dfrac{1200}{15} = 80 \text{ minutes} \]

Therefore, they will meet each other after 80 minutes. So, the correct answer is 80 minutes.

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) = 150 meters
- Time taken to pass a telegraphic post = 12 seconds

Substitute the given values into the formula:

\[ \text{Speed} = \dfrac{150 \text{ meters}}{12 \text{ seconds}} \]

Now, we need to convert the speed from meters per second to kilometers per hour because the options are given in km/hr.

To convert from meters per second to kilometers per hour, we'll use the conversion factor: 1 m/s = 3.6 km/hr.

So, the speed of the train is:

\[ \text{Speed} = \dfrac{150 \text{ meters}}{12 \text{ seconds}} \times \dfrac{3600 \text{ seconds}}{1000 \text{ meters}} \]

\[ \text{Speed} = \dfrac{150 \times 3600}{12 \times 1000} \text{ km/hr} \]

\[ \text{Speed} = \dfrac{540000}{12000} \text{ km/hr} \]

\[ \text{Speed} = 45 \text{ km/hr} \]

Therefore, the speed of the train is 45 km/hr. So, the correct option is 45.