Two trains 100 metres and 95 metres long respectively pass each other in 27 seconds when they run in the same direction and in 9 seconds when they run in opposite directions. Speed of the two trains are

Options

A. 44km/hr, 22km/hr
B. 52km/hr, 26km/hr
C. 36km/hr, 18 km/hr
D. 40km/hr, 20km/hr

Answer

To solve the problem, we need to find the speeds of the two trains based on the information provided about their lengths and the time taken to pass each other in different directions.

### Step 1: Understand the Problem
We have two trains:
- Train A: Length = 100 meters
- Train B: Length = 95 meters

When they run in the same direction, they take 27 seconds to pass each other. When they run in opposite directions, they take 9 seconds to pass each other.

### Step 2: Calculate the Total Length of the Trains
When the two trains pass each other, the distance covered is the sum of their lengths.

Total length = Length of Train A + Length of Train B
Total length = 100 m + 95 m = 195 m

### Step 3: Set Up the Equations
Let the speed of Train A be \( x \) m/s and the speed of Train B be \( y \) m/s.

**Case 1: Same Direction**
When the trains are moving in the same direction, the relative speed is \( (x - y) \) m/s. The time taken to pass each other is 27 seconds.

Using the formula:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
We can write:
\[ 195 = (x - y) \times 27 \]
So,
\[ x - y = \frac{195}{27} \]
\[ x - y = 7.222 \, \text{m/s} \] (approximately)

**Case 2: Opposite Directions**
When the trains are moving in opposite directions, the relative speed is \( (x + y) \) m/s. The time taken to pass each other is 9 seconds.

Using the same formula:
\[ 195 = (x + y) \times 9 \]
So,
\[ x + y = \frac{195}{9} \]
\[ x + y = 21.666 \, \text{m/s} \] (approximately)

### Step 4: Solve the Equations
Now we have a system of two equations:
1. \( x - y = 7.222 \)
2. \( x + y = 21.666 \)

We can solve these equations by adding them together:
\[ (x - y) + (x + y) = 7.222 + 21.666 \]
\[ 2x = 28.888 \]
\[ x = 14.444 \, \text{m/s} \]

Now, substitute \( x \) back into one of the equations to find \( y \):
Using \( x + y = 21.666 \):
\[ 14.444 + y = 21.666 \]
\[ y = 21.666 - 14.444 \]
\[ y = 7.222 \, \text{m/s} \]

### Step 5: Convert Speeds to km/h
To convert the speeds from m/s to km/h, we multiply by \( \frac{18}{5} \):
- Speed of Train A:
\[ x = 14.444 \times \frac{18}{5} \approx 52.000 \, \text{km/h} \]
- Speed of Train B:
\[ y = 7.222 \times \frac{18}{5} \approx 26.000 \, \text{km/h} \]

### Final Answer
- Speed of Train A: \( 52 \, \text{km/h} \)
- Speed of Train B: \( 26 \, \text{km/h} \)