A train overtakes two girls who are walking in the opposite direction in which the train is going at the rate of 3 km/h and 6km/hr and passes them completely in 36 seconds and 30 seconds respectively. The length of the train (in metres) is :

Options

A. 120 m
B. 150 m
C. 125 m
D. none of these

Answer

To solve the problem of finding the length of the train that overtakes two girls walking in opposite directions, we can follow these steps:

### Step 1: Define Variables
Let:
- \( x \) = length of the train in meters
- \( y \) = speed of the train in km/h

### Step 2: Calculate Relative Speed for the First Girl
The first girl walks at a speed of 3 km/h. Since she is walking in the opposite direction to the train, the relative speed of the train with respect to the first girl is:
\[ \text{Relative Speed} = y + 3 \text{ km/h} \]

### Step 3: Convert Relative Speed to m/s
To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \):
\[ \text{Relative Speed in m/s} = \left(y + 3\right) \times \frac{5}{18} \]

### Step 4: Use the Time Taken to Pass the First Girl
The time taken to pass the first girl is 36 seconds. The distance covered by the train while passing her is equal to the length of the train:
\[ x = \left(y + 3\right) \times \frac{5}{18} \times 36 \]

### Step 5: Simplify the Equation for the First Girl
Now we can simplify this equation:
\[ x = (y + 3) \times 10 \]
\[ x = 10y + 30 \quad \text{(Equation 1)} \]

### Step 6: Calculate Relative Speed for the Second Girl
The second girl walks at a speed of 6 km/h. The relative speed of the train with respect to the second girl is:
\[ \text{Relative Speed} = y + 6 \text{ km/h} \]

### Step 7: Convert Relative Speed to m/s for the Second Girl
Again, converting to m/s:
\[ \text{Relative Speed in m/s} = \left(y + 6\right) \times \frac{5}{18} \]

### Step 8: Use the Time Taken to Pass the Second Girl
The time taken to pass the second girl is 30 seconds. The distance covered by the train while passing her is also equal to the length of the train:
\[ x = \left(y + 6\right) \times \frac{5}{18} \times 30 \]

### Step 9: Simplify the Equation for the Second Girl
Simplifying this equation gives:
\[ x = (y + 6) \times \frac{25}{3} \]
\[ x = \frac{25y}{3} + 50 \quad \text{(Equation 2)} \]

### Step 10: Set the Two Equations Equal
Since both equations represent the length of the train \( x \), we can set them equal to each other:
\[ 10y + 30 = \frac{25y}{3} + 50 \]

### Step 11: Solve for \( y \)
To eliminate the fraction, multiply the entire equation by 3:
\[ 30y + 90 = 25y + 150 \]
Rearranging gives:
\[ 30y - 25y = 150 - 90 \]
\[ 5y = 60 \]
\[ y = 12 \text{ km/h} \]

### Step 12: Substitute \( y \) Back to Find \( x \)
Now substitute \( y = 12 \) into Equation 1:
\[ x = 10(12) + 30 \]
\[ x = 120 + 30 \]
\[ x = 150 \text{ meters} \]

### Final Answer
The length of the train is **150 meters**.

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