A train overtakes two persons who are walking in the same direction in which the train is going, at the rate of 2 kmph and 4 kmph and passes them completely in 9 and 10 seconds respectively. The length of the train is -----

Options

A. 72 metres
B. 54 metres
C. 50 metres
D. 45 metres

Answer

To find the length of the train, we will use the information provided about the two persons being overtaken by the train.

### Step-by-Step Solution:

1. **Identify Variables:**
- Let the speed of the train be \( x \) km/h.
- Let the length of the train be \( L \) meters.
- The speed of the first person is 2 km/h.
- The speed of the second person is 4 km/h.

2. **Convert Speeds to m/s:**
- To convert km/h to m/s, we use the conversion factor \( \frac{5}{18} \).
- Speed of the first person in m/s: \( 2 \times \frac{5}{18} = \frac{10}{18} = \frac{5}{9} \) m/s.
- Speed of the second person in m/s: \( 4 \times \frac{5}{18} = \frac{20}{18} = \frac{10}{9} \) m/s.
- Speed of the train in m/s: \( x \times \frac{5}{18} \).

3. **Set Up Equations Using Distance Formula:**
- The distance covered by the train to completely pass the first person in 9 seconds:
\[ L = (x - 2) \times \frac{5}{18} \times 9 \]
Simplifying gives:
\[ L = (x - 2) \times \frac{5}{2} \]
Rearranging gives:
\[ 2L = 5x - 10 \quad \text{(Equation 1)} \]

- The distance covered by the train to completely pass the second person in 10 seconds:
\[ L = (x - 4) \times \frac{5}{18} \times 10 \]
Simplifying gives:
\[ L = (x - 4) \times \frac{25}{9} \]
Rearranging gives:
\[ 9L = 25x - 100 \quad \text{(Equation 2)} \]

4. **Solve the System of Equations:**
- From Equation 1: \( 2L = 5x - 10 \)
- From Equation 2: \( 9L = 25x - 100 \)

- We can express \( L \) from Equation 1:
\[ L = \frac{5x - 10}{2} \]
- Substitute \( L \) in Equation 2:
\[ 9\left(\frac{5x - 10}{2}\right) = 25x - 100 \]
Simplifying gives:
\[ \frac{45x - 90}{2} = 25x - 100 \]
Multiplying through by 2 to eliminate the fraction:
\[ 45x - 90 = 50x - 200 \]
Rearranging gives:
\[ 200 - 90 = 50x - 45x \]
\[ 110 = 5x \]
\[ x = 22 \quad \text{(speed of the train)} \]

5. **Find Length of the Train:**
- Substitute \( x = 22 \) back into Equation 1:
\[ 2L = 5(22) - 10 \]
\[ 2L = 110 - 10 = 100 \]
\[ L = 50 \text{ meters} \]

### Final Answer:
The length of the train is **50 meters**.