A train passes two persons who are walking in the direction opposite to the direction of train at the rate of 10 m/s and 20 m/s respectively in 12 s and 10 s, respectively. Find the length of the train.

Options

A. a. 500m
B. b. 900m
C. c. 400m
D. d. 600m

Answer

To solve the problem of finding the length of the train, we can follow these steps:

### Step 1: Understand the scenario
The train is moving in one direction, while two persons are walking in the opposite direction. The speeds of the persons are given as 10 m/s and 20 m/s.

### Step 2: Calculate the relative speed of the train with respect to each person
When two objects are moving towards each other, their relative speed is the sum of their speeds.

1. For the first person (10 m/s):
- Let the speed of the train be \( V_t \).
- Relative speed with respect to the first person = \( V_t + 10 \) m/s.

2. For the second person (20 m/s):
- Relative speed with respect to the second person = \( V_t + 20 \) m/s.

### Step 3: Use the time taken to pass each person to find the length of the train
The length of the train can be calculated using the formula:
\[ \text{Length of Train} = \text{Relative Speed} \times \text{Time} \]

1. For the first person (12 seconds):
\[ \text{Length of Train} = (V_t + 10) \times 12 \]

2. For the second person (10 seconds):
\[ \text{Length of Train} = (V_t + 20) \times 10 \]

### Step 4: Set the two expressions for the length of the train equal to each other
Since both expressions represent the length of the train, we can set them equal:
\[ (V_t + 10) \times 12 = (V_t + 20) \times 10 \]

### Step 5: Expand and simplify the equation
Expanding both sides:
\[ 12V_t + 120 = 10V_t + 200 \]

### Step 6: Rearrange the equation to solve for \( V_t \)
\[ 12V_t - 10V_t = 200 - 120 \]
\[ 2V_t = 80 \]
\[ V_t = 40 \text{ m/s} \]

### Step 7: Substitute \( V_t \) back to find the length of the train
Using the value of \( V_t \) in one of the length equations, we can find the length of the train:
\[ \text{Length of Train} = (40 + 10) \times 12 = 50 \times 12 = 600 \text{ meters} \]

### Final Answer
The length of the train is **600 meters**.

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