A train overtakes two persons who are walking at the rate of 4km/h and8km/h in the same direction and passes them completely in 18 and 20s,respectively.Find the length of the train

Answer

To find the length of the train that overtakes two persons walking at different speeds, we can follow these steps:

### Step 1: Understand the problem
The train overtakes two persons walking at speeds of 4 km/h and 8 km/h. The time taken to completely pass the first person is 18 seconds, and for the second person, it is 20 seconds. We need to find the length of the train.

### Step 2: Convert speeds from km/h to m/s
To work with time in seconds, we need to convert the speeds of the persons from km/h to m/s.

- Speed of the first person:
\[ 4 \text{ km/h} = \frac{4 \times 1000}{3600} \text{ m/s} = \frac{4000}{3600} \text{ m/s} = \frac{10}{9} \text{ m/s} \]

- Speed of the second person:
\[ 8 \text{ km/h} = \frac{8 \times 1000}{3600} \text{ m/s} = \frac{8000}{3600} \text{ m/s} = \frac{20}{9} \text{ m/s} \]

### Step 3: Set up equations for the length of the train
Let \( L \) be the length of the train and \( S \) be the speed of the train in m/s.

1. For the first person:
\[ \text{Relative speed} = S - \frac{10}{9} \]
The distance covered (length of the train) in 18 seconds is:
\[ L = (S - \frac{10}{9}) \times 18 \]

2. For the second person:
\[ \text{Relative speed} = S - \frac{20}{9} \]
The distance covered (length of the train) in 20 seconds is:
\[ L = (S - \frac{20}{9}) \times 20 \]

### Step 4: Equate the two expressions for \( L \)
From the two equations, we can set them equal to each other:
\[ (S - \frac{10}{9}) \times 18 = (S - \frac{20}{9}) \times 20 \]

### Step 5: Simplify the equation
Expanding both sides:
\[ 18S - 20 = 20S - \frac{400}{9} \]

Rearranging gives:
\[ 20S - 18S = 20 - \frac{400}{9} \]
\[ 2S = 20 - \frac{400}{9} \]

Converting 20 to a fraction:
\[ 20 = \frac{180}{9} \]
Thus,
\[ 2S = \frac{180}{9} - \frac{400}{9} = \frac{-220}{9} \]
\[ S = \frac{-110}{9} \text{ (not possible, recheck)} \]

### Step 6: Correct the calculation
Revisiting the equation:
\[ 18S - 20 = 20S - \frac{400}{9} \]
Rearranging correctly:
\[ 2S = 20 + \frac{400}{9} \]
Convert 20 to a fraction:
\[ 20 = \frac{180}{9} \]
So,
\[ 2S = \frac{180 + 400}{9} = \frac{580}{9} \]
Thus,
\[ S = \frac{290}{9} \text{ m/s} \]

### Step 7: Calculate the length of the train
Substituting \( S \) back into one of the equations for \( L \):
Using the first equation:
\[ L = (S - \frac{10}{9}) \times 18 \]
Substituting \( S \):
\[ L = \left(\frac{290}{9} - \frac{10}{9}\right) \times 18 = \left(\frac{280}{9}\right) \times 18 = \frac{5040}{9} = 560 \text{ meters} \]

### Final Answer
The length of the train is **560 meters**.