A train overtakes two persons who are walking in the same direction as the train is moving , at the rate of 2 km/hr and 4km/hr and passes them completely in 9 and 10 seconds respectively. Find the speed and the length of the train ?

Answer

To solve the problem, we need to find the speed of the train and its length based on the information given about two persons walking in the same direction as the train. Let's break down the solution step by step.

### Step 1: Define Variables
Let the speed of the train be \( x \) km/hr.

### Step 2: Convert Speeds to m/s
Since we will be dealing with time in seconds, we need to convert the speeds of the persons from km/hr to m/s. The conversion factor is \( \frac{5}{18} \).

- Speed of the first person:
\[ 2 \text{ km/hr} = 2 \times \frac{5}{18} = \frac{10}{18} = \frac{5}{9} \text{ m/s} \]

- Speed of the second person:
\[ 4 \text{ km/hr} = 4 \times \frac{5}{18} = \frac{20}{18} = \frac{10}{9} \text{ m/s} \]

### Step 3: Relative Speed of the Train
The relative speed of the train with respect to each person is given by:
- For the first person:
\[ \text{Relative speed} = x - \frac{5}{9} \text{ m/s} \]

- For the second person:
\[ \text{Relative speed} = x - \frac{10}{9} \text{ m/s} \]

### Step 4: Use Time to Find Length of the Train
The length of the train can be calculated using the formula:
\[ \text{Length of the train} = \text{Relative speed} \times \text{Time} \]

For the first person:
\[ \text{Length of the train} = \left(x - \frac{5}{9}\right) \times 9 \]

For the second person:
\[ \text{Length of the train} = \left(x - \frac{10}{9}\right) \times 10 \]

### Step 5: Set Up the Equation
Since both expressions represent the length of the train, we can set them equal to each other:
\[ \left(x - \frac{5}{9}\right) \times 9 = \left(x - \frac{10}{9}\right) \times 10 \]

### Step 6: Simplify the Equation
Expanding both sides:
\[ 9x - 5 = 10x - \frac{100}{9} \]

Rearranging gives:
\[ 9x - 10x = -\frac{100}{9} + 5 \]
\[ -x = -\frac{100}{9} + \frac{45}{9} \]
\[ -x = -\frac{55}{9} \]
\[ x = \frac{55}{9} \text{ km/hr} \]

### Step 7: Calculate the Length of the Train
Now, substitute \( x \) back into one of the length equations. Using the first person's equation:
\[ \text{Length of the train} = \left(\frac{55}{9} - \frac{5}{9}\right) \times 9 \]
\[ = \left(\frac{50}{9}\right) \times 9 = 50 \text{ meters} \]

### Final Answer
- Speed of the train: \( \frac{55}{9} \) km/hr (approximately 6.11 km/hr)
- Length of the train: 50 meters