A train travelling at the speed of x km/h crossed a 260 m long piatform in 30 seconds and overtook a man walking in the same direction at the speed of 6 km.h in 20 seconds. What is the value of x ?

Options

A. 50
B. 56
C. 54
D. 81.6

Answer

To solve the problem step by step, we need to derive the speed of the train (x) using the information provided about the train crossing a platform and overtaking a man walking in the same direction.

### Step 1: Convert the speed of the train to meters per second
The speed of the train is given as \( x \) km/h. To convert this to meters per second, we use the conversion factor:
\[ \text{Speed in m/s} = \frac{x \times 1000}{3600} = \frac{5x}{18} \]

### Step 2: Set up the first equation using the platform crossing
When the train crosses a 260 m long platform in 30 seconds, the total distance covered by the train is the length of the train (let's denote it as \( L \)) plus the length of the platform (260 m). The equation for distance is:
\[ \text{Distance} = \text{Speed} \times \text{Time} \]
Thus, we have:
\[ L + 260 = \left(\frac{5x}{18}\right) \times 30 \]
This simplifies to:
\[ L + 260 = \frac{150x}{18} \]
\[ L + 260 = \frac{25x}{3} \]
Rearranging gives us:
\[ L = \frac{25x}{3} - 260 \quad \text{(Equation 1)} \]

### Step 3: Set up the second equation using the man overtaking
The man walks at a speed of 6 km/h, which in meters per second is:
\[ \text{Speed of man} = \frac{6 \times 1000}{3600} = \frac{5}{3} \text{ m/s} \]
When the train overtakes the man, it covers its own length \( L \) in 20 seconds. The relative speed of the train with respect to the man is:
\[ \text{Relative speed} = \frac{5x}{18} - \frac{5}{3} = \frac{5x}{18} - \frac{30}{18} = \frac{5(x - 6)}{18} \]
Thus, the distance covered (length of the train) can be expressed as:
\[ L = \left(\frac{5(x - 6)}{18}\right) \times 20 \]
This simplifies to:
\[ L = \frac{100(x - 6)}{18} = \frac{50(x - 6)}{9} \quad \text{(Equation 2)} \]

### Step 4: Equate the two expressions for \( L \)
From Equation 1 and Equation 2, we can set them equal to each other:
\[ \frac{25x}{3} - 260 = \frac{50(x - 6)}{9} \]

### Step 5: Solve for \( x \)
To eliminate the fractions, multiply through by 9:
\[ 9\left(\frac{25x}{3}\right) - 9(260) = 50(x - 6) \]
This simplifies to:
\[ 75x - 2340 = 50x - 300 \]
Rearranging gives:
\[ 75x - 50x = 2340 - 300 \]
\[ 25x = 2040 \]
\[ x = \frac{2040}{25} = 81.6 \text{ km/h} \]

### Conclusion
The speed of the train \( x \) is 81.6 km/h.