A train 150 metres long crosses a man walking at a speed of 6 km/hr, in the opposite direction in 6 seconds. The speed of the train is:

Options

A. 96 km/hr.
B. 84 km/hr.
C. 106 km/hr.
D. 66 km/hr.

Answer

To solve the problem step by step, let's break it down:

### Step 1: Understand the Problem
We have a train that is 150 meters long and crosses a man walking at a speed of 6 km/hr in the opposite direction in 6 seconds. We need to find the speed of the train.

### Step 2: Calculate the Distance
The distance that the train needs to cover to completely cross the man is equal to the length of the train, which is:
- Distance = 150 meters

### Step 3: Convert the Man's Speed to Meters per Second
The speed of the man is given in km/hr. We need to convert this speed to meters per second (m/s) for easier calculations:
- Speed of the man = 6 km/hr
- To convert km/hr to m/s, we use the conversion factor: \( \text{Speed in m/s} = \text{Speed in km/hr} \times \frac{5}{18} \)
- So, \( 6 \times \frac{5}{18} = \frac{30}{18} = \frac{5}{3} \) m/s

### Step 4: Calculate the Relative Speed
The train crosses the man in 6 seconds. We can calculate the relative speed of the train with respect to the man using the formula:
- Relative Speed = Distance / Time
- Relative Speed = \( \frac{150 \text{ meters}}{6 \text{ seconds}} = 25 \text{ m/s} \)

### Step 5: Set Up the Equation for Relative Speed
Since the train and the man are moving in opposite directions, we can express the relative speed as:
- Relative Speed = Speed of the train + Speed of the man
Let the speed of the train be \( x \) m/s. Then:
- \( 25 = x + \frac{5}{3} \)

### Step 6: Solve for the Speed of the Train
Now, we can solve for \( x \):
- Rearranging the equation gives us:
\( x = 25 - \frac{5}{3} \)
- To subtract \( \frac{5}{3} \) from 25, convert 25 to a fraction:
\( 25 = \frac{75}{3} \)
- Now, subtract:
\( x = \frac{75}{3} - \frac{5}{3} = \frac{70}{3} \) m/s

### Step 7: Convert the Speed of the Train to km/hr
Now, we need to convert the speed of the train from m/s to km/hr:
- Speed in km/hr = Speed in m/s \( \times \frac{18}{5} \)
- So, \( x = \frac{70}{3} \times \frac{18}{5} = \frac{70 \times 18}{3 \times 5} = \frac{1260}{15} = 84 \) km/hr

### Final Answer
The speed of the train is **84 km/hr**.

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