Two men are running in the same direction with a speed of 6 km/hr and `7(1)/(2)` km/hr. A train running in the same direction crosses them in 5 sec and `5(1)/(2)` sec respectively. The length and the speed of the train are

Options

A. 22.92 m (approx) and 22 km/hr
B. 22 m (approx) and 22.5 km/hr
C. 22.90 m (approx) and 20.5 km/hr
D. 22.92 m (approx) and 22.5 km/hr

Answer

To solve the problem step by step, we need to find the length and speed of the train based on the information given about the two men running in the same direction.

### Step-by-Step Solution:

1. **Convert Speeds to m/s**:
- The speeds of the two men are given in km/hr. We convert them to m/s using the conversion factor \( \frac{5}{18} \).
- Speed of the first man:
\[ 6 \text{ km/hr} = 6 \times \frac{5}{18} = \frac{30}{18} = \frac{5}{3} \text{ m/s} \]
- Speed of the second man:
\[ 7.5 \text{ km/hr} = 7.5 \times \frac{5}{18} = \frac{37.5}{18} = \frac{25}{12} \text{ m/s} \]

2. **Calculate Time Taken to Cross Each Man**:
- The train crosses the first man in 5 seconds and the second man in 5.5 seconds.

3. **Set Up the Equations**:
- Let \( L \) be the length of the train and \( V \) be the speed of the train in m/s.
- For the first man:
\[ L = (V - \frac{5}{3}) \times 5 \]
- For the second man:
\[ L = (V - \frac{25}{12}) \times 5.5 \]

4. **Equate the Two Expressions for Length**:
- Since both expressions equal \( L \), we can set them equal to each other:
\[ (V - \frac{5}{3}) \times 5 = (V - \frac{25}{12}) \times 5.5 \]

5. **Expand and Simplify**:
- Expanding both sides:
\[ 5V - \frac{25}{3} = 5.5V - \frac{137.5}{12} \]
- To eliminate fractions, multiply the entire equation by 12:
\[ 60V - 100 = 66V - 137.5 \]

6. **Rearranging the Equation**:
- Rearranging gives:
\[ 66V - 60V = 137.5 - 100 \]
\[ 6V = 37.5 \]
\[ V = \frac{37.5}{6} = 6.25 \text{ m/s} \]

7. **Calculate Length of the Train**:
- Substitute \( V \) back into one of the equations to find \( L \):
\[ L = (6.25 - \frac{5}{3}) \times 5 \]
- Convert \( \frac{5}{3} \) to a decimal:
\[ \frac{5}{3} \approx 1.67 \]
- Thus,
\[ L = (6.25 - 1.67) \times 5 = 4.58 \times 5 = 22.92 \text{ meters} \]

8. **Final Results**:
- The speed of the train is \( 6.25 \text{ m/s} \) or \( 22.5 \text{ km/hr} \).
- The length of the train is approximately \( 22.92 \text{ meters} \).