A train passes two persons walking in the same direction in which the train is going. The persons are walking at the rate of 5kmph and 8 kmph respectively passes them completely in 20 and 25 seconds respectively. Find the speed of the train.

Options

A. 20 km//hr
B. 28 km//hr
C. 18 km//hr
D. 15 km//hr

Answer

To find the speed of the train, we can follow these steps:

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

### Step 2: Convert time into hours
The time taken to pass the first person is 20 seconds, and for the second person, it is 25 seconds. We need to convert these times into hours:
- For the first person:
\[ \text{Time} = \frac{20}{3600} \text{ hours} = \frac{1}{180} \text{ hours} \]
- For the second person:
\[ \text{Time} = \frac{25}{3600} \text{ hours} = \frac{5}{720} \text{ hours} \]

### Step 3: Set up the equations using distance
The distance (length of the train) can be expressed in terms of speed and time. The distance \( D \) is the same for both persons.

For the first person (walking at 5 km/h):
\[ D = \text{Relative Speed} \times \text{Time} = (x - 5) \times \frac{1}{180} \]

For the second person (walking at 8 km/h):
\[ D = \text{Relative Speed} \times \text{Time} = (x - 8) \times \frac{5}{720} \]

### Step 4: Equate the two expressions for distance
Since both expressions represent the same distance \( D \):
\[ (x - 5) \times \frac{1}{180} = (x - 8) \times \frac{5}{720} \]

### Step 5: Clear the fractions by multiplying through by 720
\[ 720 \times (x - 5) \times \frac{1}{180} = 720 \times (x - 8) \times \frac{5}{720} \]
This simplifies to:
\[ 4(x - 5) = 5(x - 8) \]

### Step 6: Expand and simplify the equation
Expanding both sides:
\[ 4x - 20 = 5x - 40 \]

### Step 7: Rearrange to solve for \( x \)
Rearranging gives:
\[ 40 - 20 = 5x - 4x \]
\[ 20 = x \]

### Step 8: Conclusion
The speed of the train is \( x = 20 \) km/h.