A train crosses two persons going in same direction with respective speeds of 3 km./hr. and 5 km./hr. in 10 sec. and 11 sec. respectively. What is speed of train ?

Options

A. 28 km./hr.
B. 27 km./hr.
C. 25 km./hr.
D. 24 km./hr.

Answer

To find the speed of the train, we can use the information given about the two persons it crosses. Let's break it down step by step:

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

### Step 2: Understand Relative Speed
When the train crosses a person, the relative speed is the speed of the train minus the speed of the person.

- For the first person (speed = 3 km/hr):
\[ \text{Relative speed} = S - 3 \text{ km/hr} \]

- For the second person (speed = 5 km/hr):
\[ \text{Relative speed} = S - 5 \text{ km/hr} \]

### Step 3: Convert Time to Hours
The time taken to cross each person is given in seconds. We need to convert this time into hours since the speeds are in km/hr.

- For the first person:
\[ 10 \text{ seconds} = \frac{10}{3600} \text{ hours} = \frac{1}{360} \text{ hours} \]

- For the second person:
\[ 11 \text{ seconds} = \frac{11}{3600} \text{ hours} = \frac{11}{3600} \text{ hours} \]

### Step 4: Set Up Equations
The distance covered by the train while crossing each person is equal to the length of the train.

1. For the first person:
\[ \text{Distance} = \text{Relative speed} \times \text{Time} \]
\[ L = (S - 3) \times \frac{1}{360} \]

2. For the second person:
\[ L = (S - 5) \times \frac{11}{3600} \]

### Step 5: Equate the Two Distances
Since both expressions are equal to the length of the train \( L \), we can set them equal to each other:
\[ (S - 3) \times \frac{1}{360} = (S - 5) \times \frac{11}{3600} \]

### Step 6: Simplify the Equation
To eliminate the fractions, we can multiply both sides by \( 3600 \):
\[ 3600 \times (S - 3) \times \frac{1}{360} = 3600 \times (S - 5) \times \frac{11}{3600} \]
This simplifies to:
\[ 10(S - 3) = 11(S - 5) \]

### Step 7: Expand and Rearrange
Expanding both sides:
\[ 10S - 30 = 11S - 55 \]
Rearranging gives:
\[ 10S - 11S = -55 + 30 \]
\[ -S = -25 \]
Thus,
\[ S = 25 \text{ km/hr} \]

### Final Answer
The speed of the train is **25 km/hr**.