Preeti reached the metro station and found that the escalator was not working. She walked up the stationary escalator in time `t_1`. On other days, if the remains stationary on the moving escalator, then the escalator takes her up in time `t_2`. The time taken by her to walk up on the moving escalator will be :

Options

A. `(t_1 t_2)/(t_2 - t_1)`
B. `(t_1 t_2)/(t_2 + t_1)`
C. `t_1 - t_2`
D. `(t_1 + t_2)/(2)`

Explanation

To solve the problem, we need to find the time taken by Preeti to walk up on the moving escalator. We will use the information given about her walking speed on a stationary escalator and the speed of the escalator itself.

### Step-by-Step Solution:

1. **Define Variables**:
- Let \( L \) be the length of the escalator.
- Let \( v_p \) be Preeti's walking speed.
- Let \( v_e \) be the speed of the escalator.
- Let \( t_1 \) be the time taken by Preeti to walk up the stationary escalator.
- Let \( t_2 \) be the time taken by the escalator to take her up when she is stationary.

2. **Calculate Preeti's Walking Speed**:
- When Preeti walks up the stationary escalator, her speed is \( v_p = \frac{L}{t_1} \).

3. **Calculate Escalator Speed**:
- When the escalator takes her up while she is stationary, the speed of the escalator is \( v_e = \frac{L}{t_2} \).

4. **Calculate Combined Speed on Moving Escalator**:
- When Preeti walks up the moving escalator, her effective speed relative to the ground is the sum of her walking speed and the escalator speed:
\[ v_{total} = v_p + v_e = \frac{L}{t_1} + \frac{L}{t_2} \]

5. **Express Total Speed in Terms of Time**:
- Let \( t \) be the time taken by Preeti to walk up the moving escalator. The distance covered is still \( L \), so we can express the total speed as:
\[ v_{total} = \frac{L}{t} \]

6. **Set Up the Equation**:
- Equating the two expressions for total speed:
\[ \frac{L}{t} = \frac{L}{t_1} + \frac{L}{t_2} \]

7. **Simplify the Equation**:
- Cancel \( L \) from both sides (assuming \( L \neq 0 \)):
\[ \frac{1}{t} = \frac{1}{t_1} + \frac{1}{t_2} \]

8. **Find Time \( t \)**:
- Taking the reciprocal gives:
\[ t = \frac{t_1 t_2}{t_1 + t_2} \]

### Final Answer:
The time taken by Preeti to walk up on the moving escalator is:
\[ t = \frac{t_1 t_2}{t_1 + t_2} \]