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))/(2)`
B. `(t_(1)t_(2))/(t_(2)-t_(1))`
C. `(t_(1)t_(2))/(t_(2)+t_(1))`
D. `t_(1)-t_(2)`

Answer

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

### 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 Preeti up when she is stationary.

2. **Calculate Preeti's Speed**:
- When Preeti walks up the stationary escalator, she covers the length \( L \) in time \( t_1 \).
- Therefore, her speed can be expressed as:
\[ v_P = \frac{L}{t_1} \]

3. **Calculate Escalator's Speed**:
- When Preeti is stationary on the moving escalator, it takes time \( t_2 \) to cover the same length \( L \).
- Thus, the speed of the escalator is:
\[ v_E = \frac{L}{t_2} \]

4. **Determine the Time Taken on the Moving Escalator**:
- When both Preeti and the escalator are moving, her effective speed with respect to the ground is the sum of her walking speed and the escalator's speed:
\[ v_{PG} = v_P + v_E = \frac{L}{t_1} + \frac{L}{t_2} \]
- The total time \( t \) taken to cover the length \( L \) on the moving escalator can be calculated using the formula:
\[ t = \frac{L}{v_{PG}} = \frac{L}{\left(\frac{L}{t_1} + \frac{L}{t_2}\right)} \]

5. **Simplify the Expression**:
- Factor out \( L \) from the denominator:
\[ t = \frac{L}{L\left(\frac{1}{t_1} + \frac{1}{t_2}\right)} = \frac{1}{\left(\frac{1}{t_1} + \frac{1}{t_2}\right)} \]
- This can be rewritten as:
\[ t = \frac{t_1 \cdot t_2}{t_1 + t_2} \]

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