At a metro station, a girl walks up a stationary escalator in time `t_1` If she remains stationary on the escalator, then the escalator take her up in time `t_2`. The time taken by her to walk up 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`

Explanation

To solve the problem, we need to find the time taken by the girl to walk up the moving escalator. Let's break it down step by step.

### Step 1: Understand the Problem
We have two scenarios:
1. The girl walks up a stationary escalator in time \( t_1 \).
2. The escalator takes her up when she is stationary on it in time \( t_2 \).

### Step 2: Define Variables
Let:
- \( L \) = Length of the escalator
- \( v_g \) = Velocity of the girl
- \( v_e \) = Velocity of the escalator

From the first scenario, we can express the velocity of the girl as:
\[ v_g = \frac{L}{t_1} \]

From the second scenario, the velocity of the escalator can be expressed as:
\[ v_e = \frac{L}{t_2} \]

### Step 3: Determine the Total Velocity on the Moving Escalator
When the girl walks up the moving escalator, her effective velocity will be the sum of her velocity and the escalator's velocity:
\[ v_{total} = v_g + v_e = \frac{L}{t_1} + \frac{L}{t_2} \]

### Step 4: Calculate the Time Taken on the Moving Escalator
The time taken \( t \) to walk up the moving escalator can be found using the formula:
\[ t = \frac{L}{v_{total}} \]
Substituting \( v_{total} \):
\[ t = \frac{L}{\left(\frac{L}{t_1} + \frac{L}{t_2}\right)} \]

### Step 5: Simplify the Expression
We can simplify the expression:
\[ t = \frac{L}{L\left(\frac{1}{t_1} + \frac{1}{t_2}\right)} = \frac{1}{\left(\frac{1}{t_1} + \frac{1}{t_2}\right)} \]
This can be rewritten as:
\[ t = \frac{t_1 t_2}{t_1 + t_2} \]

### Conclusion
Thus, the time taken by the girl to walk up the moving escalator is:
\[ t = \frac{t_1 t_2}{t_1 + t_2} \]