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. We will break it down step by step.

### Step 1: Define the Variables
Let:
- \( t_1 \) = time taken by the girl to walk up the stationary escalator.
- \( t_2 \) = time taken by the escalator to take the stationary girl to the top.
- \( x \) = length of the escalator.
- \( v_g \) = speed of the girl.
- \( v_e \) = speed of the escalator.

### Step 2: Calculate the Speeds
From the information given:
1. When the girl walks up the stationary escalator, she covers the distance \( x \) in time \( t_1 \):
\[ v_g = \frac{x}{t_1} \]

2. When the escalator moves with the girl stationary, it covers the same distance \( x \) in time \( t_2 \):
\[ v_e = \frac{x}{t_2} \]

### Step 3: Determine the Combined Speed
When both the girl and the escalator are moving together, their speeds add up since they are moving in the same direction. The effective speed when both are moving is:
\[ v_{g+e} = v_g + v_e \]

### Step 4: Substitute the Speeds
Substituting the expressions for \( v_g \) and \( v_e \) into the equation:
\[ v_{g+e} = \frac{x}{t_1} + \frac{x}{t_2} \]

### Step 5: Find the Time Taken to Walk Up the Moving Escalator
Let \( t \) be the time taken by the girl to walk up the moving escalator. The distance \( x \) can also be expressed in terms of \( t \) and the combined speed:
\[ x = v_{g+e} \cdot t \]
Substituting for \( v_{g+e} \):
\[ x = \left(\frac{x}{t_1} + \frac{x}{t_2}\right) t \]

### Step 6: Simplify the Equation
We can cancel \( x \) from both sides (assuming \( x \neq 0 \)):
\[ 1 = \left(\frac{1}{t_1} + \frac{1}{t_2}\right) t \]

### Step 7: Solve for \( t \)
Rearranging gives:
\[ t = \frac{1}{\left(\frac{1}{t_1} + \frac{1}{t_2}\right)} \]
This can be rewritten as:
\[ t = \frac{t_1 \cdot t_2}{t_1 + t_2} \]

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