A pipe can fill a tank in 'x hours and another pipe can empty it in 'y' `(y gt x)` hours. If both the pipes are open, in how many hours will the tank be filled ?

Options

A. `(x-y)` hours
B. `(y-x)` hours
C. `(xy)/(x-y)` hours
D. `(xy)/(y-x)` hours

Answer

To solve the problem step by step, we need to determine how long it will take to fill the tank when both a filling pipe and an emptying pipe are open.

### Step-by-Step Solution:

1. **Understand the Problem**:
- We have two pipes:
- Pipe A (filling pipe) can fill the tank in 'x' hours.
- Pipe B (emptying pipe) can empty the tank in 'y' hours, where y > x.

2. **Calculate the Rates**:
- The rate of Pipe A (filling) is \( \frac{1}{x} \) of the tank per hour.
- The rate of Pipe B (emptying) is \( \frac{1}{y} \) of the tank per hour.

3. **Determine the Combined Rate**:
- When both pipes are open, the effective rate of filling the tank is:
\[ \text{Effective Rate} = \text{Rate of Pipe A} - \text{Rate of Pipe B} = \frac{1}{x} - \frac{1}{y} \]

4. **Find a Common Denominator**:
- To combine the rates, find a common denominator:
\[ \frac{1}{x} - \frac{1}{y} = \frac{y - x}{xy} \]

5. **Calculate the Time to Fill the Tank**:
- The time taken to fill the tank when both pipes are open can be calculated using the formula:
\[ \text{Time} = \frac{1}{\text{Effective Rate}} = \frac{1}{\left(\frac{y - x}{xy}\right)} = \frac{xy}{y - x} \]

6. **Final Answer**:
- Therefore, the time taken to fill the tank when both pipes are open is:
\[ \text{Time} = \frac{xy}{y - x} \text{ hours} \]