A and B are two pipes which can fill a tank individually in 20 minutes and 25 minutes respectively, however there is a leakage at the bottom of tank which can empty the filled tank in 30 minutes. If the tank is empty initialy, how much time will both the taps take to fill the tank (leakage is still there)?

Options

A. `13(9)/(17)` minutes
B. `17(3)/(16)` minutes
C. `17(11)/(17)` minutes
D. `12(6)/(17)` minutes

Answer

To solve the problem, we need to find out how long it will take for pipes A and B to fill the tank while accounting for the leakage. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the filling rates of pipes A and B and the leakage rate.

- **Pipe A** can fill the tank in 20 minutes.
- **Pipe B** can fill the tank in 25 minutes.
- The leakage can empty the tank in 30 minutes.

### Step 2: Calculate the filling rates.

1. **Filling rate of Pipe A**:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{20 \text{ minutes}} = \frac{1}{20} \text{ tanks per minute} \]

2. **Filling rate of Pipe B**:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{25 \text{ minutes}} = \frac{1}{25} \text{ tanks per minute} \]

3. **Emptying rate of the leakage**:
\[ \text{Rate of leakage} = \frac{1 \text{ tank}}{30 \text{ minutes}} = \frac{1}{30} \text{ tanks per minute} \]

### Step 3: Combine the rates to find the net filling rate.

The net filling rate when both pipes are open and considering the leakage is:
\[ \text{Net Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of leakage} \]
Substituting the values:
\[ \text{Net Rate} = \frac{1}{20} + \frac{1}{25} - \frac{1}{30} \]

### Step 4: Find a common denominator and calculate the net rate.

The least common multiple (LCM) of 20, 25, and 30 is 300. Now, we convert each rate to have a denominator of 300:
- For Pipe A:
\[ \frac{1}{20} = \frac{15}{300} \]
- For Pipe B:
\[ \frac{1}{25} = \frac{12}{300} \]
- For leakage:
\[ \frac{1}{30} = \frac{10}{300} \]

Now, substituting these into the net rate equation:
\[ \text{Net Rate} = \frac{15}{300} + \frac{12}{300} - \frac{10}{300} = \frac{15 + 12 - 10}{300} = \frac{17}{300} \text{ tanks per minute} \]

### Step 5: Calculate the time taken to fill the tank.

To find the time taken to fill one tank, we take the reciprocal of the net rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\text{Net Rate}} = \frac{1}{\frac{17}{300}} = \frac{300}{17} \text{ minutes} \]

### Step 6: Convert to a more understandable format.

Calculating \( \frac{300}{17} \):
\[ 300 \div 17 \approx 17.647 \text{ minutes} \]
This can be expressed as:
\[ 17 \text{ minutes and } \frac{11}{17} \text{ minutes} \text{ (approximately 0.647 minutes)} \]

### Final Answer:
Thus, the time taken to fill the tank with both pipes A and B open, while accounting for the leakage, is approximately:
\[ \text{17 minutes and } \frac{11}{17} \text{ minutes} \]