A can fill the tank in 6 min. and B can empty the filled tank in 10 min. If both time pipes are opened simultaneously then find the time taken by both the pipes to fill the tank.

Options

A. 12 min
B. 10 min
C. 15 min
D. 20 min

Answer

To solve the problem of how long it takes for both pipes A and B to fill the tank when opened simultaneously, we can follow these steps:

### Step-by-Step Solution:

1. **Identify the Rates of Work**:
- Pipe A can fill the tank in 6 minutes. Therefore, the rate of work of A is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{6 \text{ minutes}} = \frac{1}{6} \text{ tanks per minute} \]
- Pipe B can empty the tank in 10 minutes. Therefore, the rate of work of B is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{10 \text{ minutes}} = \frac{1}{10} \text{ tanks per minute} \]

2. **Combine the Rates**:
- Since A is filling the tank and B is emptying it, we can combine their rates:
\[ \text{Combined Rate} = \text{Rate of A} - \text{Rate of B} = \frac{1}{6} - \frac{1}{10} \]

3. **Find a Common Denominator**:
- The least common multiple (LCM) of 6 and 10 is 30. We convert the rates to have a common denominator:
\[ \frac{1}{6} = \frac{5}{30} \quad \text{and} \quad \frac{1}{10} = \frac{3}{30} \]

4. **Calculate the Combined Rate**:
- Now, substituting back into the combined rate:
\[ \text{Combined Rate} = \frac{5}{30} - \frac{3}{30} = \frac{2}{30} = \frac{1}{15} \text{ tanks per minute} \]

5. **Calculate the Time to Fill the Tank**:
- If the combined rate is \(\frac{1}{15}\) tanks per minute, then the time taken to fill one tank is the reciprocal of the rate:
\[ \text{Time} = \frac{1 \text{ tank}}{\frac{1}{15} \text{ tanks per minute}} = 15 \text{ minutes} \]

### Final Answer:
The time taken by both pipes to fill the tank when opened simultaneously is **15 minutes**.

---