Two pipes A and B can fill a cistern in 15 Fours and 10 hours respectively. A tap C can empty the full cistern in 30 hours. All the three taps were open for 2 hours, when it was remembered that the emptying tap had been left open. It was then closed. How many hours more would it take for the cistern to be filled?

Options

A. 30 min
B. 1.2 hours
C. 4.4 hours
D. 35 min

Answer

To solve the problem step by step, we will first determine the rates at which each pipe works, then calculate the net effect when all three pipes are open for 2 hours, and finally find out how much longer it will take to fill the cistern after the emptying tap is closed.

### Step 1: Determine the rates of filling and emptying
1. **Pipe A** fills the cistern in 15 hours.
- Rate of A = \( \frac{1}{15} \) of the cistern per hour.

2. **Pipe B** fills the cistern in 10 hours.
- Rate of B = \( \frac{1}{10} \) of the cistern per hour.

3. **Pipe C** empties the cistern in 30 hours.
- Rate of C = \( -\frac{1}{30} \) of the cistern per hour (negative because it empties).

### Step 2: Calculate the combined rate when all taps are open
- Combined rate when A, B, and C are open:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} \]
\[ = \frac{1}{15} + \frac{1}{10} - \frac{1}{30} \]

### Step 3: Find a common denominator and simplify
- The least common multiple (LCM) of 15, 10, and 30 is 30.
- Convert each rate to have a denominator of 30:
\[ = \frac{2}{30} + \frac{3}{30} - \frac{1}{30} = \frac{2 + 3 - 1}{30} = \frac{4}{30} = \frac{2}{15} \]

### Step 4: Calculate the amount filled in 2 hours
- In 2 hours, the amount filled by the combined rate:
\[ \text{Amount filled in 2 hours} = 2 \times \frac{2}{15} = \frac{4}{15} \]

### Step 5: Determine the remaining amount to be filled
- The total capacity of the cistern is considered as 1 (or 15/15).
- Remaining amount to be filled:
\[ \text{Remaining} = 1 - \frac{4}{15} = \frac{15 - 4}{15} = \frac{11}{15} \]

### Step 6: Calculate the new rate after closing tap C
- After closing tap C, the combined rate of A and B:
\[ \text{New Combined Rate} = \frac{1}{15} + \frac{1}{10} \]
\[ = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \]

### Step 7: Calculate the time required to fill the remaining amount
- Time required to fill the remaining \( \frac{11}{15} \) at the rate of \( \frac{1}{6} \):
\[ \text{Time} = \text{Remaining Amount} \div \text{New Rate} = \frac{11}{15} \div \frac{1}{6} = \frac{11}{15} \times 6 = \frac{66}{15} = 4.4 \text{ hours} \]

### Final Answer
It would take an additional 4.4 hours to fill the cistern after closing the emptying tap.

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