Two pipes A and B can fill a cistern in 15 hours and ’10 hours respectively. A tape 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. 4 hr 30 min
B. 1.2 hours
C. 4 hr 24 min
D. 3 hr 35 min

Answer

To solve the problem step-by-step, we will calculate the rates of filling and emptying the cistern and then determine how long it will take to fill the remaining water after the emptying tap has been closed.

### Step 1: Determine the filling rates of pipes A and B, and the emptying rate of pipe C.
- **Pipe A** can fill the cistern in 15 hours. Therefore, its rate is:
\[ \text{Rate of A} = \frac{1 \text{ cistern}}{15 \text{ hours}} = \frac{2}{30} \text{ cisterns per hour} \]

- **Pipe B** can fill the cistern in 10 hours. Therefore, its rate is:
\[ \text{Rate of B} = \frac{1 \text{ cistern}}{10 \text{ hours}} = \frac{3}{30} \text{ cisterns per hour} \]

- **Pipe C** can empty the cistern in 30 hours. Therefore, its rate is:
\[ \text{Rate of C} = \frac{1 \text{ cistern}}{30 \text{ hours}} = \frac{1}{30} \text{ cisterns per hour} \]

### Step 2: Calculate the combined rate when all pipes are open.
When all three pipes are open, the combined rate is:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} - \text{Rate of C} \]
Substituting the rates:
\[ \text{Combined Rate} = \frac{2}{30} + \frac{3}{30} - \frac{1}{30} = \frac{4}{30} = \frac{2}{15} \text{ cisterns per hour} \]

### Step 3: Calculate the amount of water filled in 2 hours.
In 2 hours, the amount of water filled is:
\[ \text{Water filled} = \text{Combined Rate} \times \text{Time} = \frac{2}{15} \times 2 = \frac{4}{15} \text{ cisterns} \]

### Step 4: Determine the remaining water to be filled.
The total capacity of the cistern is 1 cistern. Therefore, the remaining water to be filled is:
\[ \text{Remaining water} = 1 - \frac{4}{15} = \frac{15 - 4}{15} = \frac{11}{15} \text{ cisterns} \]

### Step 5: Calculate the time required to fill the remaining water with pipes A and B only.
Now, we will only consider pipes A and B since pipe C is closed. The combined rate of pipes A and B is:
\[ \text{Combined Rate of A and B} = \frac{2}{30} + \frac{3}{30} = \frac{5}{30} = \frac{1}{6} \text{ cisterns per hour} \]
To find the time required to fill the remaining \(\frac{11}{15}\) cisterns:
\[ \text{Time} = \frac{\text{Remaining water}}{\text{Combined Rate of A and B}} = \frac{\frac{11}{15}}{\frac{1}{6}} = \frac{11}{15} \times 6 = \frac{66}{15} = 4.4 \text{ hours} \]

### Step 6: Convert the time into hours and minutes.
To convert 4.4 hours into hours and minutes:
- The whole number part is 4 hours.
- The decimal part \(0.4\) hours is converted to minutes:
\[ 0.4 \times 60 = 24 \text{ minutes} \]

Thus, the total time required to fill the cistern after closing the emptying tap is **4 hours and 24 minutes**.

### Final Answer:
The cistern will take **4 hours and 24 minutes** more to be filled.

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