A tank has two inlet pipes which can fill the empty tank in 12 hours and 15 hours working alone and one outlet pipe which can empty the full tank in 8 hours working alone. The inlet pipes are kept open for all the time but the outlet pipe was opened after 2 hours for one hour and then again closed for 2 hours then once again opened for one hour. This pattern of outlet pipe continued till the tank got completely filled. In how many hours the tank has been filled, working on the given pattern?

Options

A. 8 hours 24 minutes
B. 10 hours 15 minutes
C. 9 hours 10 minutes
D. 9 hours 6 minutes

Answer

To solve the problem step by step, we will first determine the filling and emptying rates of the pipes, then analyze the pattern of operation to find out how long it takes to fill the tank completely.

### Step 1: Determine the filling rates of the inlet pipes and the emptying rate of the outlet pipe.

- Let the capacity of the tank be 120 liters (LCM of 12, 15, and 8).
- **Inlet Pipe A** can fill the tank in 12 hours:
\[ \text{Rate of A} = \frac{120 \text{ liters}}{12 \text{ hours}} = 10 \text{ liters/hour} \]

- **Inlet Pipe B** can fill the tank in 15 hours:
\[ \text{Rate of B} = \frac{120 \text{ liters}}{15 \text{ hours}} = 8 \text{ liters/hour} \]

- **Outlet Pipe C** can empty the tank in 8 hours:
\[ \text{Rate of C} = \frac{120 \text{ liters}}{8 \text{ hours}} = 15 \text{ liters/hour} \]

### Step 2: Calculate the combined filling rate when all pipes are open.

When both inlet pipes A and B are open, their combined filling rate is:
\[ \text{Combined Rate of A and B} = 10 + 8 = 18 \text{ liters/hour} \]

When the outlet pipe C is open, the effective filling rate becomes:
\[ \text{Effective Rate with C open} = 10 + 8 - 15 = 3 \text{ liters/hour} \]

### Step 3: Analyze the filling pattern.

- For the first **2 hours**, only A and B are open:
\[ \text{Water filled in 2 hours} = 2 \times 18 = 36 \text{ liters} \]

- After 2 hours, the outlet pipe C is opened for **1 hour**:
\[ \text{Water filled in 1 hour with C open} = 3 \text{ liters} \]

- This cycle (2 hours filling and 1 hour with C open) continues. In **3 hours**, the total water filled is:
\[ \text{Total in 3 hours} = 36 + 3 = 39 \text{ liters} \]

### Step 4: Calculate how many cycles are needed to fill the tank.

- In **9 hours** (3 cycles of 3 hours), the total water filled is:
\[ \text{Water filled in 9 hours} = 39 \times 3 = 117 \text{ liters} \]

- The tank capacity is 120 liters, so the remaining water to fill is:
\[ \text{Remaining water} = 120 - 117 = 3 \text{ liters} \]

### Step 5: Calculate the time needed to fill the remaining 3 liters.

- In the next hour (the 10th hour), A and B are open for 2 hours, filling:
\[ \text{Water filled in 2 hours} = 2 \times 18 = 36 \text{ liters} \]
- Then, for the next hour, C is open for 1 hour, emptying:
\[ \text{Water emptied in 1 hour} = 15 \text{ liters} \]

- The effective filling in the next 3 hours is:
\[ \text{Total in 3 hours} = 36 - 15 = 21 \text{ liters} \]

However, we only need to fill 3 liters.

- The rate of filling in the first 2 hours is 36 liters, which means in 1 hour, it fills 18 liters. Therefore, to fill 3 liters:
\[ \text{Time to fill 3 liters} = \frac{3 \text{ liters}}{18 \text{ liters/hour}} = \frac{1}{6} \text{ hours} = 10 \text{ minutes} \]

### Final Calculation: Total time taken to fill the tank.

- Total time = 9 hours + 10 minutes = 9 hours and 10 minutes.

### Conclusion:
The tank is completely filled in **9 hours and 10 minutes**.