A tank has two inlet pipes, A and B, which can fill the tank in 12 hours and 18 hours respectively, and an outlet C which can empty the tank in 9 hours. A,B and C are all opened at a time but the outlet C is blocked completely after 6 hours. Find the total time taken right from the start to fill the tank.

Answer

To solve the problem step by step, we will first determine the rates of the pipes and then calculate the total time taken to fill the tank.

### Step 1: Determine the capacity of the tank
We will take the least common multiple (LCM) of the times taken by pipes A, B, and C to fill or empty the tank.

- Pipe A fills the tank in 12 hours.
- Pipe B fills the tank in 18 hours.
- Pipe C empties the tank in 9 hours.

The LCM of 12, 18, and 9 is 36 liters.

### Step 2: Calculate the rates of each pipe
Next, we calculate how much water each pipe can fill or empty in one hour.

- **Rate of Pipe A**:
\[ \text{Rate of A} = \frac{36 \text{ liters}}{12 \text{ hours}} = 3 \text{ liters/hour} \]

- **Rate of Pipe B**:
\[ \text{Rate of B} = \frac{36 \text{ liters}}{18 \text{ hours}} = 2 \text{ liters/hour} \]

- **Rate of Pipe C**:
\[ \text{Rate of C} = -\frac{36 \text{ liters}}{9 \text{ hours}} = -4 \text{ liters/hour} \quad (\text{negative because it empties the tank}) \]

### Step 3: Calculate the net rate when all pipes are open
When all three pipes are open, the net rate of filling the tank is:
\[ \text{Net Rate} = \text{Rate of A} + \text{Rate of B} + \text{Rate of C} = 3 + 2 - 4 = 1 \text{ liter/hour} \]

### Step 4: Calculate the amount of water filled in the first 6 hours
For the first 6 hours, the net rate is 1 liter/hour. Therefore, the total amount of water filled in 6 hours is:
\[ \text{Water filled in 6 hours} = 1 \text{ liter/hour} \times 6 \text{ hours} = 6 \text{ liters} \]

### Step 5: Calculate the remaining capacity of the tank
The total capacity of the tank is 36 liters. After 6 hours, the amount of water filled is 6 liters, so the remaining capacity is:
\[ \text{Remaining capacity} = 36 \text{ liters} - 6 \text{ liters} = 30 \text{ liters} \]

### Step 6: Calculate the time taken to fill the remaining capacity with only A and B
After 6 hours, pipe C is blocked, and only pipes A and B are filling the tank. The combined rate of A and B is:
\[ \text{Combined Rate of A and B} = 3 + 2 = 5 \text{ liters/hour} \]

Now, we need to find out how long it takes to fill the remaining 30 liters:
\[ \text{Time} = \frac{\text{Remaining capacity}}{\text{Combined Rate}} = \frac{30 \text{ liters}}{5 \text{ liters/hour}} = 6 \text{ hours} \]

### Step 7: Calculate the total time taken to fill the tank
The total time taken to fill the tank is the sum of the time taken in the first 6 hours and the time taken to fill the remaining capacity:
\[ \text{Total Time} = 6 \text{ hours} + 6 \text{ hours} = 12 \text{ hours} \]

### Final Answer
The total time taken to fill the tank is **12 hours**.

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