Pipes A and B are filling pipes while pipe C is an emptying pipe A and B can fill a tank in 72 and 90 minutes respectively. When all the three pipes are opened together, the tank gets filled in 2 hours. A and B are opened together for 12 minutes, then closed and C is opened. The tank will be empty after :

Options

A. 15 minutes
B. 18 minutes
C. 12 minutes
D. 16 minutes

Answer

To solve the problem step by step, we will first find the efficiencies of pipes A, B, and C, then calculate how much water is filled by A and B together in 12 minutes, and finally determine how long it will take for pipe C to empty the remaining water in the tank.

### Step 1: Calculate the filling rates of pipes A and B
- Pipe A can fill the tank in 72 minutes.
- Pipe B can fill the tank in 90 minutes.

To find their rates:
- Rate of A = 1 tank / 72 minutes = \( \frac{1}{72} \) tanks per minute
- Rate of B = 1 tank / 90 minutes = \( \frac{1}{90} \) tanks per minute

### Step 2: Calculate the combined filling rate of pipes A and B
- Combined rate of A and B = Rate of A + Rate of B
- Combined rate = \( \frac{1}{72} + \frac{1}{90} \)

To add these fractions, we need a common denominator:
- The least common multiple (LCM) of 72 and 90 is 360.

Now, convert the rates:
- Rate of A = \( \frac{5}{360} \) tanks per minute (since \( \frac{1}{72} = \frac{5}{360} \))
- Rate of B = \( \frac{4}{360} \) tanks per minute (since \( \frac{1}{90} = \frac{4}{360} \))

Combined rate = \( \frac{5 + 4}{360} = \frac{9}{360} = \frac{1}{40} \) tanks per minute.

### Step 3: Calculate the rate of pipe C
When all three pipes are opened together, the tank fills in 2 hours (120 minutes).
- Combined rate of A, B, and C = 1 tank / 120 minutes = \( \frac{1}{120} \) tanks per minute.

Now, we can find the rate of pipe C:
- Rate of C = Combined rate of A, B, and C - Combined rate of A and B
- Rate of C = \( \frac{1}{120} - \frac{1}{40} \)

Convert \( \frac{1}{40} \) to have a common denominator of 120:
- \( \frac{1}{40} = \frac{3}{120} \)

Now, calculate the rate of C:
- Rate of C = \( \frac{1}{120} - \frac{3}{120} = -\frac{2}{120} = -\frac{1}{60} \) tanks per minute (indicating that C empties the tank).

### Step 4: Calculate the amount filled by A and B in 12 minutes
- In 12 minutes, A and B together will fill:
- Amount filled = Combined rate of A and B × Time
- Amount filled = \( \frac{9}{360} \times 12 = \frac{9 \times 12}{360} = \frac{108}{360} = \frac{3}{10} \) of the tank.

### Step 5: Calculate the remaining amount in the tank
- Total capacity of the tank = 1 tank.
- Remaining amount = 1 - \( \frac{3}{10} = \frac{7}{10} \) of the tank.

### Step 6: Calculate the time taken by pipe C to empty the remaining water
- Rate of C = \( \frac{1}{60} \) tanks per minute.
- Time taken to empty \( \frac{7}{10} \) of the tank = Remaining amount / Rate of C
- Time = \( \frac{7/10}{1/60} = \frac{7}{10} \times 60 = 42 \) minutes.

### Final Answer
The tank will be empty after **42 minutes**.

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