Two pipes A and B can fill a tank with water in 30 minutes and 45 minutes respectively. The water pipe C can empty the tank in 36 minutes. First A and B are opened. After 12 minutes C is opened. Total time (in minutes) in which the tank will be filled up is:

Options

A. 30
B. 12
C. 36
D. 24

Answer

To solve the problem step by step, we will calculate the rates at which pipes A, B, and C fill or empty the tank, and then determine the total time taken to fill the tank.

### Step 1: Determine the rates of pipes A, B, and C
- Pipe A can fill the tank in 30 minutes. Therefore, the rate of pipe A is:
\[ \text{Rate of A} = \frac{1 \text{ tank}}{30 \text{ minutes}} = \frac{1}{30} \text{ tanks per minute} \]

- Pipe B can fill the tank in 45 minutes. Therefore, the rate of pipe B is:
\[ \text{Rate of B} = \frac{1 \text{ tank}}{45 \text{ minutes}} = \frac{1}{45} \text{ tanks per minute} \]

- Pipe C can empty the tank in 36 minutes. Therefore, the rate of pipe C is:
\[ \text{Rate of C} = \frac{1 \text{ tank}}{36 \text{ minutes}} = \frac{1}{36} \text{ tanks per minute} \]

### Step 2: Calculate the combined rate of pipes A and B
To find the combined rate of pipes A and B when both are open:
\[ \text{Combined Rate of A and B} = \text{Rate of A} + \text{Rate of B} = \frac{1}{30} + \frac{1}{45} \]
To add these fractions, we need a common denominator. The least common multiple of 30 and 45 is 90.
\[ \frac{1}{30} = \frac{3}{90}, \quad \frac{1}{45} = \frac{2}{90} \]
Thus,
\[ \text{Combined Rate of A and B} = \frac{3}{90} + \frac{2}{90} = \frac{5}{90} = \frac{1}{18} \text{ tanks per minute} \]

### Step 3: Calculate the amount of water filled by A and B in 12 minutes
In 12 minutes, the amount of water filled by A and B is:
\[ \text{Water filled in 12 minutes} = \text{Combined Rate of A and B} \times 12 = \frac{1}{18} \times 12 = \frac{12}{18} = \frac{2}{3} \text{ of the tank} \]

### Step 4: Determine the remaining water to be filled
The total capacity of the tank is 1 tank. After 12 minutes, the remaining water to be filled is:
\[ \text{Remaining water} = 1 - \frac{2}{3} = \frac{1}{3} \text{ of the tank} \]

### Step 5: Calculate the effective rate when pipe C is opened
When pipe C is opened, the effective rate of filling the tank becomes:
\[ \text{Effective Rate} = \text{Combined Rate of A and B} - \text{Rate of C} = \frac{1}{18} - \frac{1}{36} \]
Finding a common denominator (which is 36):
\[ \frac{1}{18} = \frac{2}{36} \]
Thus,
\[ \text{Effective Rate} = \frac{2}{36} - \frac{1}{36} = \frac{1}{36} \text{ tanks per minute} \]

### Step 6: Calculate the time required to fill the remaining water
To fill the remaining \(\frac{1}{3}\) of the tank at the effective rate of \(\frac{1}{36}\) tanks per minute:
\[ \text{Time} = \frac{\text{Remaining water}}{\text{Effective Rate}} = \frac{\frac{1}{3}}{\frac{1}{36}} = \frac{1}{3} \times 36 = 12 \text{ minutes} \]

### Step 7: Calculate the total time taken to fill the tank
The total time taken to fill the tank is the time A and B worked alone plus the time with C:
\[ \text{Total Time} = 12 \text{ minutes (A and B)} + 12 \text{ minutes (A, B, and C)} = 24 \text{ minutes} \]

### Final Answer
The total time in which the tank will be filled up is **24 minutes**.

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