Two taps A and B can fill a tank in 20 min and 30 min, respectively. An outlet pipe C can empty the full tank in 15 min A, B and C are opened alternatively ,each for 1 min . How long will the tank take to filled ?

Options

A. 105 min
B. 120 min
C. 167 min
D. 185 min

Answer

To solve the problem of how long it will take to fill the tank with taps A, B, and outlet pipe C operating alternately, we can follow these steps:

### Step 1: Determine the filling and emptying rates of taps A, B, and C

1. **Tap A** can fill the tank in 20 minutes.
- Filling rate of A = 1 tank / 20 minutes = 1/20 tanks per minute.

2. **Tap B** can fill the tank in 30 minutes.
- Filling rate of B = 1 tank / 30 minutes = 1/30 tanks per minute.

3. **Pipe C** can empty the tank in 15 minutes.
- Emptying rate of C = 1 tank / 15 minutes = 1/15 tanks per minute.

### Step 2: Calculate the net effect of A, B, and C when they operate alternately

1. In 1 minute, when Tap A is open:
- Amount filled = 1/20 tanks.

2. In the next minute, when Tap B is open:
- Amount filled = 1/30 tanks.

3. In the third minute, when Pipe C is open:
- Amount emptied = 1/15 tanks.

### Step 3: Calculate the total amount filled in 3 minutes

1. **Total amount filled in 3 minutes**:
- Amount filled by A in 1 minute = 1/20
- Amount filled by B in 1 minute = 1/30
- Amount emptied by C in 1 minute = 1/15

2. To find the total amount filled in 3 minutes:
\[ \text{Total filled} = \left(\frac{1}{20} + \frac{1}{30} - \frac{1}{15}\right) \]

3. To add these fractions, find a common denominator, which is 60:
- \(\frac{1}{20} = \frac{3}{60}\)
- \(\frac{1}{30} = \frac{2}{60}\)
- \(\frac{1}{15} = \frac{4}{60}\)

4. Now, substituting back:
\[ \text{Total filled} = \left(\frac{3}{60} + \frac{2}{60} - \frac{4}{60}\right) = \frac{1}{60} \]

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

1. Since the tank is 1 full tank, and each cycle (3 minutes) fills \(\frac{1}{60}\) of the tank:
- To fill 1 tank, we need:
\[ \text{Number of cycles} = 1 \div \frac{1}{60} = 60 \text{ cycles} \]

### Step 5: Calculate the total time taken

1. Each cycle takes 3 minutes, so:
\[ \text{Total time} = 60 \text{ cycles} \times 3 \text{ minutes/cycle} = 180 \text{ minutes} \]

### Conclusion

The tank will take **180 minutes** to fill when taps A, B, and outlet pipe C are opened alternately.

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