A, B, C are three pipes attached to a cistern. A and B can fill it in 20 min and 30 min respectively while C can empty it in 15 min. If A, B, C kept open successively for 1 min each, how soon the cistern will be filled?

Options

A. 120 min.
B. 167 min.
C. 240 min.
D. 180 min.

Answer

To solve the problem step by step, we will first determine the work done by each pipe in one minute, then calculate the total work done in a cycle of three minutes (where each pipe is open for one minute), and finally find out how long it takes to fill the cistern completely.

### Step 1: Calculate the work done by each pipe in one minute.
- Pipe A can fill the cistern in 20 minutes, so in one minute, it fills \( \frac{1}{20} \) of the cistern.
- Pipe B can fill the cistern in 30 minutes, so in one minute, it fills \( \frac{1}{30} \) of the cistern.
- Pipe C can empty the cistern in 15 minutes, so in one minute, it empties \( \frac{1}{15} \) of the cistern.

### Step 2: Calculate the net work done in one cycle of 3 minutes (A, B, C).
In one cycle of 3 minutes:
- Work done by A in 1 minute: \( \frac{1}{20} \)
- Work done by B in 1 minute: \( \frac{1}{30} \)
- Work done by C in 1 minute (emptying): \( -\frac{1}{15} \)

Now, we need to find the total work done in one cycle:
\[ \text{Total work} = \frac{1}{20} + \frac{1}{30} - \frac{1}{15} \]

To add these fractions, we need a common denominator. The least common multiple of 20, 30, and 15 is 60.
- Convert each fraction:
- \( \frac{1}{20} = \frac{3}{60} \)
- \( \frac{1}{30} = \frac{2}{60} \)
- \( \frac{1}{15} = \frac{4}{60} \)

Now, substituting these values:
\[ \text{Total work} = \frac{3}{60} + \frac{2}{60} - \frac{4}{60} = \frac{1}{60} \]

### Step 3: Calculate how long it takes to fill the cistern.
In 3 minutes, the net work done is \( \frac{1}{60} \) of the cistern. To fill the entire cistern (1 whole), we need to find out how many cycles of 3 minutes are required.

Let \( x \) be the number of cycles needed:
\[ x \cdot \frac{1}{60} = 1 \implies x = 60 \]
This means it takes 60 cycles of 3 minutes to fill the cistern.

### Step 4: Calculate the total time taken.
Each cycle takes 3 minutes, so the total time taken is:
\[ \text{Total time} = 60 \times 3 = 180 \text{ minutes} \]

### Final Step: Add the last minute for A to fill the remaining part.
After 180 minutes, the cistern is filled completely. Since A fills the cistern in the last minute, we add this to our total time:
\[ \text{Total time} = 180 + 1 = 181 \text{ minutes} \]

Thus, the cistern will be completely filled in **181 minutes**.