Two pipes A and B can fill a cistern in 24 minutes and 30 minutes, respectively. There is also an outlet C. If all the three pipes are opened together, the cistern is filled in 20 minutes. How much time will be taken by C to empty the full cistern?

Options

A. 30 min
B. 40 min
C. 45 min
D. 50 min

Answer

To solve the problem step by step, we will follow these calculations:

### Step 1: Determine the total work
The total work can be calculated using the least common multiple (LCM) of the times taken by pipes A, B, and the time taken when all three are opened together.

- Pipe A fills the cistern in 24 minutes.
- Pipe B fills the cistern in 30 minutes.
- When all three pipes are opened together, the cistern is filled in 20 minutes.

To find the LCM of 24, 30, and 20:
- The prime factorization of 24 is \(2^3 \times 3^1\).
- The prime factorization of 30 is \(2^1 \times 3^1 \times 5^1\).
- The prime factorization of 20 is \(2^2 \times 5^1\).

The LCM is taken by using the highest power of each prime:
- For 2: \(2^3\)
- For 3: \(3^1\)
- For 5: \(5^1\)

Thus, LCM = \(2^3 \times 3^1 \times 5^1 = 120\).

**Total Work (TW) = 120 units.**

### Step 2: Calculate the efficiencies of pipes A and B
- Efficiency of pipe A = Total Work / Time taken by A = \(120 / 24 = 5\) units per minute.
- Efficiency of pipe B = Total Work / Time taken by B = \(120 / 30 = 4\) units per minute.

### Step 3: Calculate the combined efficiency of A, B, and C
When all three pipes are opened together, the efficiency is:
- Efficiency of A, B, and C = Total Work / Time taken when all are opened = \(120 / 20 = 6\) units per minute.

### Step 4: Set up the equation for the efficiency of pipe C
Since pipe C is an outlet, its efficiency will be subtracted from the combined efficiency of pipes A and B:
- Efficiency of A + Efficiency of B - Efficiency of C = Efficiency of A, B, and C
- \(5 + 4 - \text{Efficiency of C} = 6\)

### Step 5: Solve for the efficiency of pipe C
- \(9 - \text{Efficiency of C} = 6\)
- \(\text{Efficiency of C} = 9 - 6 = 3\) units per minute.

### Step 6: Calculate the time taken by pipe C to empty the cistern
To find the time taken by pipe C to empty the full cistern:
- Time = Total Work / Efficiency of C = \(120 / 3 = 40\) minutes.

### Final Answer
The time taken by pipe C to empty the full cistern is **40 minutes**.

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