A cistern has two taps which fill it in 12 minutes and 15 minutes respectively. There is also a waste pipe in the cistern. When all the three are opened, the empty cistern is full in 20 minutes. How long will the waste pipe take to empty the full cistern ?

Options

A. 10 min
B. 12 min
C. 15 min
D. None of these

Answer

To solve the problem, we need to determine how long the waste pipe will take to empty the full cistern. We have two taps that fill the cistern and one waste pipe that empties it. Let's break down the solution step by step.

### Step 1: Determine the filling rates of the taps
- **Tap A** fills the cistern in 12 minutes. Therefore, its filling rate is:
\[ \text{Rate of A} = \frac{1}{12} \text{ cisterns per minute} \]

- **Tap B** fills the cistern in 15 minutes. Therefore, its filling rate is:
\[ \text{Rate of B} = \frac{1}{15} \text{ cisterns per minute} \]

### Step 2: Calculate the combined filling rate of taps A and B
To find the combined filling rate of taps A and B, we add their rates:
\[ \text{Combined Rate of A and B} = \frac{1}{12} + \frac{1}{15} \]
To add these fractions, we need a common denominator. The least common multiple of 12 and 15 is 60:
\[ \frac{1}{12} = \frac{5}{60}, \quad \frac{1}{15} = \frac{4}{60} \]
Thus,
\[ \text{Combined Rate of A and B} = \frac{5}{60} + \frac{4}{60} = \frac{9}{60} = \frac{3}{20} \text{ cisterns per minute} \]

### Step 3: Determine the overall filling rate when all three (two taps and one waste pipe) are opened
When all three are opened, the cistern fills in 20 minutes. Therefore, the overall filling rate is:
\[ \text{Overall Rate} = \frac{1}{20} \text{ cisterns per minute} \]

### Step 4: Set up the equation to find the rate of the waste pipe
Let the rate of the waste pipe (C) be represented as a negative value since it empties the cistern. We can set up the equation:
\[ \text{Combined Rate of A and B} - \text{Rate of C} = \text{Overall Rate} \]
Substituting the known values:
\[ \frac{3}{20} - \text{Rate of C} = \frac{1}{20} \]

### Step 5: Solve for the rate of the waste pipe
Rearranging the equation gives:
\[ \text{Rate of C} = \frac{3}{20} - \frac{1}{20} = \frac{2}{20} = \frac{1}{10} \text{ cisterns per minute} \]

### Step 6: Determine the time taken by the waste pipe to empty the cistern
If the waste pipe empties at the rate of \(\frac{1}{10}\) cisterns per minute, then the time taken to empty the full cistern is:
\[ \text{Time} = \frac{1 \text{ cistern}}{\frac{1}{10} \text{ cisterns per minute}} = 10 \text{ minutes} \]

### Final Answer
The waste pipe will take **10 minutes** to empty the full cistern.

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