One of the two inlet pipes works twice as efficiently as the other. The two, working alongside a drain pipe that can empty a cistern all by itself in 12 hours, can fill the empty cistern in 12 hours. How many hours will the less efficient inlet pipe take to fill the empty cistern by itself?

Options

A. 9
B. 15
C. 12
D. 18

Answer

To solve the problem, we need to determine how long the less efficient inlet pipe takes to fill the cistern by itself. We know that one pipe works twice as efficiently as the other, and together they can fill the cistern in 12 hours while a drain pipe can empty it in 12 hours.

### Step-by-Step Solution:

1. **Define the Efficiency of the Pipes:**
Let the efficiency of the less efficient inlet pipe (Pipe B) be \( r \) (in cisterns per hour). Therefore, the efficiency of the more efficient inlet pipe (Pipe A) will be \( 2r \).

2. **Determine the Efficiency of the Drain Pipe:**
The drain pipe can empty the cistern in 12 hours, which means its efficiency is \( -\frac{1}{12} \) (since it is emptying the cistern).

3. **Set Up the Equation:**
When both inlet pipes and the drain pipe are working together, their combined efficiency is:
\[ r + 2r - \frac{1}{12} = 12 \text{ hours} \]
This means they can fill the cistern at a rate of \( \frac{1}{12} \) of the cistern per hour.

4. **Combine the Efficiencies:**
The equation can be simplified:
\[ 3r - \frac{1}{12} = \frac{1}{12} \]

5. **Solve for \( r \):**
Rearranging the equation gives:
\[ 3r = \frac{1}{12} + \frac{1}{12} = \frac{2}{12} = \frac{1}{6} \]
Therefore,
\[ r = \frac{1}{18} \]

6. **Determine the Time for Pipe B:**
Since \( r \) is the efficiency of Pipe B, the time taken by Pipe B to fill the cistern alone is the reciprocal of its efficiency:
\[ \text{Time for Pipe B} = \frac{1}{r} = \frac{1}{\frac{1}{18}} = 18 \text{ hours} \]

### Final Answer:
The less efficient inlet pipe (Pipe B) will take **36 hours** to fill the empty cistern by itself.