A, B & C are three inlet pipes. Time taken by A & B together to fill half of the tank is same as time taken by pipe C alone to fill one - sixth of the tank. If A, B & C together can fill the tank in 9 hours, then find time taken by pipe C alone to fill the tank?

Options

A. 24 hours
B. 18 hours
C. 28 hours
D. 36 hours

Answer

To solve the problem step by step, we will follow the reasoning laid out in the video transcript.

### Step-by-Step Solution:

1. **Understanding the Problem**:
We have three pipes A, B, and C. The time taken by A and B together to fill half of the tank is equal to the time taken by pipe C to fill one-sixth of the tank. We also know that A, B, and C together can fill the tank in 9 hours.

2. **Let’s Define Variables**:
- Let the time taken by A and B together to fill half the tank be \( x \).
- Thus, the time taken by C to fill one-sixth of the tank is also \( x \).

3. **Relating the Times**:
- Since A and B fill half the tank in time \( x \), they would take \( 2x \) to fill the entire tank (because half the tank takes \( x \), so the full tank takes double that time).
- Since C fills one-sixth of the tank in time \( x \), it would take \( 6x \) to fill the entire tank (because one-sixth of the tank takes \( x \), so the full tank takes six times that time).

4. **Setting Up the Equation**:
- We know that the combined time taken by A, B, and C to fill the tank is 9 hours. Therefore, we can express this as:
\[ \frac{1}{2x} + \frac{1}{6x} = \frac{1}{9} \]
- This means that the work done by A and B together is \( \frac{1}{2x} \) and the work done by C is \( \frac{1}{6x} \).

5. **Finding a Common Denominator**:
- The common denominator for \( 2x \) and \( 6x \) is \( 6x \). Thus, we can rewrite the equation:
\[ \frac{3}{6x} + \frac{1}{6x} = \frac{1}{9} \]
- This simplifies to:
\[ \frac{4}{6x} = \frac{1}{9} \]

6. **Cross-Multiplying**:
- Cross-multiplying gives:
\[ 4 \cdot 9 = 6x \]
- Therefore:
\[ 36 = 6x \]

7. **Solving for x**:
- Dividing both sides by 6:
\[ x = 6 \]

8. **Finding Time Taken by Pipe C Alone**:
- Since we established that C takes \( 6x \) to fill the tank, we substitute \( x \):
\[ \text{Time taken by C} = 6 \cdot 6 = 36 \text{ hours} \]

### Conclusion:
The time taken by pipe C alone to fill the tank is **36 hours**.