Two pipes A and B are opened together to fill a tank. Both pipes fill the tank in a certain time. If A separately takes 16 min more than the time taken by (A + B) and B takes 9 min more than the time taken by(A+B).Find the time taken byAandBto fill the tank when both the pipes are opened together.

Options

A. 10 min
B. 12 min
C. 15min
D. 8 min

Answer

To solve the problem, let's denote the time taken by both pipes A and B to fill the tank together as \( X \) minutes.

1. **Understanding the times taken by A and B:**
- According to the problem, pipe A takes 16 minutes more than the time taken by both pipes together. Therefore, the time taken by pipe A alone can be expressed as:
\[ A = X + 16 \]
- Similarly, pipe B takes 9 minutes more than the time taken by both pipes together. Thus, the time taken by pipe B alone can be expressed as:
\[ B = X + 9 \]

2. **Finding the rates of the pipes:**
- The rate of work done by pipe A is the reciprocal of the time taken by A:
\[ \text{Rate of A} = \frac{1}{A} = \frac{1}{X + 16} \]
- The rate of work done by pipe B is the reciprocal of the time taken by B:
\[ \text{Rate of B} = \frac{1}{B} = \frac{1}{X + 9} \]

3. **Combining the rates of A and B:**
- When both pipes A and B are opened together, their combined rate of work is the sum of their individual rates:
\[ \text{Combined Rate} = \text{Rate of A} + \text{Rate of B} = \frac{1}{X + 16} + \frac{1}{X + 9} \]

4. **Setting up the equation:**
- The combined rate when both pipes are open together is also equal to the rate of filling the tank when both pipes are open, which is:
\[ \text{Combined Rate} = \frac{1}{X} \]
- Therefore, we can set up the equation:
\[ \frac{1}{X + 16} + \frac{1}{X + 9} = \frac{1}{X} \]

5. **Finding a common denominator and solving the equation:**
- The common denominator for the left side is \((X + 16)(X + 9)\):
\[ \frac{(X + 9) + (X + 16)}{(X + 16)(X + 9)} = \frac{1}{X} \]
- Simplifying the numerator:
\[ \frac{2X + 25}{(X + 16)(X + 9)} = \frac{1}{X} \]
- Cross-multiplying gives:
\[ (2X + 25)X = (X + 16)(X + 9) \]
- Expanding both sides:
\[ 2X^2 + 25X = X^2 + 25X + 144 \]
- Rearranging the equation:
\[ 2X^2 + 25X - X^2 - 25X - 144 = 0 \]
\[ X^2 - 144 = 0 \]
- This simplifies to:
\[ X^2 = 144 \]
- Taking the square root of both sides:
\[ X = 12 \]

6. **Conclusion:**
- The time taken by both pipes A and B to fill the tank together is \( \boxed{12} \) minutes.