Pipe A can fill an empty cistern in 15 hours while pipe B take 25 hours to fill it . Initially pipe A is left open for some time and then closed and pipe B is switched on immediately . In all the cistern takes takes 21 hours to be full. How long was pipe A open ?
A)3 hours
B)9 hours
C)6 hours
D)7.5 hours

Options

A. 3 hours
B. 9 hours
C. 6 hours
D. 7.5 hours

Answer

To solve the problem, we need to determine how long Pipe A was open before it was closed and Pipe B was turned on. Here’s a step-by-step breakdown of the solution:

### Step 1: Determine the filling rates of the pipes
- Pipe A can fill the cistern in 15 hours. Therefore, its filling rate is:
\[ \text{Rate of Pipe A} = \frac{1 \text{ cistern}}{15 \text{ hours}} = \frac{1}{15} \text{ cistern per hour} \]
- Pipe B can fill the cistern in 25 hours. Therefore, its filling rate is:
\[ \text{Rate of Pipe B} = \frac{1 \text{ cistern}}{25 \text{ hours}} = \frac{1}{25} \text{ cistern per hour} \]

### Step 2: Find the combined filling rate when both pipes are used
To find the combined filling rate of both pipes, we add their individual rates:
\[ \text{Combined rate} = \frac{1}{15} + \frac{1}{25} \]
To add these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 25 is 75:
\[ \frac{1}{15} = \frac{5}{75}, \quad \frac{1}{25} = \frac{3}{75} \]
Thus,
\[ \text{Combined rate} = \frac{5}{75} + \frac{3}{75} = \frac{8}{75} \text{ cistern per hour} \]

### Step 3: Set up the equation for the total time to fill the cistern
Let \( x \) be the time (in hours) that Pipe A was open. After Pipe A is closed, Pipe B is turned on. The total time taken to fill the cistern is given as 21 hours. Therefore, the time Pipe B is on will be \( 21 - x \) hours.

The amount of cistern filled by Pipe A in \( x \) hours is:
\[ \text{Amount filled by Pipe A} = \frac{x}{15} \]
The amount filled by Pipe B in \( 21 - x \) hours is:
\[ \text{Amount filled by Pipe B} = \frac{21 - x}{25} \]

### Step 4: Set up the equation for the total amount filled
The total amount filled by both pipes must equal 1 (the entire cistern):
\[ \frac{x}{15} + \frac{21 - x}{25} = 1 \]

### Step 5: Solve the equation
To solve the equation, we first find a common denominator, which is 75:
\[ \frac{5x}{75} + \frac{3(21 - x)}{75} = 1 \]
Multiplying through by 75 gives:
\[ 5x + 3(21 - x) = 75 \]
Expanding this:
\[ 5x + 63 - 3x = 75 \]
Combining like terms:
\[ 2x + 63 = 75 \]
Subtracting 63 from both sides:
\[ 2x = 12 \]
Dividing by 2:
\[ x = 6 \]

### Conclusion
Pipe A was open for **6 hours**.