Pipes A and B can fill a cistern in 15 hours together. But ifthese pipes operate separately A takes 40 hours less than Bto fill the tank. In how many hours the pipe A will fill thecistern working alone?

Options

A. A) 60
B. B) 20
C. C) 40
D. D) 15

Answer

To solve the problem step by step, we will define the variables and set up equations based on the information provided.

### Step 1: Define Variables
Let:
- \( x \) = the number of hours Pipe A takes to fill the cistern alone.
- \( x + 40 \) = the number of hours Pipe B takes to fill the cistern alone (since A takes 40 hours less than B).

### Step 2: Write the Rate of Work
The rate of work for each pipe can be expressed as:
- Rate of Pipe A = \( \frac{1}{x} \) (cisterns per hour)
- Rate of Pipe B = \( \frac{1}{x + 40} \) (cisterns per hour)

### Step 3: Combined Rate of Work
When both pipes A and B work together, they can fill the cistern in 15 hours. Therefore, their combined rate is:
\[ \frac{1}{x} + \frac{1}{x + 40} = \frac{1}{15} \]

### Step 4: Clear the Denominators
To eliminate the fractions, we can multiply through by \( 15x(x + 40) \):
\[ 15(x + 40) + 15x = x(x + 40) \]

### Step 5: Expand and Simplify
Expanding both sides gives:
\[ 15x + 600 + 15x = x^2 + 40x \]
Combining like terms:
\[ 30x + 600 = x^2 + 40x \]

### Step 6: Rearrange to Form a Quadratic Equation
Rearranging the equation gives:
\[ x^2 + 40x - 30x - 600 = 0 \]
This simplifies to:
\[ x^2 + 10x - 600 = 0 \]

### Step 7: Factor the Quadratic Equation
Now we will factor the quadratic equation:
\[ (x + 30)(x - 20) = 0 \]

### Step 8: Solve for x
Setting each factor equal to zero gives:
1. \( x + 30 = 0 \) → \( x = -30 \) (not valid since time cannot be negative)
2. \( x - 20 = 0 \) → \( x = 20 \)

### Conclusion
Thus, Pipe A will fill the cistern alone in **20 hours**.