One pipe can fill a cistren in 3 hours less than the other. The two pipes together can fill the cistern in 6 hours 40 minutes. Find the time that each pipe will take to fill the cistern.

Answer

To solve the problem, we need to find the time taken by each of the two pipes to fill the cistern. Let's break down the solution step by step.

### Step 1: Define Variables
Let the time taken by Pipe A to fill the cistern be \( x \) hours.
Then, the time taken by Pipe B to fill the cistern will be \( x - 3 \) hours (since Pipe A takes 3 hours less than Pipe B).

### Step 2: Convert Time Together into Hours
The two pipes together can fill the cistern in 6 hours and 40 minutes.
Convert this time into hours:
\[ 6 \text{ hours } 40 \text{ minutes} = 6 + \frac{40}{60} = 6 + \frac{2}{3} = \frac{20}{3} \text{ hours} \]

### Step 3: Write the Equation for Work Done
The work done by each pipe in one hour can be expressed as:
- Work done by Pipe A in one hour = \( \frac{1}{x} \)
- Work done by Pipe B in one hour = \( \frac{1}{x - 3} \)

Together, their work done in one hour is:
\[ \frac{1}{x} + \frac{1}{x - 3} = \frac{3}{20} \]

### Step 4: Solve the Equation
Now, we can set up the equation:
\[ \frac{1}{x} + \frac{1}{x - 3} = \frac{3}{20} \]

To solve this, we find a common denominator:
\[ \frac{(x - 3) + x}{x(x - 3)} = \frac{3}{20} \]
\[ \frac{2x - 3}{x^2 - 3x} = \frac{3}{20} \]

Cross-multiplying gives:
\[ 20(2x - 3) = 3(x^2 - 3x) \]
\[ 40x - 60 = 3x^2 - 9x \]

Rearranging the equation:
\[ 3x^2 - 49x + 60 = 0 \]

### Step 5: Apply the Quadratic Formula
Using the quadratic formula \( x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \):
Here, \( a = 3 \), \( b = -49 \), and \( c = 60 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-49)^2 - 4 \cdot 3 \cdot 60 = 2401 - 720 = 1681 \]

Now, substituting into the quadratic formula:
\[ x = \frac{49 \pm \sqrt{1681}}{6} \]
Since \( \sqrt{1681} = 41 \):
\[ x = \frac{49 \pm 41}{6} \]

Calculating the two possible values for \( x \):
1. \( x = \frac{90}{6} = 15 \)
2. \( x = \frac{8}{6} = \frac{4}{3} \) (not valid since it leads to negative time for Pipe B)

### Step 6: Determine Time for Each Pipe
Thus, the time taken by Pipe A is \( 15 \) hours, and the time taken by Pipe B is:
\[ x - 3 = 15 - 3 = 12 \text{ hours} \]

### Final Answer
- Time taken by Pipe A: **15 hours**
- Time taken by Pipe B: **12 hours**