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, let's denote the time taken by the first pipe to fill the cistern as \( x \) hours. According to the problem, the second pipe takes 3 hours longer than the first pipe, so the time taken by the second pipe will be \( x + 3 \) hours.

Next, we know that both pipes together can fill the cistern in 6 hours and 40 minutes. We need to convert this time into hours:
\[ 6 \text{ hours } 40 \text{ minutes} = 6 + \frac{40}{60} = 6 + \frac{2}{3} = \frac{20}{3} \text{ hours} \]

Now, we can express the rates of filling the cistern for both pipes. The rate of the first pipe is \( \frac{1}{x} \) (cisterns per hour), and the rate of the second pipe is \( \frac{1}{x + 3} \).

When both pipes work together, their combined rate is:
\[ \frac{1}{x} + \frac{1}{x + 3} = \frac{1}{\frac{20}{3}} \]

This can be simplified to:
\[ \frac{1}{x} + \frac{1}{x + 3} = \frac{3}{20} \]

Now, we will find a common denominator for the left side:
\[ \frac{(x + 3) + x}{x(x + 3)} = \frac{3}{20} \]
\[ \frac{2x + 3}{x(x + 3)} = \frac{3}{20} \]

Next, we cross-multiply:
\[ 20(2x + 3) = 3x(x + 3) \]
\[ 40x + 60 = 3x^2 + 9x \]

Rearranging the equation gives us:
\[ 3x^2 + 9x - 40x - 60 = 0 \]
\[ 3x^2 - 31x - 60 = 0 \]

Now we will use the quadratic formula to solve for \( x \):
\[ x = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]
where \( a = 3 \), \( b = -31 \), and \( c = -60 \).

Calculating the discriminant:
\[ b^2 - 4ac = (-31)^2 - 4 \cdot 3 \cdot (-60) = 961 + 720 = 1681 \]

Now substituting back into the quadratic formula:
\[ x = \frac{31 \pm \sqrt{1681}}{6} \]
\[ x = \frac{31 \pm 41}{6} \]

Calculating the two possible values:
1. \( x = \frac{72}{6} = 12 \)
2. \( x = \frac{-10}{6} \) (not valid since time cannot be negative)

Thus, the time taken by the first pipe is \( 12 \) hours, and the time taken by the second pipe is:
\[ x + 3 = 12 + 3 = 15 \text{ hours} \]

**Final Answer:**
- The first pipe takes 12 hours to fill the cistern.
- The second pipe takes 15 hours to fill the cistern.