A filling pipe can fill a swimming pool in 15 hours, and a draining pipe can empty the same pool in 25 hours. If a system of n pipes, including both filling and draining pipes, can fill an empty pool in 5 hours, what could be the potential value of r?

To solve this problem, let's analyze the rates of the filling and draining pipes and then find the possible value of \( r \) (presumably the number of filling pipes). Here’s a step-by-step solution:

1. **Determine the rates of the pipes**:
   - The rate of the filling pipe is \(\frac{1}{15}\) of the pool per hour, as it fills the pool in 15 hours.
   - The rate of the draining pipe is \(-\frac{1}{25}\) of the pool per hour, as it empties the pool in 25 hours.

2. **Set up the equation with \( r \) filling pipes and \( (n - r) \) draining pipes**:
   - With \( r \) filling pipes, the combined filling rate would be \(\frac{r}{15}\).
   - With \( (n - r) \) draining pipes, the combined draining rate would be \(-\frac{n - r}{25}\).
   - The net rate of filling the pool with all \( n \) pipes would then be:
     \[
     \frac{r}{15} - \frac{n - r}{25}
     \]
   
3. **Use the given condition**:
   - The system of \( n \) pipes fills the pool in 5 hours, so the net rate is \(\frac{1}{5}\) of the pool per hour.
   - Set up the equation:
     \[
     \frac{r}{15} - \frac{n - r}{25} = \frac{1}{5}
     \]

4. **Clear the denominators**:
   - Multiply through by 75 (the least common multiple of 15, 25, and 5) to eliminate fractions:
     \[
     5r - 3(n - r) = 15
     \]
   - Distribute and simplify:
     \[
     5r - 3n + 3r = 15
     \]
     \[
     8r - 3n = 15
     \]

5. **Express \( n \) in terms of \( r \)**:
   - Solve for \( n \):
     \[
     3n = 8r - 15
     \]
     \[
     n = \frac{8r - 15}{3}
     \]
   - For \( n \) to be an integer, \( 8r - 15 \) must be divisible by 3.

6. **Determine the potential values of \( r \)**:
   - Test values of \( r \) that make \( \frac{8r - 15}{3} \) an integer.

   Let's try a few values of \( r \):

   - If \( r = 3 \):
     \[
     n = \frac{8 \cdot 3 - 15}{3} = \frac{24 - 15}{3} = 3
     \]
     So, one solution is \( r = 3 \) and \( n = 3 \).

   - If \( r = 6 \):
     \[
     n = \frac{8 \cdot 6 - 15}{3} = \frac{48 - 15}{3} = 11
     \]
     So, another solution is \( r = 6 \) and \( n = 11 \).

Thus, the potential values of \( r \) are **3 or 6**.

Would you like more detailed explanation on any step, or have further questions? 

Here are 5 related questions for practice:
1. What if the pool needed to be filled in 8 hours instead? How would that change the result?
2. How many draining pipes would there be if \( r = 3 \)?
3. If both pipes work together without a limit on the number of pipes, what’s the fastest filling time?
4. How would you modify the calculation if the draining pipe emptied the pool in 20 hours instead?
5. What happens if all pipes (filling and draining) are running at the same rate?

**Tip**: For mixture problems, consider breaking down each component’s rate individually, then set up equations based on their combined effect.

Learn how to solve a rate problem involving filling and draining pipes. This problem explores how many filling and draining pipes (r and n) are needed to fill a pool in 5 hours. Suitable for students in Grades 9-10, this example covers key algebraic methods and the setup of equations based on rates.

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