Math Problem Statement

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?

Solution

To solve this problem, let's analyze the rates of the filling and draining pipes and then find the possible value of rr (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 115\frac{1}{15} of the pool per hour, as it fills the pool in 15 hours.
    • The rate of the draining pipe is 125-\frac{1}{25} of the pool per hour, as it empties the pool in 25 hours.
  2. Set up the equation with rr filling pipes and (nr)(n - r) draining pipes:

    • With rr filling pipes, the combined filling rate would be r15\frac{r}{15}.
    • With (nr)(n - r) draining pipes, the combined draining rate would be nr25-\frac{n - r}{25}.
    • The net rate of filling the pool with all nn pipes would then be: r15nr25\frac{r}{15} - \frac{n - r}{25}
  3. Use the given condition:

    • The system of nn pipes fills the pool in 5 hours, so the net rate is 15\frac{1}{5} of the pool per hour.
    • Set up the equation: r15nr25=15\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: 5r3(nr)=155r - 3(n - r) = 15
    • Distribute and simplify: 5r3n+3r=155r - 3n + 3r = 15 8r3n=158r - 3n = 15
  5. Express nn in terms of rr:

    • Solve for nn: 3n=8r153n = 8r - 15 n=8r153n = \frac{8r - 15}{3}
    • For nn to be an integer, 8r158r - 15 must be divisible by 3.
  6. Determine the potential values of rr:

    • Test values of rr that make 8r153\frac{8r - 15}{3} an integer.

    Let's try a few values of rr:

    • If r=3r = 3: n=83153=24153=3n = \frac{8 \cdot 3 - 15}{3} = \frac{24 - 15}{3} = 3 So, one solution is r=3r = 3 and n=3n = 3.

    • If r=6r = 6: n=86153=48153=11n = \frac{8 \cdot 6 - 15}{3} = \frac{48 - 15}{3} = 11 So, another solution is r=6r = 6 and n=11n = 11.

Thus, the potential values of rr 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=3r = 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.

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Math Problem Analysis

Mathematical Concepts

Rate Problems
Systems of Equations
Algebra

Formulas

Rate of filling pipe: 1/15 per hour
Rate of draining pipe: -1/25 per hour
Combined rate: r/15 - (n - r)/25
Net rate equation: combined rate = 1/5

Theorems

Basic Rate Theorem in Work Problems

Suitable Grade Level

Grades 9-10