Filling a Pool with Two Pipes: A Comprehensive Analysis

Imagine you are faced with the task of filling an empty pool using two pipes. The first pipe can fill the pool in 15 hours, while the second takes 21 hours. This intriguing problem requires a mathematical approach to determine the optimal time for both pipes to work together. In this article, we will explore the solution step-by-step and provide clear explanations to help you understand the underlying concepts.

Understanding the Problem

Let's define the rates at which the two pipes fill the pool:

Pipe A: 1/15 of the pool per hour.

Pipe B: 1/21 of the pool per hour.

Step-by-Step Calculation

To determine how long it takes for the two pipes to fill the pool together, we need to add their rates of filling:

Rate of filling by both pipes together 1/15 1/21

Let's find a common denominator to add these fractions:

1/15 1/21 (21 15) / (15 * 21) 36 / 315 12 / 105

We can simplify this fraction:

12 / 105 4 / 35

This fraction represents the portion of the pool that can be filled by both pipes in one hour.

To find the total time required to fill the pool, we take the reciprocal:

Time required 35 / 4 hours

Which simplifies to:

Time required 8.75 hours, or 8 hours and 45 minutes.

Use of Equations

A more formal mathematical approach involves setting up an equation:

1/15 1/21 1/x

This equation can be solved by finding a common denominator and equating the numerators:

21/15 times 21 15/15 times 21 12/15 times 21

Simplifying the right side of the equation:

1/x 4/35

Solving for x:

x 8.75 hours or 8 hours and 45 minutes.

Conclusion

Using two pipes together, it takes 8 hours and 45 minutes to fill the pool. This problem demonstrates the power of basic arithmetic and algebra in solving real-world problems. Understanding the rates at which tasks can be completed, and combining them to find the total time, is a valuable skill in many areas, including engineering, construction, and project management.