Solving The Sink Filling Problem A Step-by-Step Guide
In this exploration, we delve into a classic mathematical problem involving the time it takes to fill a sink using different water sources. This problem is an excellent example of how we can use algebraic equations to model and solve real-world situations. The core of the problem lies in understanding rates of work and how they combine when multiple sources contribute to a task. Our main objective is to determine the time it takes to fill a sink using hot water alone, given the times it takes to fill the sink using cold water alone and both hot and cold water together. We will dissect the problem, establish the fundamental concepts, formulate the equations, and ultimately arrive at the solution. This detailed guide aims to provide a clear, step-by-step approach that not only solves the problem at hand but also equips you with the skills to tackle similar rate-of-work problems. By the end of this discussion, you'll have a solid grasp of the underlying principles and the practical application of these principles in solving mathematical puzzles. So, let's dive in and unravel the intricacies of this sink-filling scenario!
Problem Statement
The sink-filling problem presented to us involves a scenario where it takes 5 minutes to fill a sink using only cold water. When both hot and cold water are turned on, the sink fills in just 2 minutes. The challenge is to find the correct approach to determine x, which represents the time in minutes it takes to fill the sink using only hot water. To solve this, we need to understand the concept of rates. The rate at which the cold water fills the sink is 1/5 of the sink per minute, and the combined rate of hot and cold water is 1/2 of the sink per minute. By setting up an equation that represents the sum of the individual rates equaling the combined rate, we can solve for the rate of the hot water alone. From there, we can easily find the time it takes for the hot water to fill the sink by taking the reciprocal of its rate. This problem highlights the practical application of mathematical concepts in everyday situations and demonstrates how algebraic equations can be used to model and solve real-world problems. The key is to break down the problem into manageable parts, identify the relevant variables, and establish the relationships between them.
Understanding Rates of Work
To effectively solve the sink-filling problem, it's crucial to grasp the concept of rates of work. In this context, the "work" is the act of filling the sink, and the "rate" is how quickly this work is completed. We measure this rate as the fraction of the sink filled per unit of time, typically minutes in this scenario. For instance, if the cold water fills the sink in 5 minutes, its rate of work is 1/5 of the sink per minute. This means that in one minute, the cold water fills one-fifth of the sink. Similarly, if both hot and cold water together fill the sink in 2 minutes, their combined rate is 1/2 of the sink per minute. The rate of work is a fundamental concept in many mathematical problems, particularly those involving time, speed, and efficiency. It allows us to quantify how much of a task is completed in a given period. When multiple entities are working together, their rates of work are additive. That is, the combined rate is the sum of the individual rates. This principle is the cornerstone of solving problems where different agents contribute to the same task at varying speeds. By understanding and applying this concept, we can translate real-world scenarios into mathematical equations, making complex problems more approachable and solvable. The ability to work with rates is not only valuable in mathematics but also in various fields like physics, engineering, and economics.
Setting Up the Equation
The crucial step in solving this sink-filling problem is setting up the correct equation. The equation should accurately represent the relationship between the rates of the cold water, the hot water, and their combined effect. We know that the cold water fills the sink at a rate of 1/5 per minute. Let's denote the rate at which the hot water fills the sink as 1/x per minute, where x is the time we are trying to find. When both hot and cold water are turned on, their rates combine, and the sink fills in 2 minutes. This means their combined rate is 1/2 per minute. The fundamental principle here is that the sum of the individual rates equals the combined rate. Therefore, we can express this relationship as an equation: (Rate of cold water) + (Rate of hot water) = (Combined rate). Substituting the known values and the variable, we get: 1/5 + 1/x = 1/2. This equation is the cornerstone of our solution. It encapsulates the entire problem in a concise mathematical form. Solving this equation for x will give us the time it takes for the hot water alone to fill the sink. The ability to translate a word problem into a mathematical equation is a critical skill in problem-solving. It requires a clear understanding of the problem's context, identifying the relevant variables, and expressing the relationships between these variables in a mathematical form. This equation not only allows us to solve the specific problem but also provides a framework for tackling similar problems in the future.
Solving for x
Now that we have established the equation 1/5 + 1/x = 1/2, the next step is to solve for x. This involves a series of algebraic manipulations to isolate x on one side of the equation. Our first step is to eliminate the fraction 1/5 from the left side. We can do this by subtracting 1/5 from both sides of the equation, which gives us: 1/x = 1/2 - 1/5. Next, we need to simplify the right side of the equation. To subtract the fractions, we need a common denominator. The least common multiple of 2 and 5 is 10, so we rewrite the fractions with a denominator of 10: 1/x = 5/10 - 2/10. Now, we can subtract the fractions: 1/x = 3/10. At this point, we have a simple equation where the reciprocal of x is equal to 3/10. To solve for x, we take the reciprocal of both sides of the equation. This gives us: x = 10/3. Converting this improper fraction to a mixed number, we get x = 3 1/3. Therefore, it takes 3 1/3 minutes for the hot water alone to fill the sink. This process demonstrates the power of algebraic manipulation in solving equations. By applying basic algebraic principles, we can transform a complex equation into a simple one and ultimately find the value of the unknown variable. This skill is essential not only in mathematics but also in various fields that involve quantitative analysis and problem-solving.
The Correct Table
To determine the correct table that can be used to find x, we need to understand how the equation 1/5 + 1/x = 1/2 translates into a tabular representation. A table, in this context, would typically organize the information in a way that highlights the relationship between the rates of the cold water, the hot water, and their combined rate. One possible table structure could have columns for "Water Source," "Rate (Sink per Minute)," and "Time (Minutes)." The rows would then represent the cold water, the hot water, and the combined hot and cold water. In the row for cold water, the "Rate" would be 1/5, and the "Time" would be 5. In the row for hot water, the "Rate" would be 1/x, and the "Time" would be x. In the row for combined hot and cold water, the "Rate" would be 1/2, and the "Time" would be 2. Another way to structure the table is to focus on the equation itself. In this case, the table might have columns representing the terms of the equation: "Cold Water Rate (1/5)," "Hot Water Rate (1/x)," and "Combined Rate (1/2)." The rows could then represent different scenarios or steps in solving the equation. The key is that the table should clearly and logically present the information in a way that makes the relationship between the rates and the unknown time x evident. The table should serve as a visual aid in understanding the problem and the equation used to solve it. A well-constructed table can greatly enhance comprehension and facilitate problem-solving, particularly in complex scenarios. It allows for a clear and organized presentation of data, making it easier to identify patterns and relationships.
Practical Applications
The practical applications of solving problems like this sink-filling scenario extend far beyond the realm of mathematics classrooms. This type of problem-solving is directly applicable to various real-world situations, particularly in fields such as engineering, construction, and project management. For instance, consider a scenario where multiple pumps are working to drain a flooded area. Each pump has a different pumping rate, and the goal is to determine how long it will take to drain the area completely. This problem is analogous to the sink-filling problem, where the pumps are the water sources, the pumping rates are the rates of work, and the area to be drained is the sink. Similarly, in construction, tasks often involve multiple workers or machines working together. Understanding how their individual work rates combine to achieve a common goal is crucial for efficient project planning and execution. Project managers frequently use similar calculations to estimate project completion times, allocate resources, and manage deadlines. In software development, multiple programmers might work on different modules of a project. Estimating the time it takes to complete the project requires understanding the individual coding rates and how they contribute to the overall progress. Even in everyday life, we encounter situations where understanding rates of work is beneficial. For example, when cooking a meal, we might need to coordinate the cooking times of different dishes to ensure they are ready simultaneously. This involves considering the cooking rates of each dish and adjusting cooking times accordingly. The ability to analyze and solve rate-of-work problems is a valuable skill that enhances our problem-solving capabilities in various aspects of life.
Conclusion
In conclusion, the sink-filling problem serves as an excellent illustration of how mathematical principles can be applied to solve real-world challenges. By understanding the concept of rates of work, setting up the correct equation, and employing algebraic techniques, we can effectively determine the time it takes to complete a task when multiple entities contribute at different rates. This problem not only reinforces fundamental mathematical skills but also highlights the importance of logical reasoning and analytical thinking. The process of breaking down a complex problem into manageable parts, identifying the relevant variables, and establishing the relationships between them is a valuable skill that transcends the boundaries of mathematics. The ability to translate real-world scenarios into mathematical models and solve them is crucial in various fields, including engineering, project management, and everyday decision-making. Moreover, the problem underscores the significance of clear and organized problem-solving approaches. Setting up a table to visualize the information, formulating the equation step-by-step, and carefully performing the algebraic manipulations are all essential components of successful problem-solving. By mastering these techniques, we can confidently tackle a wide range of challenges and make informed decisions in various aspects of our lives. The sink-filling problem, therefore, is not just a mathematical exercise but a valuable learning experience that equips us with essential problem-solving skills applicable in diverse contexts.
