Finding The Equation Connecting Q And R In Inverse Variation

In the realm of mathematics, inverse variation plays a crucial role in describing relationships where one quantity decreases as another increases, and vice versa. To truly grasp this concept, let's dive into the world of inverse variation and see how it applies in real-world situations. Inverse variation, also known as inverse proportionality or reciprocal variation, is a mathematical relationship between two variables where their product is a constant. In simpler terms, if one variable increases, the other variable decreases proportionally, and vice versa. This relationship is fundamental in various scientific and mathematical contexts, such as physics, engineering, and economics. Inverse variation can be represented mathematically by the equation $y = k/x$, where $y$ and $x$ are the variables, and $k$ is the constant of variation. This constant, $k$, is the product of $x$ and $y$, and it remains the same regardless of the values of $x$ and $y$. Understanding this constant is key to solving problems involving inverse variation. For instance, consider the relationship between the speed at which you travel and the time it takes to reach your destination. If you increase your speed, the time taken will decrease, assuming the distance remains constant. This is a classic example of inverse variation. The faster you go, the less time it takes; the slower you go, the more time it takes. In this scenario, the distance is the constant of variation. Think about other real-world examples like the number of workers on a job and the time it takes to complete the job. More workers mean less time, and fewer workers mean more time, assuming each worker contributes equally. This principle applies in many aspects of daily life and scientific calculations, making inverse variation a crucial concept to understand. Another example can be seen in electrical circuits. The current flowing through a circuit is inversely proportional to the resistance. If you increase the resistance, the current decreases, and if you decrease the resistance, the current increases, assuming the voltage remains constant. This relationship is critical in designing and troubleshooting electrical systems. Inverse variation is also evident in the behavior of gases. According to Boyle's Law, the pressure of a gas is inversely proportional to its volume, provided the temperature and the amount of gas remain constant. If you compress a gas (decrease its volume), the pressure increases. Conversely, if you allow a gas to expand (increase its volume), the pressure decreases. This principle is fundamental in understanding thermodynamics and the behavior of gases in various applications. Understanding inverse variation allows us to predict how changes in one variable will affect another. It enables us to solve practical problems and make informed decisions based on the relationships between quantities. Whether it's planning a journey, managing resources, or designing systems, the principles of inverse variation provide a valuable framework for analysis and problem-solving. In summary, inverse variation is a fundamental mathematical concept that describes how two variables relate when their product is constant. By understanding this relationship, we can better interpret and predict the behavior of various systems and phenomena in the world around us.

Problem Statement: q Varies Inversely as r

The problem at hand presents a classic scenario of inverse variation. We are told that the variable q varies inversely as the variable r. To break this down, let's dissect the meaning of "varies inversely." In mathematical terms, when one quantity varies inversely with another, it means that as one quantity increases, the other quantity decreases proportionally, and vice versa. This relationship can be expressed using an equation where the product of the two quantities is a constant. For instance, if we say that "q varies inversely as r," we can write this relationship as $q = k/r$, where k is a constant of proportionality. This constant, k, is crucial because it represents the fixed relationship between q and r. The value of k remains the same regardless of the values of q and r, as long as they are inversely related. Understanding this constant is essential for solving problems involving inverse variation. To further clarify, consider a practical example. Imagine you are distributing a fixed amount of work among a group of people. The more people you have, the less work each person needs to do. Conversely, the fewer people you have, the more work each person needs to do. This is an inverse relationship, and the total amount of work is the constant, k. In our specific problem, we are given that q = 10 when r = 2.5. These values provide us with the key to finding the constant of proportionality, k. By substituting these values into the inverse variation equation, $q = k/r$, we can solve for k. Once we find the value of k, we can establish the specific equation that connects q and r for this particular scenario. This equation will allow us to determine the value of q for any given value of r, and vice versa, as long as they maintain the same inverse relationship. The ability to translate real-world scenarios into mathematical equations is a fundamental skill in mathematics and science. Inverse variation is a common type of relationship that appears in various contexts, from physics and engineering to economics and everyday life. By mastering the concept of inverse variation, you can gain a deeper understanding of how quantities relate to each other and make predictions based on these relationships. The problem's setup is straightforward: we have two variables, q and r, that are inversely related. We are given a specific pair of values for q and r, and our goal is to find the equation that represents their relationship. This involves identifying the constant of proportionality and expressing the inverse variation in a clear, concise equation. The equation will then serve as a model for how q and r interact, allowing us to analyze and solve related problems. In summary, the problem statement sets the stage for an exploration of inverse variation, requiring us to determine the equation that connects q and r based on the given information. This task highlights the practical application of mathematical concepts in describing and understanding real-world relationships.

Step-by-Step Solution: Finding the Equation

To find the equation that connects q and r, we will follow a systematic, step-by-step approach that leverages the principles of inverse variation. This method ensures a clear and logical progression towards the solution, making it easier to understand each stage of the process. First, we start with the fundamental concept of inverse variation. As stated earlier, when q varies inversely as r, we can express this relationship using the equation $q = k/r$, where k is the constant of variation. This equation is the cornerstone of our solution, as it mathematically represents the inverse relationship between q and r. The next crucial step is to determine the value of the constant k. We are given that q = 10 when r = 2.5. These values provide us with a specific data point that allows us to solve for k. To find k, we substitute these values into the equation $q = k/r$. This gives us $10 = k/2.5$. Now, we need to isolate k to find its value. To do this, we multiply both sides of the equation by 2.5: $10 * 2.5 = k$. This calculation yields $k = 25$. Therefore, the constant of variation for this particular relationship is 25. With the value of k determined, we can now write the specific equation that connects q and r. We substitute k = 25 back into the general inverse variation equation, $q = k/r$. This results in the equation $q = 25/r$. This equation is the solution to the problem, as it expresses the exact relationship between q and r given the initial conditions. To verify that our equation is correct, we can plug in the original values of q and r to ensure they satisfy the equation. If we substitute r = 2.5 into $q = 25/r$, we get $q = 25/2.5 = 10$, which matches the given value of q. This confirms that our equation accurately represents the inverse variation between q and r. In summary, our step-by-step solution involved: 1. Expressing the inverse variation relationship as $q = k/r$. 2. Substituting the given values of q and r to find the constant of variation k. 3. Substituting the value of k back into the equation to obtain the specific equation connecting q and r. 4. Verifying the equation by plugging in the original values. This structured approach not only leads to the correct answer but also enhances understanding of the underlying concepts. By breaking down the problem into manageable steps, we can confidently solve inverse variation problems and apply these skills to various mathematical and real-world scenarios. The final equation, $q = 25/r$, provides a clear and concise representation of the inverse relationship between q and r, demonstrating the power of mathematical equations in describing and predicting the behavior of variables.

Identifying the Correct Option

Now that we have found the equation that connects q and r, the next step is to identify the correct option from the given choices. This involves comparing our derived equation with the options provided and selecting the one that matches. The equation we found through our step-by-step solution is $q = 25/r$. This equation expresses the inverse relationship between q and r, where q is equal to 25 divided by r. Let's examine the options provided: A. $q = 4/r$ B. $q = r/4$ C. $q = r/25$ D. $q = 25/r$ By comparing our equation, $q = 25/r$, with the options, it is clear that option D, $q = 25/r$, exactly matches our solution. The other options do not represent the same relationship. Option A, $q = 4/r$, has a different constant of variation. Option B, $q = r/4$, represents a direct variation rather than an inverse variation. Similarly, option C, $q = r/25$, also represents a direct variation. Therefore, only option D accurately represents the inverse relationship we derived from the given information. To further confirm our choice, we can consider the properties of inverse variation. In an inverse relationship, as r increases, q decreases, and vice versa. Our equation, $q = 25/r$, demonstrates this property. If r increases, the denominator of the fraction increases, causing q to decrease. Conversely, if r decreases, the denominator decreases, causing q to increase. This behavior aligns with the definition of inverse variation, providing additional confidence in our solution. The process of identifying the correct option highlights the importance of accuracy in mathematical problem-solving. It is essential to carefully derive the solution and then meticulously compare it with the given options to ensure a perfect match. A small error in the solution or a misinterpretation of the options can lead to an incorrect answer. In this case, our methodical approach and attention to detail have allowed us to confidently identify the correct option. In summary, we found the correct option by: 1. Deriving the equation $q = 25/r$ through a step-by-step solution. 2. Comparing our equation with the provided options. 3. Identifying option D, $q = 25/r$, as the exact match. 4. Confirming our choice by considering the properties of inverse variation. This systematic process underscores the importance of both problem-solving skills and careful verification in mathematics. Choosing the correct option is the final step in demonstrating a complete understanding of the problem and its solution.

Conclusion: The Equation Connecting q and r

In conclusion, we have successfully determined the equation that connects q and r when q varies inversely as r, and q = 10 when r = 2.5. Through a step-by-step approach, we established the fundamental concept of inverse variation, calculated the constant of variation, derived the specific equation, and verified our solution against the given conditions. The equation that connects q and r is $q = 25/r$. This equation clearly represents the inverse relationship between q and r, where q is equal to 25 divided by r. This means that as r increases, q decreases proportionally, and vice versa. This inverse relationship is a fundamental concept in mathematics and has numerous applications in various fields, including physics, engineering, and economics. Understanding inverse variation allows us to model and predict how changes in one quantity will affect another, providing valuable insights in problem-solving and decision-making. Our solution process began with recognizing the inverse variation relationship and expressing it mathematically as $q = k/r$, where k is the constant of variation. We then used the given values of q and r to solve for k, finding that k = 25. Substituting this value back into the equation, we obtained the specific equation $q = 25/r$. This equation is not only the solution to the problem but also a powerful tool for analyzing the relationship between q and r. To ensure the accuracy of our solution, we verified it by plugging in the original values of q and r. Substituting r = 2.5 into the equation, we found that q = 25/2.5 = 10, which matches the given value. This verification step is crucial in mathematical problem-solving, as it confirms that our solution is consistent with the problem's conditions. Furthermore, we compared our derived equation with the provided options and identified option D, $q = 25/r$, as the correct choice. This process highlights the importance of meticulous comparison and attention to detail in mathematics. A slight error in the equation or a misinterpretation of the options can lead to an incorrect answer. In summary, our comprehensive approach to solving this problem demonstrates the key steps involved in tackling inverse variation problems. We: 1. Established the inverse variation relationship. 2. Calculated the constant of variation. 3. Derived the specific equation connecting q and r. 4. Verified our solution. 5. Identified the correct option. This structured method not only leads to the correct answer but also enhances understanding of the underlying mathematical concepts. By mastering the principles of inverse variation, you can confidently solve a wide range of problems and apply these skills in various practical contexts. The equation $q = 25/r$ serves as a testament to the power of mathematical equations in representing and understanding relationships between quantities.