Inverse Proportionality Equation And Value Calculation
If y is inversely proportional to x, and when y = 7, x = 9: a) Determine the equation that relates y and x. b) Calculate the value of y when x = 21.
In mathematics, inverse proportionality describes a relationship between two variables where one variable decreases as the other increases, and vice versa. This concept is fundamental in various scientific and engineering applications. This article, we will delve into a problem involving inverse proportionality, step-by-step determine the equation connecting the variables, and calculate the value of one variable given the other.
Understanding Inverse Proportionality
Before we dive into the problem, let's solidify our understanding of inverse proportionality. Two variables, typically denoted as x and y, are said to be inversely proportional if their product is constant. Mathematically, this relationship is expressed as:
y ∝ 1/x
This proportionality can be converted into an equation by introducing a constant of proportionality, often represented by k:
y = k/x
Where k is a non-zero constant. This equation tells us that y is equal to the constant k divided by x. As x increases, y decreases proportionally, and as x decreases, y increases proportionally. The constant k determines the strength of this inverse relationship.
To truly grasp inverse proportionality, it's crucial to connect it to real-world examples. Think about the relationship between the speed of a car and the time it takes to travel a certain distance. If you double the speed, you halve the time, assuming the distance remains constant. This is a classic example of inverse proportionality. Similarly, the number of workers on a project and the time it takes to complete it are inversely proportional. More workers mean less time, and fewer workers mean more time. Understanding these real-world applications will help you visualize and solve problems involving inverse proportionality more effectively. The key takeaway is that the product of the two variables remains constant, highlighting the inverse nature of their relationship.
Problem Statement
We are given that y is inversely proportional to x. This is the core concept we need to address, the relationship between the two variables. We are also provided with a specific condition: when y = 7, x = 9. Our goal is twofold:
a) First, we need to work out an equation connecting y and x. This involves determining the constant of proportionality, which will allow us to express the relationship between y and x in a precise mathematical form. This is a fundamental step in solving any problem involving proportionality, as it establishes the specific rule governing the relationship between the variables.
b) Second, we need to work out the value of y when x = 21. This requires us to apply the equation we derived in part (a) to find the corresponding value of y for a given value of x. This is a practical application of the equation, demonstrating how we can use it to predict the behavior of the variables under different conditions. This is a critical step, it requires careful substitution and calculation to arrive at the correct answer.
This problem exemplifies a typical scenario in inverse proportionality, where we are given some initial conditions and asked to find the general relationship and then use it to solve for specific values. By tackling this problem step-by-step, we'll reinforce our understanding of inverse proportionality and develop our problem-solving skills in this area.
Part a) Finding the Equation Connecting y and x
To find the equation connecting y and x, we start with the general form of inverse proportionality:
y = k/x
where k is the constant of proportionality. This is our starting point, the fundamental relationship that we know holds true for inversely proportional variables. The next step is crucial: determining the value of k. We are given that when y = 7, x = 9. We can use this information to solve for k. Substituting these values into the equation, we get:
7 = k/9
To isolate k, we multiply both sides of the equation by 9:
7 * 9 = k
63 = k
Therefore, the constant of proportionality, k, is 63. Now that we have found the value of k, we can substitute it back into the general equation to obtain the specific equation connecting y and x for this particular problem. Replacing k with 63 in the equation y = k/x, we get:
y = 63/x
This is the equation that connects y and x in this specific scenario. It tells us that y is equal to 63 divided by x. This equation is the key to solving the second part of the problem, where we need to find the value of y for a given value of x. It's important to remember that this equation is specific to the given conditions (when y = 7, x = 9) and the inverse relationship between y and x. Understanding how to derive this equation from the general form of inverse proportionality is a fundamental skill in solving these types of problems.
Part b) Working Out the Value of y when x = 21
Now that we have the equation connecting y and x, which we found to be y = 63/x, we can proceed to find the value of y when x = 21. This is a straightforward application of the equation we derived in the previous step. The process involves substituting the given value of x into the equation and then solving for y. This is a common technique in mathematics, where we use a known equation to find the value of an unknown variable. Substituting x = 21 into the equation y = 63/x, we get:
y = 63/21
Now, we simply need to perform the division to find the value of y. 63 divided by 21 is 3. Therefore:
y = 3
So, when x = 21, the value of y is 3. This result makes sense in the context of inverse proportionality. Since x has increased from its initial value of 9 to 21, we expect y to decrease proportionally. Indeed, y has decreased from 7 to 3. This confirms our understanding of the inverse relationship between x and y. We can also check our answer by verifying that the product of x and y remains constant. When x = 9 and y = 7, x * y* = 9 * 7 = 63. When x = 21 and y = 3, x * y* = 21 * 3 = 63. The product is the same, which is consistent with the principle of inverse proportionality. This step reinforces the importance of checking our answers whenever possible to ensure accuracy and to deepen our understanding of the underlying concepts.
Conclusion
In this article, we have successfully tackled a problem involving inverse proportionality. We started by understanding the fundamental concept of inverse proportionality, which states that the product of two inversely proportional variables is constant. This understanding is crucial for solving any problem related to inverse proportionality. We then moved on to the specific problem, where we were given that y is inversely proportional to x and that when y = 7, x = 9. The problem had two parts: first, we needed to find the equation connecting y and x; and second, we needed to find the value of y when x = 21.
To find the equation connecting y and x, we used the general form of inverse proportionality, y = k/x, where k is the constant of proportionality. We substituted the given values of x and y into this equation to solve for k, which we found to be 63. This allowed us to write the specific equation connecting y and x as y = 63/x. The ability to determine the constant of proportionality is a key skill in solving inverse proportionality problems. It allows us to move from a general relationship to a specific equation that applies to the given situation.
Next, we used this equation to find the value of y when x = 21. We simply substituted x = 21 into the equation y = 63/x and solved for y, which we found to be 3. This demonstrated how we can use the equation we derived to predict the value of one variable given the value of the other. This is a practical application of inverse proportionality, and it highlights the power of mathematical equations in modeling real-world relationships.
By solving this problem step-by-step, we have reinforced our understanding of inverse proportionality and developed our problem-solving skills in this area. Remember, the key to solving inverse proportionality problems is to understand the concept, identify the constant of proportionality, and use the resulting equation to solve for unknown values. Practice is essential to master these skills, so be sure to work through a variety of problems to solidify your understanding.
Keywords
Inverse Proportionality, Equation, Constant of Proportionality, Value, Variables
Repair Input Keyword
If y is inversely proportional to x, and when y = 7, x = 9:
a) Determine the equation that relates y and x.
b) Calculate the value of y when x = 21.
Title
Inverse Proportionality Equation and Value Calculation