Equation For Direct Variation In Amir's Stamp Collection

#main-title Selecting the correct equation to represent a direct variation is a fundamental concept in mathematics, particularly in algebra. This article delves into the problem of finding the equation that models Amir's stamp collection growth, providing a detailed explanation and addressing the underlying principles of direct variation. We will explore the given data, analyze the options, and ultimately determine the accurate equation. This comprehensive guide is designed to help students and enthusiasts alike grasp the core concepts and apply them effectively to similar problems.

Understanding Direct Variation

Direct variation is a mathematical relationship between two variables where one variable is a constant multiple of the other. In simpler terms, as one variable increases, the other increases proportionally, and as one decreases, the other decreases proportionally. This relationship can be expressed in the form y = kx, where:

• y is the dependent variable,
• x is the independent variable,
• k is the constant of variation.

The constant of variation, k, represents the ratio between y and x. It signifies the rate at which y changes with respect to x. In the context of Amir's stamp collection, the number of stamps collected (y) varies directly with the number of weeks (x). To find the equation that represents this direct variation, we need to determine the value of k. Understanding direct variation is crucial not only for solving mathematical problems but also for recognizing and interpreting proportional relationships in real-world scenarios. From calculating distances based on speed and time to understanding how costs increase with the quantity of items purchased, direct variation plays a significant role in various aspects of our lives. Recognizing this relationship allows us to make predictions, solve problems, and gain a deeper insight into the interconnectedness of different variables. In the subsequent sections, we will apply this understanding to Amir's stamp collection problem and identify the correct equation that models his progress.

Problem Statement: Amir's Stamp Collection

Our central problem revolves around Amir's stamp collection. We are given that Amir starts a stamp collection and, after 3 weeks, he has collected 35 different stamps. After 9 weeks, his collection grows to 105 different stamps. The core question we aim to answer is: Which equation accurately represents this direct variation? This problem embodies the concept of direct variation, where the number of stamps collected is directly proportional to the number of weeks. To solve this, we need to identify the constant of variation, which will allow us to formulate the correct equation. The provided information gives us two data points: (3 weeks, 35 stamps) and (9 weeks, 105 stamps). These data points are crucial for determining the relationship between the number of weeks and the number of stamps collected. Understanding the problem statement is the first step towards finding the correct solution. It involves recognizing the variables, identifying the relationship between them, and understanding the goal of the problem. In this case, the variables are the number of weeks (x) and the number of stamps (y), the relationship is direct variation, and the goal is to find the equation that represents this relationship. By carefully analyzing the problem statement and the given data, we can proceed to the next step, which involves applying the principles of direct variation to determine the correct equation.

Analyzing the Given Data

To determine the equation that represents the direct variation in Amir's stamp collection, we need to carefully analyze the given data. We have two key data points:

• After 3 weeks, Amir has 35 stamps.
• After 9 weeks, Amir has 105 stamps.

These data points can be represented as ordered pairs (3, 35) and (9, 105), where the first value represents the number of weeks (x) and the second value represents the number of stamps (y). In a direct variation relationship, the ratio between y and x remains constant. This constant ratio is the constant of variation, k. To find k, we can use either of the data points. Let's use the first data point (3, 35):

k = y / x = 35 / 3

This calculation gives us the constant of variation, which is approximately 11.67. Now, let's verify this constant using the second data point (9, 105):

k = y / x = 105 / 9 = 35 / 3

As we can see, the constant of variation remains the same, confirming that this is indeed a direct variation. By analyzing the given data, we have successfully calculated the constant of variation, which is crucial for formulating the correct equation. This constant represents the rate at which Amir collects stamps per week. Understanding the significance of this constant is essential for both solving the problem and gaining a deeper understanding of direct variation. In the following sections, we will use this constant to identify the correct equation from the given options and further explore the implications of this direct variation relationship.

Identifying the Correct Equation

Now that we have determined the constant of variation, k, to be 35/3, we can identify the correct equation that represents the direct variation in Amir's stamp collection. Recall that the general form of a direct variation equation is:

y = kx

where y is the number of stamps, x is the number of weeks, and k is the constant of variation. Substituting the value of k we calculated earlier, we get:

y = (35/3)x

This equation represents the relationship between the number of weeks and the number of stamps collected. For every week that passes, Amir collects approximately 11.67 stamps. This equation allows us to predict the number of stamps Amir will have in his collection after any given number of weeks, assuming he continues to collect stamps at the same rate. To further validate this equation, we can plug in the given data points and see if they satisfy the equation. For the first data point (3, 35):

35 = (35/3) * 3 35 = 35

This confirms that the equation holds true for the first data point. For the second data point (9, 105):

105 = (35/3) * 9 105 = 105

This also confirms that the equation holds true for the second data point. By identifying the constant of variation and substituting it into the general form of a direct variation equation, we have successfully determined the correct equation that represents Amir's stamp collection growth. This equation provides a mathematical model of the relationship between the number of weeks and the number of stamps collected, allowing us to make predictions and further analyze this direct variation.

Conclusion: The Equation for Amir's Stamp Collection

In conclusion, after a thorough analysis of the given data and the principles of direct variation, we have successfully identified the correct equation that represents Amir's stamp collection growth. The problem presented us with a scenario where Amir's stamp collection increased proportionally with the number of weeks. By understanding the concept of direct variation and calculating the constant of variation, we were able to formulate the equation.

The correct equation is: y = (35/3)x

This equation accurately models the relationship between the number of weeks (x) and the number of stamps (y) in Amir's collection. The constant of variation, 35/3, signifies the rate at which Amir collects stamps per week. This comprehensive guide has not only provided the solution to the problem but also delved into the underlying concepts of direct variation, the importance of analyzing data, and the process of formulating mathematical equations to represent real-world scenarios. Understanding these principles is crucial for success in mathematics and for applying mathematical concepts to various aspects of life. We hope this explanation has provided clarity and enhanced your understanding of direct variation and its applications. By mastering these fundamental concepts, you will be well-equipped to tackle similar problems and gain a deeper appreciation for the power of mathematics in modeling and understanding the world around us.