Find The Correct Equation For Means And Extremes In Proportion 7.2 / 9.6 = 21.6 / 28.8

When dealing with proportions, understanding the relationship between means and extremes is crucial. A proportion is an equation that states that two ratios are equal. In the proportion ${ a/b = c/d }$, the terms a and d are called the extremes, while the terms b and c are called the means. A fundamental property of proportions is that the product of the means equals the product of the extremes. This principle is essential for solving various mathematical problems, and in this article, we will delve deeply into this concept and apply it to a specific problem.

Defining Proportions, Means, and Extremes

Before we tackle the specific question, let’s clarify the foundational concepts. A proportion is essentially a statement of equality between two ratios. Ratios, in turn, are comparisons of two quantities by division. For example, if we say the ratio of apples to oranges is 2 to 3, we are expressing a proportional relationship that can be written as ${ 2/3 }$. When two such ratios are equated, we form a proportion, such as ${ a/b = c/d }$.

In this proportional relationship, each term has a specific designation: a and d are the extremes, while b and c are the means. The extremes are the first and last terms of the proportion, while the means are the inner terms. This nomenclature is derived from their positions when the proportion is written in its standard form.

The significance of understanding means and extremes lies in the property that their products are always equal in a valid proportion. This property, often referred to as the cross-product property, is a cornerstone of proportional reasoning. It allows us to solve for unknown quantities within a proportion and verify the correctness of a proportional relationship. For instance, if we have a proportion ${ a/b = c/d }$, then it must be true that ${ a imes d = b imes c }$. This equation forms the basis for solving many problems involving proportions.

To illustrate this, consider the proportion ${ 3/4 = 6/8 }$. Here, 3 and 8 are the extremes, and 4 and 6 are the means. According to the cross-product property, ${ 3 imes 8 }$ should equal ${ 4 imes 6 }$. Indeed, ${ 3 imes 8 = 24 }$ and ${ 4 imes 6 = 24 }$, confirming that the proportion is valid. This fundamental rule is not just a mathematical curiosity; it is a practical tool used in various fields, from basic algebra to complex engineering calculations.

The importance of this concept extends to real-world applications. Proportions are used extensively in scaling recipes, converting units, calculating financial ratios, and determining geometric similarities. Understanding how to correctly identify and use means and extremes can simplify problem-solving in these areas. For example, if a recipe calls for 2 cups of flour for every 1 cup of sugar, and you want to make a larger batch using 6 cups of flour, you can set up a proportion to find out how much sugar you need. The proportion would be ${ 2/1 = 6/x }$, where x is the unknown amount of sugar. By applying the cross-product property, you can quickly solve for x and ensure your recipe turns out perfectly.

Furthermore, the concept of means and extremes is crucial in understanding more advanced mathematical concepts such as similar triangles and trigonometric ratios. In geometry, similar triangles have corresponding sides that are in proportion, and the properties of means and extremes help in determining unknown side lengths. In trigonometry, ratios like sine, cosine, and tangent are essentially proportions, and understanding these proportional relationships is key to solving problems involving angles and distances.

In summary, the concepts of proportions, means, and extremes are fundamental in mathematics and have far-reaching applications in various fields. The principle that the product of the means equals the product of the extremes is a powerful tool for solving problems and verifying proportional relationships. By mastering these concepts, one can build a strong foundation for more advanced mathematical studies and practical applications.

Analyzing the Given Proportion: 7.2 / 9.6 = 21.6 / 28.8

Now, let's apply our understanding of means and extremes to the given proportion: ${ 7.2 / 9.6 = 21.6 / 28.8 }$. Our primary goal is to identify the means and the extremes correctly and then determine which equation accurately represents the multiplication of these terms according to the fundamental property of proportions. This involves a meticulous examination of the given ratios and their components to ensure that we correctly apply the rule that the product of the means equals the product of the extremes.

In the given proportion, ${ 7.2 / 9.6 = 21.6 / 28.8 }$, we can clearly identify the extremes and the means. The extremes are the first and last terms, which are 7.2 and 28.8. The means are the inner terms, which are 9.6 and 21.6. To correctly apply the property of proportions, we need to multiply the extremes together and the means together and then equate the two products. This process ensures that we are adhering to the basic rule that governs proportional relationships.

The rule that the product of the means equals the product of the extremes is a direct consequence of the properties of equality and multiplication. To understand why this rule holds, consider the proportion ${ a/b = c/d }$. To eliminate the fractions, we can multiply both sides of the equation by the product of the denominators, which is ${ b imes d }$. This gives us:

${ (a/b) imes (b imes d) = (c/d) imes (b imes d) }$

Simplifying both sides, we get:

${ a imes d = c imes b }$

This equation, ${ a imes d = b imes c }$, clearly shows that the product of the extremes (a and d) is equal to the product of the means (b and c). This algebraic manipulation provides a concrete justification for the rule and highlights its mathematical validity. Understanding this derivation can help in remembering the rule and applying it correctly in various problem-solving scenarios.

Now, applying this understanding to our specific proportion, ${ 7.2 / 9.6 = 21.6 / 28.8 }$, we need to multiply the extremes (7.2 and 28.8) and the means (9.6 and 21.6). This gives us the equation:

${ 7.2 imes 28.8 = 9.6 imes 21.6 }$

This equation represents the correct application of the cross-product property to the given proportion. It states that the product of 7.2 and 28.8 should be equal to the product of 9.6 and 21.6. This equation is crucial for verifying the correctness of the proportion and for solving problems related to proportional relationships.

When presented with multiple options, it is essential to carefully examine each one and compare it to the correct equation derived from the principle of means and extremes. Common errors include misidentifying the means and extremes or incorrectly setting up the multiplication. By understanding the underlying principle and practicing its application, we can avoid these mistakes and accurately solve problems involving proportions. The ability to correctly identify and manipulate means and extremes is a fundamental skill in mathematics, with applications ranging from basic algebra to advanced calculus.

In conclusion, the process of analyzing a proportion involves identifying the means and extremes, understanding the principle that their products are equal, and correctly setting up the equation. This foundational knowledge is essential for solving a wide range of mathematical problems and for developing a deeper understanding of proportional relationships.

Evaluating the Options: Identifying the Correct Equation

Having established the correct equation that represents the multiplication of means and extremes in the proportion ${ 7.2 / 9.6 = 21.6 / 28.8 }$, we now need to evaluate the given options to identify the one that matches our derived equation. This step is crucial in problem-solving as it involves a careful comparison of different mathematical statements to determine their accuracy. The correct equation, as we determined, is ${ 7.2 imes 28.8 = 9.6 imes 21.6 }$. Now, let's examine the options provided and see which one aligns with this equation.

The options are:

A) ${ 7.2 imes 21.6 = 28.8 imes 9.6 }$ B) ${ 7.2 imes 28.8 = 21.6 imes 28.8 }$ C) ${ 9.6 imes 21.6 = 28.8 imes 7.2 }$ D) ${ 7.2 imes 9.6 = 21.6 }$

To correctly identify the matching equation, we need to compare each option with our derived equation, ${ 7.2 imes 28.8 = 9.6 imes 21.6 }$. This involves checking whether the products on both sides of the equation are correctly formed using the means and extremes of the original proportion. A systematic approach to this comparison can prevent errors and ensure accuracy.

Let’s start with Option A: ${ 7.2 imes 21.6 = 28.8 imes 9.6 }$. In this equation, 7.2 and 21.6 are multiplied together, and 28.8 and 9.6 are multiplied together. Comparing this to our derived equation, we see that the extremes (7.2 and 28.8) are not multiplied together on the same side of the equation, and neither are the means (9.6 and 21.6). Therefore, Option A does not correctly represent the multiplication of the means and extremes.

Next, we consider Option B: ${ 7.2 imes 28.8 = 21.6 imes 28.8 }$. This equation has 7.2 and 28.8 on one side, which are the extremes, but it incorrectly equates their product with the product of 21.6 and 28.8. The number 28.8 appears twice, which is not consistent with the principle of means and extremes. Thus, Option B is also incorrect.

Now, let’s look at Option C: ${ 9.6 imes 21.6 = 28.8 imes 7.2 }$. This equation has the means (9.6 and 21.6) multiplied together on one side and the extremes (7.2 and 28.8) multiplied together on the other side. This matches our derived equation, ${ 7.2 imes 28.8 = 9.6 imes 21.6 }$, since the order of multiplication does not affect the product (i.e., ${ a imes b = b imes a }$). Therefore, Option C correctly represents the multiplication of the means and extremes.

Finally, we examine Option D: ${ 7.2 imes 9.6 = 21.6 }$. This equation is incomplete as it only provides the product of 7.2 and 9.6 on one side and the single number 21.6 on the other side. It does not represent the multiplication of means and extremes in the form of an equation where two products are equated. Therefore, Option D is incorrect.

By systematically comparing each option to the correct equation, we can confidently conclude that Option C is the correct answer. This process highlights the importance of not only understanding the underlying mathematical principles but also applying careful evaluation techniques to arrive at the correct solution.

In summary, evaluating the options involved comparing each given equation with the derived equation that correctly represents the multiplication of the means and extremes. This process required a thorough understanding of the proportional relationship and the ability to identify the correct arrangement of terms. Option C was the only equation that accurately matched the principle that the product of the means equals the product of the extremes, making it the correct choice.

Conclusion: Option C Correctly Shows the Multiplication of Means and Extremes

In conclusion, after a detailed analysis of the proportion ${ 7.2 / 9.6 = 21.6 / 28.8 }$ and the given options, we have determined that Option C, which states ${ 9.6 imes 21.6 = 28.8 imes 7.2 }$, correctly shows the multiplication of the means and extremes. This determination was made by first identifying the means (9.6 and 21.6) and the extremes (7.2 and 28.8) in the proportion and then applying the fundamental property that the product of the means equals the product of the extremes. This property is a cornerstone of proportional reasoning and is essential for solving a wide variety of mathematical problems.

The process of arriving at the correct answer involved several key steps. First, we clarified the concepts of proportions, means, and extremes, emphasizing the significance of the cross-product property. This property, derived from the basic principles of algebra, provides a reliable method for verifying proportional relationships and solving for unknown quantities. Understanding the algebraic basis of this property helps in remembering and applying it correctly.

Next, we applied our understanding to the given proportion, identifying the means and extremes and setting up the correct equation that represents their multiplication. This involved a careful examination of the proportion to ensure that we accurately identified each term. The correct equation, ${ 7.2 imes 28.8 = 9.6 imes 21.6 }$, was derived directly from the definition of means and extremes and the principle that their products are equal.

Finally, we evaluated each of the provided options, comparing them to our derived equation. This step required a systematic approach, where each option was scrutinized to determine whether it correctly represented the multiplication of the means and extremes. By eliminating the incorrect options one by one, we were able to confidently identify Option C as the correct answer. This process highlights the importance of both understanding the underlying mathematical principles and applying careful evaluation techniques.

The correct equation, ${ 9.6 imes 21.6 = 28.8 imes 7.2 }$, is a direct application of the principle that the product of the means equals the product of the extremes. This principle is not only useful in solving problems involving proportions but also forms the basis for more advanced mathematical concepts, such as similar triangles and trigonometric ratios. A strong understanding of proportions and their properties is therefore essential for success in mathematics.

The ability to correctly identify and manipulate means and extremes is also valuable in real-world applications. Proportions are used in various fields, including cooking, construction, and finance. For example, when scaling a recipe, understanding proportions ensures that the ingredients are in the correct ratio, resulting in a successful dish. In construction, proportions are used to create accurate scale models and ensure that building materials are used efficiently. In finance, proportions are used to calculate ratios and analyze financial data.

In summary, the question of which equation correctly shows the multiplication of the means and extremes in the proportion ${ 7.2 / 9.6 = 21.6 / 28.8 }$ has been thoroughly addressed. By understanding the concepts of proportions, means, and extremes, applying the cross-product property, and carefully evaluating the given options, we have confidently concluded that Option C is the correct answer. This process underscores the importance of a solid foundation in mathematical principles and the application of systematic problem-solving techniques. Mastering these skills is essential for success in mathematics and various real-world applications.