Evaluating The Expression -6(4 2/3) A Step-by-Step Guide

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In the realm of mathematics, evaluating expressions accurately is a fundamental skill. This article delves into the evaluation of the expression $-6(4 \frac{2}{3})$, exploring the correct method and dissecting why certain approaches are valid while others are not. Our primary focus will be on identifying the appropriate technique to solve this problem, ensuring a clear understanding of the underlying mathematical principles.

Understanding the Expression and the Distributive Property

Before diving into the options, it's crucial to understand the expression $-6(4 \frac{2}{3})$. This expression involves multiplying -6 by a mixed number, $4 \frac{2}{3}$. The core concept for correctly evaluating this expression is the distributive property. The distributive property states that a(b + c) = ab + ac. This property allows us to multiply a single term by a sum or difference of terms within parentheses.

In our case, we can rewrite the mixed number $4 \frac{2}{3}$ as the sum of its whole number part and its fractional part: 4 + $\frac{2}{3}$. Therefore, the expression becomes $-6(4 + \frac{2}{3})$. Applying the distributive property, we multiply -6 by both 4 and $\frac{2}{3}$. This means we need to perform two separate multiplications: (-6)(4) and (-6)($\frac{2}{3}$), and then add the results together.

To truly grasp this concept, let's delve deeper into why the distributive property is so essential in this context. When we have a number multiplied by a sum, we are essentially scaling the entire sum by that number. Think of it like having six groups of (4 + $\frac{2}{3}$). Each group contains 4 whole units and two-thirds of another unit. To find the total, we need to account for the scaling effect on both the whole units and the fractional part. The distributive property provides a systematic way to do this, ensuring that every component within the parentheses is correctly multiplied by the factor outside.

Furthermore, consider the implications of not using the distributive property correctly. If we were to simply multiply -6 by 4 and then by $\frac{2}{3}$ without adding them appropriately, we would be neglecting the fundamental principle of how multiplication interacts with addition. This could lead to a drastically different and incorrect result. For instance, if we were to multiply -6 by 4 and then multiply the result by $\frac{2}{3}$, we would be performing a completely different mathematical operation, one that doesn't accurately represent the original expression.

Therefore, a thorough understanding of the distributive property is not just a matter of following a rule; it's about grasping the underlying logic of how numbers interact in mathematical expressions. It allows us to break down complex problems into simpler steps, ensuring that we account for every component and arrive at the correct solution. In the context of our expression, $-6(4 \frac{2}{3})$, the distributive property serves as the key to unlocking the correct evaluation, providing a pathway to accurately multiply -6 by the mixed number.

Analyzing the Answer Choices

Now, let's analyze the given answer choices in light of the distributive property:

• A. (-6)(4) + (-6)($\frac{2}{3}$)

  This option correctly applies the distributive property. It multiplies -6 by both 4 and $\frac{2}{3}$ and then adds the results. This aligns perfectly with the distributive property a(b + c) = ab + ac, where a = -6, b = 4, and c = $\frac{2}{3}$. Thus, this option appears to be the correct one.

  To further solidify our understanding, let's break down this option step by step. First, we multiply -6 by 4, which gives us -24. Then, we multiply -6 by $\frac{2}{3}$. To do this, we can think of -6 as $\frac{-6}{1}$ and multiply the numerators and denominators: $\frac{-6}{1} * \frac{2}{3}$ = $\frac{-12}{3}$, which simplifies to -4. Finally, we add the two results: -24 + (-4) = -28. This demonstrates how the distributive property allows us to systematically break down the expression and arrive at a numerical answer.

  Moreover, this option mirrors the fundamental principle of the distributive property, which is to ensure that the term outside the parentheses is multiplied by every term inside the parentheses. By multiplying -6 by both 4 and $\frac{2}{3}$, we are adhering to this principle and correctly scaling the entire expression within the parentheses. This approach leaves no room for ambiguity and guarantees that we are accounting for every component of the original expression.

• B. (-6)(4) × (-6)($\frac{2}{3}$)

  This option incorrectly multiplies the results of (-6)(4) and (-6)($\frac{2}{3}$). The distributive property involves addition, not multiplication, between the two products. This choice misinterprets the fundamental operation required to correctly distribute -6 across the sum within the parentheses.

  The error in this option stems from a misunderstanding of the distributive property's core mechanism. Instead of adding the scaled components, it multiplies them, which fundamentally alters the mathematical relationship. To illustrate this, let's consider a simpler example: 2(1 + 2). The distributive property dictates that this equals (2)(1) + (2)(2) = 2 + 4 = 6. If we were to incorrectly apply the logic of option B, we would calculate (2)(1) × (2)(2) = 2 × 4 = 8, which is clearly incorrect. This simple example highlights the critical difference between addition and multiplication in the context of distribution.

• C. (-6 + 4) + (-6 + $\frac{2}{3}$)

  This option incorrectly adds -6 to both 4 and $\frac{2}{3}$. The distributive property involves multiplication, not addition, of the term outside the parentheses with the terms inside. This choice completely disregards the order of operations and the principles of distribution.

  The fallacy in this option lies in the fact that it treats multiplication as addition. In the original expression, -6 is a factor that multiplies the entire quantity (4 + $\frac{2}{3}$). Adding -6 to each term within the parentheses fundamentally changes the mathematical operation and yields an incorrect result. It's akin to saying that 2(1 + 2) is the same as (2 + 1) + (2 + 2), which is clearly not true. The distributive property is all about scaling, and addition does not achieve the same scaling effect as multiplication.

• D. (-6 + 4) × (-6 + $\frac{2}{3}$)

  This option also incorrectly adds -6 to both 4 and $\frac{2}{3}$ and then multiplies the results. Similar to option C, this misinterprets the distributive property by using addition instead of multiplication. Additionally, it compounds the error by multiplying the results of the incorrect additions, further deviating from the correct application of the distributive property.

  This option represents a double misapplication of mathematical principles. Not only does it incorrectly substitute addition for multiplication in the distributive process, but it also compounds the error by multiplying the results of these incorrect additions. This leads to a result that is far removed from the correct evaluation of the original expression. To understand the magnitude of the error, consider our earlier example: 2(1 + 2). If we were to apply the logic of option D, we would calculate (2 + 1) × (2 + 2) = 3 × 4 = 12, which is significantly different from the correct answer of 6. This highlights how seemingly small deviations from the correct procedure can lead to substantial errors in the final result.

The Correct Answer: Option A

Based on our analysis, option A, (-6)(4) + (-6)($\frac{2}{3}$), is the correct answer. It accurately applies the distributive property, ensuring that -6 is correctly multiplied by both the whole number and fractional parts of the mixed number.

Option A stands out as the only choice that faithfully adheres to the distributive property's rules. By multiplying -6 by both 4 and $\frac{2}{3}$ and then adding the results, it perfectly mirrors the principle of scaling the entire sum within the parentheses by the factor outside. This approach not only guarantees the correct result but also showcases a deep understanding of the distributive property's underlying logic. In contrast, the other options deviate from this principle in various ways, either by incorrectly using addition instead of multiplication or by misinterpreting the order of operations, ultimately leading to erroneous conclusions.

Conclusion

In conclusion, understanding and correctly applying the distributive property is crucial for evaluating expressions like $-6(4 \frac{2}{3})$. Option A demonstrates the proper application of this property, while the other options showcase common errors in mathematical reasoning. By mastering the distributive property, you can confidently tackle a wide range of mathematical problems and ensure accurate results.

This exploration of the expression $-6(4 \frac{2}{3})$ serves as a valuable lesson in the importance of adhering to fundamental mathematical principles. The distributive property, in particular, is a cornerstone of algebraic manipulation and is essential for simplifying and evaluating expressions correctly. By carefully analyzing the given options and understanding the underlying mathematical operations, we can confidently identify the correct approach and avoid common pitfalls. This not only enhances our problem-solving skills but also deepens our appreciation for the elegance and precision of mathematics.