Simplifying The Mathematical Expression 3 × 6 + 4 Of 6 - 6 + 2 × (4 - 6)
In this article, we will delve into simplifying a complex mathematical expression. Mastering the order of operations is crucial for solving such expressions accurately. The expression we aim to simplify is: 3 × 6 + 4 of 6 - 6 + 2 × (4 - 6) + 4 - 2 × 3 + 6 of 1/3. This problem requires a solid understanding of the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Each step in the simplification process will be thoroughly explained, ensuring a clear understanding of how to arrive at the correct answer. We will begin by addressing the 'of' operations, which often cause confusion but are essentially a form of multiplication. Then, we will handle the parentheses, followed by multiplication and division, and finally, addition and subtraction. Let's embark on this mathematical journey to demystify the expression and find its simplified value. Understanding each step not only helps in solving this particular problem but also builds a strong foundation for tackling more complex mathematical challenges in the future. Through this detailed exploration, you will gain confidence in your ability to simplify any mathematical expression effectively.
Breaking Down the Expression
The given mathematical expression is: 3 × 6 + 4 of 6 - 6 + 2 × (4 - 6) + 4 - 2 × 3 + 6 of 1/3. To simplify this, we need to follow the order of operations (PEMDAS/BODMAS). First, let's address the 'of' operations, which signify multiplication. '4 of 6' means 4 multiplied by 6, and '6 of 1/3' means 6 multiplied by 1/3. Next, we'll handle the parenthetical operation (4 - 6). After dealing with these preliminary steps, we'll focus on multiplications and divisions from left to right, and finally, we'll perform additions and subtractions, also from left to right. This methodical approach ensures that we don't miss any steps and reduces the chance of errors. Each operation will be performed with careful attention to detail, and the intermediate results will be noted to maintain clarity. This structured approach is essential in simplifying complex expressions and arriving at the correct solution. By breaking down the problem into manageable parts, we can tackle it systematically and efficiently. Remember, accuracy and precision are key in mathematics, and following the correct order of operations is paramount.
Step-by-Step Simplification
Now, let's simplify the expression step-by-step. First, we tackle the 'of' operations:
4 of 6 = 4 × 6 = 246 of 1/3 = 6 × (1/3) = 2
Substituting these values back into the expression, we get: 3 × 6 + 24 - 6 + 2 × (4 - 6) + 4 - 2 × 3 + 2. Next, we deal with the parentheses: (4 - 6) = -2. The expression now becomes: 3 × 6 + 24 - 6 + 2 × (-2) + 4 - 2 × 3 + 2. Now, we perform the multiplications:
3 × 6 = 182 × (-2) = -42 × 3 = 6
Substituting these values, the expression is now: 18 + 24 - 6 - 4 + 4 - 6 + 2. Finally, we perform the additions and subtractions from left to right:
18 + 24 = 4242 - 6 = 3636 - 4 = 3232 + 4 = 3636 - 6 = 3030 + 2 = 32
Thus, the simplified value of the expression is 32. This step-by-step approach ensures that each operation is performed in the correct order, leading to the accurate final result. Attention to detail and methodical execution are essential in simplifying complex mathematical expressions.
Detailed Calculation Breakdown
To further illustrate the simplification process, let's break down the calculation in even greater detail. We started with the expression: 3 × 6 + 4 of 6 - 6 + 2 × (4 - 6) + 4 - 2 × 3 + 6 of 1/3. First, we addressed the 'of' operations: 4 of 6 = 24 and 6 of 1/3 = 2. Substituting these, we have: 3 × 6 + 24 - 6 + 2 × (4 - 6) + 4 - 2 × 3 + 2. Next, we simplified the parentheses: (4 - 6) = -2, resulting in: 3 × 6 + 24 - 6 + 2 × (-2) + 4 - 2 × 3 + 2. Now, we perform the multiplications:
3 × 6 = 182 × (-2) = -42 × 3 = 6
Substituting these values back into the expression gives us: 18 + 24 - 6 - 4 + 4 - 6 + 2. Finally, we perform the additions and subtractions from left to right:
18 + 24 = 4242 - 6 = 3636 - 4 = 3232 + 4 = 3636 - 6 = 3030 + 2 = 32
Therefore, the final simplified value of the expression is 32. This detailed breakdown highlights the importance of following the order of operations meticulously to avoid errors. Each step builds upon the previous one, and precision at each stage is crucial for obtaining the correct result.
Analyzing the Answer Choices and Identifying the Correct Solution
After simplifying the expression 3 × 6 + 4 of 6 - 6 + 2 × (4 - 6) + 4 - 2 × 3 + 6 of 1/3, we arrived at the result 32. Now, let's analyze the provided answer choices to identify the correct solution. The given options are:
1 3/47 3/413 3/48 1/3
Comparing our result (32) with the given options, we can clearly see that none of the choices match our calculated value. This discrepancy indicates a potential error in the provided answer choices or a misunderstanding of the question's intent. It's crucial to double-check our calculations to ensure accuracy. However, based on our detailed step-by-step simplification, the result of 32 appears to be correct. Therefore, we can conclude that the correct answer is not among the given options. This situation underscores the importance of not only performing calculations accurately but also critically evaluating the results in the context of the given choices. In cases like this, it might be necessary to seek clarification or re-evaluate the problem statement to ensure a correct interpretation. Accuracy in calculation and critical evaluation are both essential skills in mathematical problem-solving.
Conclusion: The Importance of Order of Operations
In conclusion, simplifying the mathematical expression 3 × 6 + 4 of 6 - 6 + 2 × (4 - 6) + 4 - 2 × 3 + 6 of 1/3 demonstrates the critical importance of following the order of operations (PEMDAS/BODMAS). Through a step-by-step approach, we first addressed the 'of' operations, then the parentheses, followed by multiplications, and finally, additions and subtractions. Our detailed calculations led us to the simplified value of 32. However, upon analyzing the provided answer choices, we found that none of them matched our result, indicating a possible issue with the options themselves. This exercise highlights not only the necessity of accurate calculation but also the importance of critical thinking and evaluation in mathematical problem-solving. Mastering the order of operations is fundamental for success in mathematics, as it ensures consistency and accuracy in simplifying complex expressions. Furthermore, the ability to analyze and interpret results in the context of the given problem is equally crucial. By combining these skills, we can confidently tackle a wide range of mathematical challenges. The systematic approach used in this article serves as a valuable framework for simplifying any mathematical expression, emphasizing the significance of each step in the process.