Evaluating $2+3[1/2+{1/2-3(1/2-1/4)}]$ A Step-by-Step Guide

Jul 20, 2025 by ADMIN 60 views

$Evaluating the Expression: $2+3\left[\frac{1}{2}+\left{\frac{1}{2}-3\left(\frac{1}{2}-\frac{1}{4}\right)\right}\right]$$

Introduction to Order of Operations

In the realm of mathematics, evaluating expressions accurately hinges on adhering to the order of operations. This fundamental principle ensures consistency and clarity in mathematical calculations, preventing ambiguity and yielding correct results. The expression $2+3\left[\frac{1}{2}+\left{\frac{1}{2}-3\left(\frac{1}{2}-\frac{1}{4}\right)\right}\right]$ exemplifies the need for a systematic approach, where parentheses, brackets, and braces nest multiple operations. To dissect this expression, we navigate through these layers, adhering to the conventional order: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right), often remembered by the acronym PEMDAS or BODMAS. This method provides a roadmap for simplifying complex arithmetic structures. The step-by-step simplification not only yields the final numerical answer but also enhances comprehension of how different mathematical operators interact within an expression. By mastering this order, one gains the ability to tackle more intricate mathematical problems with assurance, making it an indispensable skill for anyone venturing into algebra, calculus, and beyond.

Step-by-Step Evaluation

The evaluation of the expression $2+3\left[\frac{1}{2}+\left{\frac{1}{2}-3\left(\frac{1}{2}-\frac{1}{4}\right)\right}\right]$ requires a meticulous, step-by-step approach to ensure accuracy. Initially, we focus on the innermost parentheses, a crucial first step in untangling the complexities of the expression. Within these parentheses, we encounter the subtraction $\left(\frac{1}{2}-\frac{1}{4}\right)$ . Finding a common denominator allows us to rewrite $\frac{1}{2}$ as $\frac{2}{4}$ , thus transforming the subtraction into $\frac{2}{4}-\frac{1}{4}$ , which simplifies to $\frac{1}{4}$ . This seemingly small step is paramount as it begins to unravel the layers of operations. Next, we address the multiplication within the braces. We multiply the result from the parentheses, $\frac{1}{4}$ , by 3, yielding $\frac{3}{4}$ . The expression within the braces now reads $\left{\frac{1}{2}-\frac{3}{4}\right}$ . Again, a common denominator is needed to perform the subtraction. We rewrite $\frac{1}{2}$ as $\frac{2}{4}$ , resulting in $\frac{2}{4}-\frac{3}{4}$ , which simplifies to $-\frac{1}{4}$ . This progression highlights the importance of carefully managing fractions and signs to maintain precision throughout the calculation. The meticulous execution of each step not only minimizes the risk of error but also fosters a deeper understanding of the arithmetic processes involved. As we advance through the subsequent operations, this foundational clarity becomes increasingly valuable, ultimately leading to the correct final result.

Working Through Brackets and Braces

Having simplified the innermost parentheses, the next crucial step in evaluating the expression $2+3\left[\frac{1}{2}+\left{\frac{1}{2}-3\left(\frac{1}{2}-\frac{1}{4}\right)\right}\right]$ is to tackle the brackets and braces. The expression now stands at $2+3\left[\frac{1}{2}+\left{-\frac{1}{4}\right}\right]$ , following our previous simplification. The focus shifts to the expression within the square brackets: $\frac{1}{2}+\left{-\frac{1}{4}\right}$ . Adding a negative fraction is akin to subtraction, so we rewrite the expression as $\frac{1}{2}-\frac{1}{4}$ . Once again, finding a common denominator is essential for performing this operation accurately. Converting $\frac{1}{2}$ to $\frac{2}{4}$ , we have $\frac{2}{4}-\frac{1}{4}$ , which simplifies to $\frac{1}{4}$ . This step effectively reduces the complexity of the expression, bringing us closer to the final solution. The ability to fluidly navigate between addition and subtraction of fractions is a key skill in simplifying complex mathematical expressions. By methodically working through the brackets and braces, we ensure that each operation is performed in the correct sequence, thereby minimizing errors and maintaining the integrity of the calculation. This careful approach not only yields the correct answer but also reinforces the importance of precision in mathematical manipulations, a skill that is invaluable in more advanced mathematical contexts.

Finalizing the Calculation

With the brackets and braces simplified, the expression $2+3\left[\frac{1}{2}+\left{\frac{1}{2}-3\left(\frac{1}{2}-\frac{1}{4}\right)\right}\right]$ has been reduced to a more manageable form: $2 + 3\left(\frac{1}{4}\right)$ . Now, we address the remaining multiplication. Multiplying 3 by $\frac{1}{4}$ gives us $\frac{3}{4}$ . The expression now reads $2 + \frac{3}{4}$ . The final step involves adding 2 and $\frac{3}{4}$ . To do this, we can convert 2 into a fraction with a denominator of 4, which gives us $\frac{8}{4}$ . Adding $\frac{8}{4}$ and $\frac{3}{4}$ results in $\frac{11}{4}$ . This fraction represents the final simplified value of the original expression. We can also express this result as a mixed number, which is 2 $\frac{3}{4}$ . The completion of this calculation underscores the significance of following the order of operations meticulously. Each step, from the innermost parentheses to the final addition, builds upon the previous one, and any deviation from the correct sequence could lead to an incorrect result. This systematic approach not only provides the solution but also demonstrates the elegance and precision inherent in mathematical problem-solving. The final result, $\frac{11}{4}$ or 2 $\frac{3}{4}$ , is a testament to the power of methodical simplification and a clear understanding of arithmetic principles.

Detailed Solution Breakdown

To fully appreciate the solution to the expression $2+3\left[\frac{1}{2}+\left{\frac{1}{2}-3\left(\frac{1}{2}-\frac{1}{4}\right)\right}\right]$ , let's break down each step in exhaustive detail.

Innermost Parentheses:
- Begin by simplifying the expression inside the innermost parentheses: $\left(\frac{1}{2}-\frac{1}{4}\right)$ .
- To subtract these fractions, we need a common denominator, which is 4. So, we convert $\frac{1}{2}$ to $\frac{2}{4}$ .
- The expression becomes $\frac{2}{4} - \frac{1}{4}$ , which simplifies to $\frac{1}{4}$ .
Multiplication within Braces:
- Next, we address the multiplication within the curly braces: $-3\left(\frac{1}{4}\right)$ .
- Multiplying 3 by $\frac{1}{4}$ gives us $\frac{3}{4}$ , so the expression becomes $-\frac{3}{4}$ .
Subtraction within Braces:
- Now, we simplify the expression inside the curly braces: $\frac{1}{2} - \frac{3}{4}$ .
- Again, we need a common denominator, which is 4. So, we convert $\frac{1}{2}$ to $\frac{2}{4}$ .
- The expression becomes $\frac{2}{4} - \frac{3}{4}$ , which simplifies to $-\frac{1}{4}$ .
Addition within Brackets:
- Moving to the square brackets, we have $\frac{1}{2} + \left(-\frac{1}{4}\right)$ .
- This is equivalent to $\frac{1}{2} - \frac{1}{4}$ .
- Converting $\frac{1}{2}$ to $\frac{2}{4}$ , we get $\frac{2}{4} - \frac{1}{4}$ , which simplifies to $\frac{1}{4}$ .
Final Multiplication:
- Now, we multiply 3 by the result from the brackets: $3\left(\frac{1}{4}\right)$ .
- This gives us $\frac{3}{4}$ .
Final Addition:
- Finally, we add 2 to the result: $2 + \frac{3}{4}$ .
- To add these, we convert 2 to a fraction with a denominator of 4, which is $\frac{8}{4}$ .
- The expression becomes $\frac{8}{4} + \frac{3}{4}$ , which simplifies to $\frac{11}{4}$ .

Therefore, the detailed step-by-step solution confirms that the final answer is $\frac{11}{4}$ , or 2 $\frac{3}{4}$ as a mixed number. Each step meticulously follows the order of operations, ensuring accuracy and clarity in the calculation. This breakdown serves as a comprehensive guide for understanding the process and reinforces the importance of precision in mathematical problem-solving.

Conclusion: The Importance of Order of Operations

In conclusion, the evaluation of the expression $2+3\left[\frac{1}{2}+\left{\frac{1}{2}-3\left(\frac{1}{2}-\frac{1}{4}\right)\right}\right]$ serves as a quintessential example of why adhering to the order of operations is paramount in mathematics. The step-by-step simplification process, from the innermost parentheses to the final addition, showcases how each operation builds upon the previous one, ultimately leading to the correct result of $\frac{11}{4}$ or 2 $\frac{3}{4}$ . This journey through nested parentheses, brackets, and braces underscores the potential for errors when the order of operations is disregarded. The correct sequence—Parentheses, Exponents, Multiplication and Division, Addition and Subtraction (PEMDAS/BODMAS)—acts as a roadmap, guiding us through the complexities of arithmetic expressions. The meticulous approach required in this evaluation not only provides the final answer but also reinforces the importance of precision and clarity in mathematical problem-solving. A thorough understanding of the order of operations is not merely a procedural skill; it is a foundational element for advanced mathematical concepts. As mathematical expressions become more intricate, the ability to systematically simplify them becomes increasingly crucial. This principle extends beyond basic arithmetic, playing a pivotal role in algebra, calculus, and various other branches of mathematics. Therefore, mastering the order of operations is an indispensable skill for anyone pursuing mathematical proficiency, ensuring accuracy and fostering a deeper comprehension of mathematical structures. This detailed exploration highlights the practical application and fundamental importance of this mathematical principle.