Solving 1 - (1/2) ÷ (2/3) A Step-by-Step Guide
In the realm of mathematics, accurately evaluating expressions is a foundational skill. This article delves into the process of evaluating the expression 1 - (1/2) ÷ (2/3), offering a step-by-step guide that emphasizes the crucial role of the order of operations. By understanding and applying these principles, we can confidently navigate mathematical challenges and arrive at correct solutions. This exploration will not only provide the answer but also illuminate the underlying mathematical concepts, making it a valuable resource for students and anyone seeking to sharpen their arithmetic skills. Whether you're a student tackling homework or simply a math enthusiast, this guide will equip you with the knowledge and confidence to solve similar problems effectively.
Understanding the Order of Operations: PEMDAS/BODMAS
To accurately evaluate mathematical expressions like 1 - (1/2) ÷ (2/3), we must adhere to the order of operations, often remembered by the acronyms PEMDAS or BODMAS. These acronyms serve as a roadmap, guiding us through the steps necessary to unravel complex calculations in the correct sequence. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). BODMAS, a common variant, represents Brackets, Orders (exponents and roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Although the acronyms differ slightly, the underlying principle remains the same: certain operations take precedence over others. Ignoring this hierarchy can lead to drastically different, and incorrect, results. In the case of our expression, 1 - (1/2) ÷ (2/3), the order of operations dictates that we must address the division before we can perform the subtraction. This seemingly simple rule is the cornerstone of accurate mathematical computation, ensuring consistency and clarity in our calculations. By following PEMDAS/BODMAS, we not only arrive at the correct answer but also demonstrate a fundamental understanding of mathematical principles. This section will lay the groundwork for a deeper dive into the specific steps involved in solving the expression, emphasizing the importance of each operation's placement within the hierarchy.
Step-by-Step Evaluation of the Expression
Let's embark on a detailed, step-by-step evaluation of the expression 1 - (1/2) ÷ (2/3), ensuring clarity and precision at each stage. As we established earlier, the order of operations mandates that we tackle the division operation before we address the subtraction. This principle is paramount to arriving at the correct solution. The division in our expression is (1/2) ÷ (2/3). Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal. The reciprocal of a fraction is obtained by simply swapping the numerator and the denominator. Therefore, the reciprocal of (2/3) is (3/2). Consequently, our division problem transforms into a multiplication problem: (1/2) ÷ (2/3) becomes (1/2) × (3/2). Now, we can easily multiply these fractions by multiplying the numerators together and the denominators together. This yields (1 × 3) / (2 × 2), which simplifies to 3/4. Having successfully navigated the division, our expression now reads 1 - (3/4). The next step involves subtracting the fraction (3/4) from the whole number 1. To perform this subtraction, we need to express 1 as a fraction with the same denominator as (3/4). In this case, we can rewrite 1 as 4/4. Our expression now becomes (4/4) - (3/4). Subtracting fractions with a common denominator is straightforward: we simply subtract the numerators and keep the denominator the same. Therefore, (4/4) - (3/4) equals (4 - 3) / 4, which simplifies to 1/4. This meticulous, step-by-step approach not only provides the solution but also reinforces the importance of methodical calculation in mathematics.
Dividing Fractions: A Detailed Explanation
Delving deeper into the process of dividing fractions, a crucial step in evaluating the expression 1 - (1/2) ÷ (2/3), requires a clear understanding of the underlying mathematical principles. As we've established, dividing by a fraction is the same as multiplying by its reciprocal. But why is this the case? The concept of a reciprocal is central to this understanding. The reciprocal of a number is simply 1 divided by that number. For a fraction like (2/3), its reciprocal is (3/2). The product of a number and its reciprocal always equals 1, which is a fundamental property that underpins the division process. When we encounter a division problem involving fractions, such as (1/2) ÷ (2/3), we are essentially asking: "How many times does (2/3) fit into (1/2)?" Instead of directly performing the division, we transform it into a multiplication problem by using the reciprocal. Multiplying by the reciprocal is a clever mathematical maneuver that allows us to express the division in terms of multiplication, which is often a more intuitive operation to perform. In our example, (1/2) ÷ (2/3) becomes (1/2) × (3/2). This transformation is not merely a trick; it's a reflection of the deep connection between multiplication and division. By multiplying by the reciprocal, we are effectively undoing the multiplication that (2/3) represents. This in-depth exploration of dividing fractions not only clarifies the process but also highlights the elegance and interconnectedness of mathematical concepts. Mastering this principle is essential for confidently tackling more complex arithmetic problems.
Subtracting Fractions: Finding a Common Denominator
The final step in evaluating 1 - (1/2) ÷ (2/3) involves subtracting fractions, a process that hinges on the concept of a common denominator. After performing the division, we arrived at the expression 1 - (3/4). To subtract (3/4) from 1, we need to express both numbers as fractions with the same denominator. This is because we can only directly add or subtract fractions that share a common denominator. The denominator represents the number of equal parts into which a whole is divided, and to combine fractions, we need to ensure that these parts are of the same size. In our case, the fraction (3/4) has a denominator of 4. Therefore, we need to rewrite the whole number 1 as a fraction with a denominator of 4. This is achieved by recognizing that any number divided by itself equals 1. So, we can express 1 as 4/4. Now, our subtraction problem becomes (4/4) - (3/4). With a common denominator in place, the subtraction becomes straightforward: we subtract the numerators while keeping the denominator the same. This means we subtract 3 from 4, resulting in 1. The denominator remains 4. Therefore, (4/4) - (3/4) equals 1/4. This detailed explanation underscores the importance of a common denominator in fraction subtraction and highlights the logical steps involved in the process. Without a common denominator, we would be attempting to subtract fractions representing different-sized parts, leading to an inaccurate result. Mastering this principle is crucial for confidently handling a wide range of arithmetic problems involving fractions.
Final Result: Expressing the Answer
After meticulously navigating the order of operations and performing the necessary calculations, we arrive at the final result for the expression 1 - (1/2) ÷ (2/3). Through a step-by-step process, we first addressed the division, converting it into multiplication by using the reciprocal. This yielded the fraction 3/4. Subsequently, we tackled the subtraction, ensuring a common denominator before combining the fractions. This process led us to the final answer of 1/4. The result, 1/4, is a proper fraction, meaning the numerator (1) is smaller than the denominator (4). It represents one-quarter of a whole. This fraction is already in its simplest form, as the numerator and denominator share no common factors other than 1. Therefore, no further simplification is required. Expressing the answer in its simplest form is a standard practice in mathematics, ensuring clarity and conciseness. In this case, 1/4 is the most concise and accurate representation of the solution. This final result encapsulates the entire evaluation process, demonstrating the power of methodical calculation and adherence to mathematical principles. By understanding each step and the reasoning behind it, we not only arrive at the correct answer but also solidify our grasp of fundamental arithmetic concepts. The journey from the initial expression to the final result is a testament to the elegance and precision of mathematics.
Common Mistakes to Avoid
When evaluating mathematical expressions, particularly those involving fractions and multiple operations like 1 - (1/2) ÷ (2/3), it's crucial to be aware of common pitfalls that can lead to incorrect results. One of the most frequent errors is disregarding the order of operations. Failing to adhere to PEMDAS/BODMAS can completely alter the outcome of the calculation. For instance, if one were to subtract (1/2) from 1 before performing the division, the result would be drastically different. Another common mistake lies in the division of fractions. Forgetting to multiply by the reciprocal, or incorrectly applying the reciprocal, is a frequent source of error. Instead of multiplying by (3/2), a student might mistakenly multiply by (2/3) or attempt to divide numerators and denominators directly, which is not mathematically sound. When subtracting fractions, neglecting to find a common denominator is a significant error. Subtracting fractions with different denominators is akin to subtracting apples from oranges; the parts are not comparable. It's essential to rewrite the fractions with a common denominator before performing the subtraction. Careless arithmetic errors, such as miscalculations in multiplication or subtraction, can also derail the process. Double-checking each step is a simple yet effective way to mitigate these errors. By being mindful of these common pitfalls and adopting a meticulous approach, we can significantly enhance our accuracy in evaluating mathematical expressions. Recognizing these potential errors is the first step in avoiding them, leading to more confident and correct mathematical problem-solving.
Conclusion: The Importance of Precision in Mathematical Evaluations
In conclusion, the evaluation of the expression 1 - (1/2) ÷ (2/3) serves as a powerful illustration of the importance of precision and adherence to fundamental mathematical principles. Through a step-by-step analysis, we have demonstrated how the order of operations, the rules of fraction division, and the necessity of a common denominator in subtraction all converge to produce the correct answer: 1/4. This journey through the calculation highlights that mathematics is not merely about arriving at a solution but also about understanding the underlying logic and processes. Each step, from recognizing the precedence of division to finding a common denominator, is a testament to the interconnectedness of mathematical concepts. The final result, 1/4, is not just a number; it's the culmination of a series of logical deductions and precise calculations. Moreover, this exercise underscores the value of a methodical approach. By breaking down a complex expression into smaller, manageable steps, we can minimize errors and enhance our understanding. The common mistakes discussed serve as a reminder that vigilance and careful attention to detail are paramount in mathematical evaluations. Ultimately, mastering the art of evaluating expressions like 1 - (1/2) ÷ (2/3) equips us with essential skills that extend far beyond the classroom. It fosters critical thinking, problem-solving abilities, and a deeper appreciation for the elegance and precision of mathematics. This article aims to provide a comprehensive guide for any similar problem in the future.
