Solving The Expression 3/4 + 1/4 × (1/2 Of 2) A Step-by-Step Guide
Introduction
In the realm of mathematics, mastering the order of operations is crucial for accurately solving expressions. This article delves into a specific mathematical expression: 3/4 + 1/4 × (1/2 of 2). We will break down each step, ensuring a clear understanding of how to arrive at the correct solution. Our primary focus will be on adhering to the PEMDAS/BODMAS rule, which dictates the sequence of operations: Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). By meticulously following this rule, we will navigate through the expression, simplifying it progressively until we reach the final answer. This exercise is not just about solving a single problem; it’s about reinforcing fundamental mathematical principles that are essential for tackling more complex calculations in the future. Understanding the precedence of different operations is a cornerstone of mathematical literacy, and this article aims to provide a comprehensive guide to that understanding.
The beauty of mathematics lies in its precision and the logical progression it demands. Each step in solving an equation or expression is built upon the previous one, creating a chain of reasoning that leads to the solution. The expression 3/4 + 1/4 × (1/2 of 2) serves as an excellent example of this principle. By carefully dissecting the expression and addressing each component in the correct order, we can demonstrate the power and elegance of mathematical problem-solving. This article will not only provide the solution but also illuminate the underlying logic, making it a valuable resource for anyone looking to enhance their mathematical skills. Whether you are a student grappling with basic arithmetic or someone looking to refresh your understanding of mathematical concepts, this article will offer a clear and concise explanation of how to approach and solve such expressions. Let’s embark on this mathematical journey together, unraveling the intricacies of this expression and solidifying our understanding of the order of operations.
Moreover, this article emphasizes the importance of clarity and accuracy in mathematical calculations. Errors can easily creep in if steps are skipped or the order of operations is misunderstood. Therefore, we will meticulously document each step, providing a transparent and easy-to-follow solution. This detailed approach will not only help in solving the current expression but also equip readers with the tools to tackle similar problems with confidence. The goal is to foster a deep understanding of the mathematical principles involved, rather than simply memorizing a set of rules. By grasping the underlying concepts, readers will be able to apply their knowledge to a wide range of mathematical challenges. The expression 3/4 + 1/4 × (1/2 of 2) is a stepping stone to more advanced mathematical concepts, and a solid understanding of its solution is crucial for future success in mathematics.
Step-by-Step Solution
To solve the expression 3/4 + 1/4 × (1/2 of 2), we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). This ensures that we perform the operations in the correct sequence, leading to the accurate result. The first step in our expression involves addressing the parentheses, which contain the phrase “1/2 of 2.” In mathematical terms, “of” often translates to multiplication. Therefore, we need to calculate 1/2 multiplied by 2. This step is crucial as it simplifies the expression within the parentheses, paving the way for the subsequent operations. By focusing on the parentheses first, we adhere to the fundamental principles of order of operations, ensuring that our solution is both logical and correct.
The calculation of 1/2 multiplied by 2 is a straightforward multiplication of fractions. We can express 2 as 2/1, making the multiplication clearer: (1/2) × (2/1). When multiplying fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. So, 1 multiplied by 2 gives us 2, and 2 multiplied by 1 gives us 2. This results in the fraction 2/2. This fraction simplifies to 1, as any number divided by itself equals 1. Therefore, the value of “1/2 of 2” is 1. Now that we have simplified the expression within the parentheses to 1, we can rewrite the original expression as 3/4 + 1/4 × 1. This simplification is a significant step forward, as it reduces the complexity of the expression and allows us to proceed with the next operation in the correct order. The meticulous attention to detail in this step underscores the importance of accuracy in mathematical calculations.
Having simplified the parentheses, the next operation according to PEMDAS/BODMAS is multiplication. In our simplified expression, 3/4 + 1/4 × 1, we have a multiplication operation: 1/4 multiplied by 1. Multiplying any number by 1 results in the same number. Therefore, 1/4 multiplied by 1 is simply 1/4. This means that we can further simplify the expression to 3/4 + 1/4. This step demonstrates the elegance of mathematical operations, where seemingly complex expressions can be broken down into simpler components. By correctly applying the order of operations, we have transformed the original expression into a straightforward addition problem involving two fractions. This simplification is a testament to the power of mathematical principles and their ability to make complex problems manageable.
Now that we have reached the addition stage, we need to add the two fractions: 3/4 and 1/4. Adding fractions requires a common denominator, which in this case is already present – both fractions have a denominator of 4. When fractions have the same denominator, we can simply add the numerators together while keeping the denominator the same. So, we add 3 and 1, which gives us 4. The resulting fraction is 4/4. This fraction represents the sum of the two fractions in our simplified expression. The process of adding fractions with a common denominator is a fundamental skill in arithmetic, and it highlights the importance of understanding the properties of fractions. By mastering this skill, we can confidently tackle more complex mathematical problems involving fractions.
The final step in solving the expression is to simplify the fraction 4/4. A fraction represents a division, where the numerator is divided by the denominator. In this case, we have 4 divided by 4. Any number divided by itself equals 1. Therefore, the fraction 4/4 simplifies to 1. This is the final answer to our expression 3/4 + 1/4 × (1/2 of 2). The entire process, from addressing the parentheses to simplifying the final fraction, showcases the importance of following the order of operations. By adhering to PEMDAS/BODMAS, we have successfully navigated through the expression, arriving at the correct solution. This exercise not only provides the answer but also reinforces the fundamental principles of mathematical problem-solving.
Detailed Breakdown of Each Step
Step 1: Parentheses (1/2 of 2)
The first step in solving 3/4 + 1/4 × (1/2 of 2) is to address the expression within the parentheses: (1/2 of 2). The term “of” in this context signifies multiplication. Therefore, we are essentially calculating 1/2 multiplied by 2. This step is crucial because, according to the order of operations (PEMDAS/BODMAS), operations within parentheses or brackets take precedence over other operations. By focusing on this part of the expression first, we ensure that we are following the correct mathematical procedure. This emphasis on the order of operations is a cornerstone of accurate mathematical calculations.
To perform the multiplication, we can rewrite 2 as a fraction, 2/1. Now, we have (1/2) × (2/1). Multiplying fractions involves multiplying the numerators (the top numbers) together and the denominators (the bottom numbers) together. In this case, 1 multiplied by 2 equals 2, and 2 multiplied by 1 equals 2. So, the result of the multiplication is 2/2. This fraction represents the value of the expression within the parentheses. Simplifying this fraction is the next step in our process.
The fraction 2/2 can be simplified by dividing both the numerator and the denominator by their greatest common divisor, which is 2. Dividing 2 by 2 gives us 1, and dividing 2 by 2 also gives us 1. Therefore, the simplified fraction is 1/1, which is equal to 1. So, the value of the expression within the parentheses, (1/2 of 2), is 1. This result is essential for the next steps in solving the overall expression. By meticulously calculating and simplifying this part, we have laid a solid foundation for the subsequent operations. The clarity and precision in this step are vital for ensuring the accuracy of the final answer.
Step 2: Multiplication (1/4 × 1)
After simplifying the parentheses, our expression now looks like this: 3/4 + 1/4 × 1. According to the order of operations (PEMDAS/BODMAS), multiplication and division take precedence over addition and subtraction. Therefore, the next step is to perform the multiplication: 1/4 × 1. This step is straightforward but crucial for correctly solving the expression. Understanding the properties of multiplication, especially the role of 1 as the multiplicative identity, is key to simplifying this part of the problem.
When any number is multiplied by 1, the result is the number itself. This is a fundamental property of multiplication. Therefore, 1/4 multiplied by 1 equals 1/4. This simplifies the expression further, allowing us to move on to the final operation. The simplicity of this step highlights the importance of recognizing and applying basic mathematical principles. By correctly performing the multiplication, we have reduced the expression to a simple addition problem involving two fractions.
Now, the expression is simplified to 3/4 + 1/4. This transformation is a direct result of applying the order of operations and correctly performing the multiplication. The clarity and simplicity of this step are a testament to the power of mathematical rules and their ability to break down complex problems into manageable parts. We are now ready to perform the final operation, which is the addition of two fractions. The meticulous attention to detail in the previous steps has paved the way for a smooth and accurate final calculation.
Step 3: Addition (3/4 + 1/4)
The expression has now been simplified to 3/4 + 1/4. This is an addition problem involving two fractions, both of which have the same denominator. This makes the addition process relatively simple. When adding fractions with a common denominator, we add the numerators (the top numbers) together and keep the denominator (the bottom number) the same. Understanding how to add fractions is a fundamental skill in arithmetic, and it is essential for solving a wide range of mathematical problems.
In this case, we have 3/4 + 1/4. The denominators are the same, so we can proceed to add the numerators. Adding 3 and 1 gives us 4. Therefore, the result of the addition is 4/4. This fraction represents the sum of the two fractions in our expression. The simplicity of this step underscores the importance of having a solid understanding of basic arithmetic operations. By correctly adding the fractions, we are one step closer to the final solution.
The fraction 4/4 can be simplified further. A fraction represents a division, where the numerator is divided by the denominator. In this case, we have 4 divided by 4. Any number divided by itself equals 1. Therefore, the fraction 4/4 simplifies to 1. This simplification is the final step in solving the expression. The entire process, from simplifying the parentheses to adding the fractions, has demonstrated the importance of following the order of operations and applying basic mathematical principles. The result, 1, is the final answer to our problem.
Step 4: Simplification (4/4 = 1)
We have reached the final stage of our calculation, where the expression has been simplified to 4/4. This fraction represents the final result of the addition operation we performed in the previous step. However, it is crucial to recognize that fractions can often be simplified to their lowest terms. Simplifying fractions is an essential skill in mathematics, as it allows us to express results in their most concise and easily understandable form. This step demonstrates the importance of not only performing calculations correctly but also presenting the answer in its simplest form.
The fraction 4/4 represents a division: 4 divided by 4. When any number is divided by itself, the result is 1. This is a fundamental mathematical principle. Therefore, the fraction 4/4 simplifies to 1. This final simplification provides us with the ultimate answer to the original expression. The clarity and conciseness of this result highlight the power of simplification in mathematics. By reducing the fraction to its simplest form, we have arrived at a clear and unambiguous solution.
Therefore, the final answer to the expression 3/4 + 1/4 × (1/2 of 2) is 1. This solution is the culmination of a series of steps, each of which has been carefully executed according to the order of operations (PEMDAS/BODMAS). The process has demonstrated the importance of understanding mathematical principles, following the correct order of operations, and simplifying results to their lowest terms. This exercise not only provides the answer to a specific problem but also reinforces the fundamental skills necessary for tackling a wide range of mathematical challenges.
Conclusion
In conclusion, the solution to the mathematical expression 3/4 + 1/4 × (1/2 of 2) is 1. This result was achieved by meticulously following the order of operations (PEMDAS/BODMAS), which dictates that we first address the parentheses, then perform multiplication and division, and finally carry out addition and subtraction. The step-by-step breakdown of the solution has highlighted the importance of each operation and its place in the overall calculation. The process began with simplifying the expression within the parentheses, (1/2 of 2), which translates to 1/2 multiplied by 2, resulting in 1. Then, we performed the multiplication, 1/4 × 1, which equals 1/4. Finally, we added the fractions, 3/4 + 1/4, which resulted in 4/4, which simplifies to 1. This journey through the expression demonstrates the elegance and precision of mathematical principles.
The exercise of solving this expression serves as a valuable lesson in mathematical problem-solving. It underscores the necessity of understanding the order of operations, as deviating from this order would lead to an incorrect result. Furthermore, it reinforces the importance of accuracy in each step, as errors in any part of the calculation would propagate through the rest of the solution. The detailed breakdown of each step, from simplifying the parentheses to adding the fractions, provides a clear and comprehensive guide to solving similar mathematical expressions. This understanding is crucial for building a strong foundation in mathematics and for tackling more complex problems in the future. The expression 3/4 + 1/4 × (1/2 of 2), while seemingly simple, encapsulates fundamental mathematical concepts that are essential for mathematical literacy.
Moreover, this exercise highlights the interconnectedness of different mathematical operations. Multiplication, addition, and simplification are all integral parts of the solution, and each operation plays a crucial role in arriving at the correct answer. The ability to seamlessly integrate these operations is a hallmark of mathematical proficiency. By mastering the order of operations and understanding the properties of different mathematical operations, individuals can confidently approach and solve a wide range of mathematical problems. The solution to 3/4 + 1/4 × (1/2 of 2) serves as a microcosm of the broader world of mathematics, where precision, logic, and a systematic approach are key to success. This article has aimed to provide not only the solution but also a deeper understanding of the underlying principles, empowering readers to tackle future mathematical challenges with confidence.
