Solving P = 4/3/2 - 1/3 Divided By 1/2 A Step-by-Step Explanation
Introduction to Order of Operations
In this article, we will dive into the step-by-step solution of the mathematical expression P = 4/3/2 - 1/3 : 1/2. This expression involves multiple arithmetic operations, and to solve it correctly, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding and applying the correct order of operations is crucial in mathematics to ensure accurate results. When dealing with complex expressions, it's like following a roadmap; each operation has its place and time. Neglecting this order can lead to incorrect solutions, which is why mastering PEMDAS is a fundamental skill in arithmetic and algebra. Think of it as the grammar of mathematics; just as sentences need proper syntax to be understood, mathematical expressions need the right order of operations to be solved correctly. This article aims to break down each step, providing a clear and concise explanation that will help readers not only understand the solution but also apply the principles to similar problems in the future. From simple arithmetic to more complex algebraic equations, the order of operations remains a constant, reliable tool. Whether you are a student learning the basics or someone looking to brush up on their math skills, understanding PEMDAS is an invaluable asset. So, let’s embark on this mathematical journey, unraveling the expression P = 4/3/2 - 1/3 : 1/2 and reinforcing our understanding of the order of operations.
Breaking Down the Expression
The expression we aim to solve is P = 4/3/2 - 1/3 : 1/2. At first glance, it might seem intimidating, but by breaking it down systematically, we can simplify and solve it with ease. The expression involves division and subtraction, and according to the order of operations (PEMDAS), we address division before subtraction. This means we first need to perform the divisions from left to right and then carry out the subtraction. It's essential to recognize that division and multiplication hold equal precedence, as do addition and subtraction. When operations of the same precedence appear, we solve them from left to right. This left-to-right approach ensures consistency and avoids ambiguity in mathematical calculations. The expression contains fractions, which adds another layer of complexity. To handle fractions effectively, we need to remember the rules of fraction division and subtraction. Dividing by a fraction is the same as multiplying by its reciprocal, and subtraction requires finding a common denominator. By understanding these principles, we can confidently tackle the expression. We will also explore how to simplify fractions along the way, making the calculations more manageable. Each step will be clearly explained, ensuring that you not only understand the process but also grasp the underlying concepts. This detailed breakdown is crucial for anyone looking to improve their mathematical skills and gain confidence in solving similar problems. The goal is not just to arrive at the correct answer but to develop a solid understanding of the mathematical principles involved. So, let’s begin the step-by-step journey of solving P = 4/3/2 - 1/3 : 1/2, transforming it from a daunting expression into a simple, manageable equation.
Step 1: Performing the First Division
The first operation we encounter in the expression P = 4/3/2 - 1/3 : 1/2 is division. Specifically, we need to perform the division 4/3/2 first. When dealing with multiple divisions in a row, it’s crucial to remember that we perform these operations from left to right. This is a key principle in the order of operations and ensures we arrive at the correct answer. So, we start by dividing 4/3 by 2. To divide a fraction by a whole number, we can think of the whole number as a fraction with a denominator of 1. Thus, we are dividing 4/3 by 2/1. Dividing fractions involves multiplying by the reciprocal of the divisor. The reciprocal of 2/1 is 1/2. Therefore, 4/3 divided by 2 is the same as 4/3 multiplied by 1/2. When multiplying fractions, we multiply the numerators (the top numbers) and the denominators (the bottom numbers). So, 4/3 * 1/2 = (4 * 1) / (3 * 2) = 4/6. Now we have the fraction 4/6, which can be simplified. Both the numerator and the denominator are divisible by 2. Dividing both by 2, we get 2/3. So, the result of the first division, 4/3/2, is 2/3. This step-by-step breakdown is essential for understanding how to handle multiple divisions in an expression. By performing the divisions from left to right and simplifying the fractions, we ensure accuracy and clarity in our calculations. This initial step sets the foundation for the rest of the solution, highlighting the importance of mastering the rules of fraction division and simplification. With this first division successfully completed, we can move on to the next operation, bringing us closer to the final answer.
Step 2: Performing the Second Division
Having completed the first division, our expression now looks like P = 2/3 - 1/3 : 1/2. The next operation we need to tackle is the division 1/3 : 1/2. This involves dividing one fraction by another, a common operation in arithmetic. To divide fractions, we multiply the first fraction by the reciprocal of the second fraction. The reciprocal of a fraction is obtained by swapping its numerator and denominator. In this case, the reciprocal of 1/2 is 2/1, which is simply 2. So, the division 1/3 : 1/2 becomes the multiplication 1/3 * 2. To multiply a fraction by a whole number, we can treat the whole number as a fraction with a denominator of 1. Thus, we are multiplying 1/3 by 2/1. When multiplying fractions, we multiply the numerators and the denominators. Therefore, 1/3 * 2/1 = (1 * 2) / (3 * 1) = 2/3. The result of the division 1/3 : 1/2 is 2/3. This step demonstrates the fundamental principle of dividing fractions and reinforces the importance of using reciprocals. By converting division into multiplication by the reciprocal, we simplify the calculation and make it easier to solve. This method is applicable to any division problem involving fractions and is a crucial skill for anyone studying mathematics. With this second division completed, we have further simplified the expression, bringing us closer to the final solution. The expression now stands as P = 2/3 - 2/3, a straightforward subtraction problem that we can easily solve in the next step. Each step in this process builds upon the previous one, highlighting the logical progression of solving mathematical expressions.
Step 3: Performing the Subtraction
After performing the divisions, our expression has been simplified to P = 2/3 - 2/3. This final step involves subtracting one fraction from another. Subtracting fractions is a relatively straightforward process when the fractions have the same denominator, which is exactly what we have here. Both fractions, 2/3 and 2/3, have a denominator of 3. When subtracting fractions with the same denominator, we simply subtract the numerators and keep the denominator the same. In this case, we subtract 2 from 2, which gives us 0. So, 2/3 - 2/3 = (2 - 2) / 3 = 0/3. Now we have the fraction 0/3. Any fraction with a numerator of 0 is equal to 0, regardless of the denominator (as long as the denominator is not 0). Therefore, 0/3 simplifies to 0. This means that the value of P is 0. The subtraction step is crucial as it brings us to the final solution of the expression. It highlights the importance of understanding how to subtract fractions, especially when they share a common denominator. This basic arithmetic operation is a building block for more complex mathematical problems and is essential for anyone looking to improve their math skills. With this final step completed, we have successfully solved the expression P = 4/3/2 - 1/3 : 1/2, arriving at the answer P = 0. This journey through the order of operations has not only provided us with the solution but has also reinforced our understanding of fundamental mathematical principles.
Final Answer: P = 0
After meticulously following the order of operations, we have arrived at the final answer: P = 0. This solution is the result of a step-by-step process, where we first performed the divisions from left to right and then completed the subtraction. The expression P = 4/3/2 - 1/3 : 1/2 initially presented a series of operations that needed to be handled in the correct sequence. By adhering to the rules of PEMDAS, we ensured that each operation was performed in its proper order, leading us to the accurate solution. The process involved dividing fractions, simplifying fractions, and subtracting fractions with a common denominator. Each of these individual steps is a fundamental concept in arithmetic, and mastering them is crucial for tackling more complex mathematical problems. The journey to the final answer not only demonstrates the importance of the order of operations but also reinforces the practical application of basic arithmetic skills. The solution P = 0 is a concise and definitive answer, representing the result of all the calculations performed. This final answer serves as a testament to the power of systematic problem-solving and the importance of understanding the underlying principles of mathematics. Whether you are a student learning the basics or someone looking to refresh their math skills, this step-by-step solution provides a clear and comprehensive guide to solving similar expressions. The value of P being 0 highlights the concept of mathematical balance and cancellation, where the positive and negative components of the expression ultimately negate each other. This exploration reinforces the elegance and precision inherent in mathematical solutions.
