Solving 7/2 X 1/4 - 7/2 + 9/2 * 1/2 A Step-by-Step Guide

This comprehensive guide breaks down the process of solving the mathematical expression 7/2 x 1/4 - 7/2 + 9/2 * 1/2. We'll go through each step in detail, ensuring clarity and understanding for anyone looking to master order of operations and fraction arithmetic. By the end of this article, you'll not only be able to solve this specific problem but also have a solid foundation for tackling similar mathematical challenges. Understanding the order of operations is crucial in mathematics, and this guide will help you grasp this fundamental concept. This problem involves fractions, multiplication, subtraction, and addition, making it an excellent example to illustrate the importance of following the correct sequence of steps. Let's dive in and simplify this expression together!

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we even begin to crunch numbers, it's paramount that we understand the order of operations. This is a set of rules that dictates the sequence in which mathematical operations should be performed. Often remembered by the acronyms PEMDAS or BODMAS, it ensures we arrive at the correct answer every time. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), Addition and Subtraction (from left to right). BODMAS is Brackets, Orders, Division and Multiplication (from left to right), Addition and Subtraction (from left to right). Both acronyms convey the same principle: some operations have precedence over others. In our specific problem, 7/2 x 1/4 - 7/2 + 9/2 * 1/2, we don't have parentheses or exponents, but we do have multiplication, subtraction, and addition. Therefore, according to PEMDAS/BODMAS, we will perform the multiplication operations first, working from left to right. Once the multiplications are handled, we proceed with addition and subtraction, also from left to right. Ignoring this order can lead to vastly different and incorrect results. Understanding this fundamental principle is the key to solving not just this problem, but many others in mathematics. It provides a structured approach, ensuring consistency and accuracy in calculations. The correct application of the order of operations is what separates a right answer from a wrong one, and it's a cornerstone of mathematical literacy. So, keep PEMDAS/BODMAS in mind as we proceed through the solution.

Step 1: Multiplication (7/2 x 1/4)

The first operation we tackle, as dictated by the order of operations, is the multiplication of 7/2 and 1/4. Multiplying fractions is a straightforward process: we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. In this case, we have (7/2) * (1/4). Multiplying the numerators, 7 * 1, gives us 7. Multiplying the denominators, 2 * 4, gives us 8. Therefore, 7/2 multiplied by 1/4 equals 7/8. This result now replaces the original multiplication part of our expression, transforming it into 7/8 - 7/2 + 9/2 * 1/2. It's crucial to maintain this result accurately as we proceed, as it forms the basis for the subsequent calculations. When dealing with fractions, precision is key. A small error in the multiplication step can propagate through the rest of the solution, leading to an incorrect final answer. Double-checking your work at each stage, particularly during fraction multiplication, is always a good practice. Remember, we're not just aiming for the right answer; we're aiming for a clear, methodical understanding of the process. So, with 7/8 securely calculated, we move on to the next multiplication in our expression.

Step 2: Multiplication (9/2 * 1/2)

Following the order of operations, we now address the second multiplication in the expression: 9/2 multiplied by 1/2. Just like before, we multiply the numerators and the denominators separately. The numerators are 9 and 1, so 9 * 1 = 9. The denominators are 2 and 2, so 2 * 2 = 4. Thus, 9/2 multiplied by 1/2 equals 9/4. Our expression now looks like this: 7/8 - 7/2 + 9/4. We've successfully completed both multiplication operations, simplifying the initial expression and paving the way for the addition and subtraction steps. At this stage, it’s worth pausing to appreciate how the order of operations has guided us. By prioritizing multiplication, we've transformed a seemingly complex expression into a more manageable form. This methodical approach is the essence of problem-solving in mathematics. Now that we have a series of additions and subtractions involving fractions, the next step is to ensure they all share a common denominator. This is a necessary prerequisite before we can combine them effectively. We're making steady progress, step by methodical step, towards our final answer.

Step 3: Finding a Common Denominator

Before we can perform the addition and subtraction, we need a common denominator for all the fractions in our expression: 7/8 - 7/2 + 9/4. The denominators we have are 8, 2, and 4. The least common multiple (LCM) of these numbers will be our common denominator. To find the LCM, we can list the multiples of each number: Multiples of 8: 8, 16, 24, ... Multiples of 2: 2, 4, 6, 8, 10, ... Multiples of 4: 4, 8, 12, ... The smallest number that appears in all three lists is 8. Therefore, 8 is our least common multiple and the common denominator we'll use. Now we need to convert each fraction to have a denominator of 8. 7/8 already has the correct denominator. For 7/2, we multiply both the numerator and denominator by 4 (since 2 * 4 = 8) to get (7 * 4) / (2 * 4) = 28/8. For 9/4, we multiply both the numerator and denominator by 2 (since 4 * 2 = 8) to get (9 * 2) / (4 * 2) = 18/8. Our expression now becomes: 7/8 - 28/8 + 18/8. With a common denominator secured, we can now proceed with the addition and subtraction in a straightforward manner. This step is crucial because it allows us to combine the fractions accurately. Without a common denominator, adding or subtracting fractions is like trying to add apples and oranges – it simply doesn't work. The LCM provides the perfect unit for our calculations.

Step 4: Subtraction and Addition (Left to Right)

With the common denominator in place, we can now perform the subtraction and addition, working from left to right, as dictated by the order of operations. Our expression is 7/8 - 28/8 + 18/8. First, we subtract 28/8 from 7/8. When subtracting fractions with a common denominator, we subtract the numerators and keep the denominator the same. So, 7/8 - 28/8 = (7 - 28) / 8 = -21/8. Now our expression is -21/8 + 18/8. Next, we add 18/8 to -21/8. Again, we add the numerators and keep the denominator the same: -21/8 + 18/8 = (-21 + 18) / 8 = -3/8. Therefore, the result of the subtraction and addition is -3/8. It's important to remember to work from left to right when performing addition and subtraction. Changing the order could lead to a different and incorrect answer. In this case, the negative sign is crucial. It indicates that the final result is less than zero. We've now successfully combined the fractions, arriving at a simplified answer. The step-by-step approach, paying close attention to the signs and order, has been instrumental in achieving this result.

Final Answer: -3/8

After meticulously following the order of operations and performing each step with care, we arrive at the final answer: -3/8. This fraction represents the simplified result of the original expression, 7/2 x 1/4 - 7/2 + 9/2 * 1/2. We started by understanding the importance of PEMDAS/BODMAS, then tackled the multiplication operations, found a common denominator, and finally performed the subtraction and addition from left to right. Each step was crucial in ensuring the accuracy of the final answer. The negative sign in -3/8 is significant, indicating that the overall value of the expression is negative. This result demonstrates the power of a structured approach to problem-solving in mathematics. By breaking down a complex expression into smaller, manageable steps, we can systematically arrive at the correct solution. This process not only yields the answer but also reinforces a deeper understanding of mathematical principles. The journey from the initial expression to the final answer highlights the interconnectedness of different mathematical operations and the importance of precision in each step. We have successfully navigated the intricacies of fraction arithmetic and the order of operations, culminating in the solution: -3/8.

Conclusion

In conclusion, solving the expression 7/2 x 1/4 - 7/2 + 9/2 * 1/2 exemplifies the importance of the order of operations and careful execution in mathematics. By adhering to PEMDAS/BODMAS, we first addressed the multiplication operations, then found a common denominator to facilitate addition and subtraction. The final result, -3/8, showcases the culmination of these steps. This exercise not only provides a specific answer but also reinforces fundamental mathematical principles applicable to a wide range of problems. The step-by-step approach demonstrated here is a valuable tool for anyone looking to improve their mathematical skills and confidence. Breaking down complex problems into smaller, manageable parts makes them less daunting and more accessible. Remember, mathematics is not just about finding the right answer; it's about understanding the process and the logic behind it. This understanding empowers you to tackle new challenges and build a solid foundation for further learning. The journey from the initial expression to the final solution is a testament to the power of methodical thinking and the beauty of mathematical precision. Keep practicing, keep exploring, and keep applying these principles to unlock the world of mathematics.