Solving Fraction Arithmetic A Step-by-Step Guide To (-5/8 - 11/20) - (3/2 + 3/10)

In this comprehensive guide, we will delve into the intricacies of fraction arithmetic by meticulously solving the expression: (-5/8 - 11/20) - (3/2 + 3/10). Fractions form a fundamental part of mathematics, appearing in various contexts from everyday calculations to advanced scientific computations. Therefore, mastering operations with fractions is crucial for building a strong mathematical foundation. This guide is designed to provide a step-by-step approach, ensuring clarity and understanding at each stage. We will explore the concepts of finding common denominators, performing addition and subtraction with fractions, and simplifying the final result. Whether you are a student looking to enhance your skills or simply someone keen on refreshing your knowledge, this guide will equip you with the necessary tools to confidently tackle fraction arithmetic problems.

Throughout this discussion, we aim to provide a clear and concise explanation of each step, ensuring that the reader not only understands the mechanics of solving the problem but also grasps the underlying principles. By breaking down the problem into manageable parts, we will illustrate how to handle complex expressions involving fractions. The goal is to transform what may seem like a daunting task into a straightforward and easily understandable process. So, let's embark on this journey of mastering fraction arithmetic and unlock the secrets to solving expressions like the one presented above.

Understanding fractions is crucial in various fields, from cooking and baking to engineering and finance. In this article, we will break down each step involved in solving the expression, making it easy to follow and understand. We will start by addressing the individual components within the parentheses and then combine them to arrive at the final answer. Remember, the key to success with fractions is to ensure a common denominator before performing addition or subtraction. This foundational principle will be emphasized throughout our step-by-step solution. So, let's begin our exploration and master the art of fraction arithmetic together!

The given expression is (-5/8 - 11/20) - (3/2 + 3/10). To solve this, we will first simplify each set of parentheses separately. This approach allows us to manage the complexity of the expression by breaking it down into smaller, more manageable parts. Within each parenthesis, we will perform the addition or subtraction operation by finding a common denominator. This is a crucial step in fraction arithmetic, as fractions can only be added or subtracted if they share the same denominator. Let's begin by addressing the first set of parentheses: (-5/8 - 11/20).

When dealing with fractions, identifying the common denominator is paramount. The common denominator is the least common multiple (LCM) of the denominators of the fractions involved. In this case, we need to find the LCM of 8 and 20. The multiples of 8 are 8, 16, 24, 32, 40, and so on. The multiples of 20 are 20, 40, 60, and so on. The least common multiple of 8 and 20 is 40. Therefore, we will convert both fractions in the first parenthesis to have a denominator of 40. This involves multiplying both the numerator and the denominator of each fraction by a suitable factor to achieve the desired denominator.

By finding the least common multiple and converting the fractions accordingly, we ensure that we can perform the subtraction accurately. This method not only simplifies the calculation but also helps in understanding the fundamental principles of fraction arithmetic. Now, let's proceed with converting the fractions and performing the subtraction within the first set of parentheses. This methodical approach will lay the groundwork for solving the entire expression step by step.

To solve (-5/8 - 11/20), our first step is to find a common denominator for the fractions 5/8 and 11/20. As established earlier, the least common multiple (LCM) of 8 and 20 is 40. We will now convert each fraction to have a denominator of 40. To convert 5/8 to an equivalent fraction with a denominator of 40, we multiply both the numerator and the denominator by 5 (since 8 * 5 = 40). This gives us (5 * 5) / (8 * 5) = 25/40. Thus, -5/8 becomes -25/40.

Next, we convert 11/20 to an equivalent fraction with a denominator of 40. To do this, we multiply both the numerator and the denominator by 2 (since 20 * 2 = 40). This gives us (11 * 2) / (20 * 2) = 22/40. Now, our expression within the first parenthesis becomes (-25/40 - 22/40). With a common denominator in place, we can now perform the subtraction of the numerators. When subtracting fractions with a common denominator, we simply subtract the numerators while keeping the denominator the same.

Subtracting the numerators, we have -25 - 22 = -47. Therefore, the result of the subtraction is -47/40. This fraction represents the simplified value of the expression within the first parenthesis. It's important to note that the negative sign applies to the entire fraction, as we are subtracting a positive value from a negative one. Now that we have solved the first parenthesis, we can proceed to the second parenthesis and follow a similar process to simplify it. This step-by-step approach ensures accuracy and clarity in solving the overall expression.

Now, let's address the second parenthesis in our expression: (3/2 + 3/10). Similar to our approach with the first parenthesis, we need to find a common denominator for the fractions 3/2 and 3/10. The least common multiple (LCM) of 2 and 10 is 10. Therefore, we will convert both fractions to have a denominator of 10. To convert 3/2 to an equivalent fraction with a denominator of 10, we multiply both the numerator and the denominator by 5 (since 2 * 5 = 10). This gives us (3 * 5) / (2 * 5) = 15/10.

The fraction 3/10 already has a denominator of 10, so no conversion is needed for this fraction. Now, our expression within the second parenthesis becomes (15/10 + 3/10). With a common denominator in place, we can now perform the addition of the numerators. When adding fractions with a common denominator, we simply add the numerators while keeping the denominator the same. Adding the numerators, we have 15 + 3 = 18. Therefore, the result of the addition is 18/10. This fraction can be further simplified by dividing both the numerator and the denominator by their greatest common divisor (GCD), which is 2. Dividing both by 2, we get (18 ÷ 2) / (10 ÷ 2) = 9/5.

Thus, the simplified value of the expression within the second parenthesis is 9/5. This simplification step is crucial as it helps in reducing the fraction to its simplest form, making it easier to work with in subsequent calculations. Now that we have solved both parentheses individually, we can proceed to the next step, which involves combining the results and completing the overall expression. By breaking down the problem into smaller parts and addressing each one methodically, we ensure accuracy and a clear understanding of the process.

Having solved both parentheses, we now have the simplified expressions -47/40 from the first parenthesis and 9/5 from the second parenthesis. The original expression was (-5/8 - 11/20) - (3/2 + 3/10), which we have now reduced to -47/40 - 9/5. To complete the calculation, we need to subtract the second fraction from the first. As with addition and subtraction within parentheses, we must first find a common denominator.

The denominators we are dealing with now are 40 and 5. The least common multiple (LCM) of 40 and 5 is 40. This means we only need to convert the fraction 9/5 to an equivalent fraction with a denominator of 40. To do this, we multiply both the numerator and the denominator by 8 (since 5 * 8 = 40). This gives us (9 * 8) / (5 * 8) = 72/40. Now, our expression becomes -47/40 - 72/40.

With a common denominator in place, we can subtract the numerators. Subtracting 72 from -47, we get -47 - 72 = -119. Therefore, the result of the subtraction is -119/40. This fraction is already in its simplest form, as 119 and 40 have no common factors other than 1. Thus, the final result of the expression (-5/8 - 11/20) - (3/2 + 3/10) is -119/40. This completes our step-by-step solution, demonstrating how to solve complex fraction arithmetic problems by breaking them down into manageable parts and applying fundamental principles.

In conclusion, we have successfully navigated the intricacies of fraction arithmetic to solve the expression (-5/8 - 11/20) - (3/2 + 3/10). By systematically breaking down the problem into smaller, more manageable steps, we were able to simplify the expression and arrive at the final answer of -119/40. This journey through fraction arithmetic has highlighted the importance of several key concepts, including finding common denominators, converting fractions, and performing addition and subtraction with fractions. Each step was carefully explained to ensure a clear understanding of the underlying principles.

Throughout this guide, we emphasized the significance of finding the least common multiple (LCM) to establish a common denominator, which is crucial for adding or subtracting fractions. We also demonstrated how to convert fractions to equivalent forms with the common denominator, making the arithmetic operations straightforward. By addressing each parenthesis separately and then combining the results, we showed a methodical approach to handling complex expressions. This step-by-step process not only simplifies the calculation but also enhances comprehension and retention.

Mastering fraction arithmetic is a fundamental skill in mathematics, and this guide has provided a comprehensive approach to achieving that mastery. Whether you are a student seeking to improve your grades or an individual looking to refresh your mathematical skills, the principles and techniques discussed here will undoubtedly prove valuable. Remember, practice is key to proficiency in mathematics, so continue to apply these concepts to various problems and build your confidence in fraction arithmetic. With a solid understanding of these principles, you will be well-equipped to tackle more advanced mathematical challenges in the future.

The final answer to the expression (-5/8 - 11/20) - (3/2 + 3/10) is -119/40.