Simplifying Mathematical Expressions A Step-by-Step Guide
In the realm of mathematics, simplifying expressions is a fundamental skill. It allows us to take complex-looking equations and reduce them to their most basic, understandable forms. This not only makes the expressions easier to work with, but it also helps us to grasp the underlying mathematical relationships. In this comprehensive guide, we will delve into the process of simplifying the expression (2/15 + 2/5) + (9/10 × 4/3), breaking it down step-by-step to ensure clarity and understanding. Whether you're a student grappling with fractions or simply looking to refresh your mathematical skills, this guide will provide you with the tools and knowledge you need to confidently tackle similar problems. By mastering the techniques outlined here, you'll be well-equipped to navigate the world of mathematical expressions with greater ease and precision. This exploration into simplifying expressions will not only enhance your mathematical abilities but also cultivate a deeper appreciation for the elegance and logic inherent in mathematical operations. Let's embark on this mathematical journey together, demystifying the process of simplification and empowering you to approach such problems with confidence and accuracy.
Understanding the Order of Operations
Before we dive into the specifics of simplifying our expression, it's crucial to understand the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform mathematical operations to ensure we arrive at the correct answer. Understanding PEMDAS is the cornerstone of simplifying any mathematical expression. It provides a clear roadmap, preventing confusion and ensuring accuracy. For instance, multiplication and division take precedence over addition and subtraction, and operations within parentheses are always performed first. This hierarchical structure is not arbitrary; it's a fundamental principle that maintains the consistency and logic of mathematical calculations. Without a standardized order of operations, the same expression could yield different results, leading to ambiguity and chaos in mathematical problem-solving. Therefore, mastering PEMDAS is not just about following a set of rules; it's about grasping the underlying principles of mathematical reasoning. This understanding empowers you to approach complex expressions with a systematic and confident approach, knowing that you have a reliable framework to guide your calculations. As we delve deeper into simplifying our expression, we'll see how PEMDAS comes into play, ensuring that each step is performed in the correct order, ultimately leading us to the accurate solution. So, let's keep PEMDAS in mind as we embark on this mathematical journey, and we'll see how it serves as our guiding star in navigating the complexities of simplifying expressions.
Step 1: Simplifying within the Parentheses
Our expression, (2/15 + 2/5) + (9/10 × 4/3), contains two sets of parentheses. Following PEMDAS, we must first simplify the expressions within these parentheses. Let's begin with the first set: (2/15 + 2/5). To add these fractions, we need a common denominator. The least common multiple (LCM) of 15 and 5 is 15. So, we'll rewrite 2/5 with a denominator of 15. Multiplying both the numerator and denominator of 2/5 by 3, we get 6/15. Now, we can add the fractions: 2/15 + 6/15 = 8/15. This completes the simplification of the first set of parentheses. Now, let's move on to the second set of parentheses: (9/10 × 4/3). This involves multiplying two fractions. To multiply fractions, we simply multiply the numerators together and the denominators together. So, 9/10 × 4/3 = (9 × 4) / (10 × 3) = 36/30. However, this fraction can be simplified further. Both 36 and 30 are divisible by 6. Dividing both the numerator and denominator by 6, we get 6/5. Thus, the second set of parentheses simplifies to 6/5. By systematically addressing the expressions within the parentheses, we've taken a significant step towards simplifying the overall expression. This methodical approach not only ensures accuracy but also breaks down the problem into manageable chunks, making it less daunting and more approachable. As we continue to simplify the expression, we'll see how each step builds upon the previous one, gradually leading us to the final solution. So, with the parentheses now simplified, we're well-positioned to tackle the remaining operations and arrive at the simplified form of our expression. This meticulous attention to detail and adherence to the order of operations are key to mastering the art of simplifying mathematical expressions.
Step 2: Performing the Multiplication
As we've already simplified the expression within the second set of parentheses in the previous step, the multiplication operation is essentially complete. We found that (9/10 × 4/3) simplifies to 6/5. This step highlights the importance of following the order of operations (PEMDAS), where multiplication and division take precedence over addition and subtraction. By addressing the multiplication within the parentheses first, we ensured that we were working with the correct intermediate values, setting the stage for the subsequent addition operation. The simplified fraction, 6/5, now represents the result of this multiplication, and we'll use it in the next step to complete the simplification of the entire expression. This meticulous approach, where each operation is performed in the correct sequence, is the cornerstone of accurate mathematical calculations. By breaking down the problem into manageable steps, we minimize the risk of errors and gain a clearer understanding of the underlying mathematical relationships. So, with the multiplication step successfully completed, we're now one step closer to the final simplified form of our expression. This methodical progression, guided by the order of operations, is the key to confidently tackling complex mathematical problems.
Step 3: Performing the Addition
Now that we've simplified the expressions within the parentheses, our expression looks like this: 8/15 + 6/5. To add these fractions, we once again need a common denominator. The least common multiple (LCM) of 15 and 5 is 15. So, we need to rewrite 6/5 with a denominator of 15. Multiplying both the numerator and denominator of 6/5 by 3, we get 18/15. Now, we can add the fractions: 8/15 + 18/15 = 26/15. This completes the addition operation. The result, 26/15, is an improper fraction, meaning the numerator is larger than the denominator. While it's a perfectly valid answer, it's often helpful to convert it to a mixed number for better understanding and interpretation. To do this, we divide 26 by 15. The quotient is 1, and the remainder is 11. Therefore, 26/15 is equal to 1 and 11/15. This mixed number representation gives us a clearer sense of the value of the fraction. The addition step, like the previous steps, demonstrates the importance of understanding fraction operations and finding common denominators. By systematically applying these principles, we can confidently add fractions and arrive at accurate results. This step also highlights the flexibility of representing fractions in different forms – as improper fractions or mixed numbers – depending on the context and desired level of clarity. So, with the addition step successfully completed, we've arrived at the simplified form of our expression, showcasing the power of methodical problem-solving and the beauty of mathematical precision. The journey of simplifying this expression has not only yielded a numerical answer but also reinforced our understanding of fundamental mathematical concepts.
Final Answer: 26/15 or 1 11/15
Therefore, the simplified form of the expression (2/15 + 2/5) + (9/10 × 4/3) is 26/15, which can also be expressed as the mixed number 1 11/15. This final answer represents the culmination of our step-by-step simplification process, where we meticulously followed the order of operations and applied our knowledge of fraction arithmetic. From identifying the need for common denominators to performing multiplication and addition, each step played a crucial role in arriving at this result. The journey of simplifying this expression has not only provided us with a numerical answer but also reinforced the importance of precision, attention to detail, and a systematic approach to problem-solving in mathematics. The final answer, whether expressed as an improper fraction or a mixed number, represents the most concise and understandable form of the original expression. It encapsulates the essence of the mathematical relationships within the expression, making it easier to interpret and use in further calculations. This process of simplification is not just about finding an answer; it's about gaining a deeper understanding of the mathematical structures and operations involved. By mastering these techniques, we empower ourselves to confidently tackle more complex mathematical problems and appreciate the elegance and logic inherent in the world of numbers and equations. So, with our final answer in hand, we can look back at the journey of simplification and recognize the power of methodical problem-solving and the beauty of mathematical precision.
Conclusion
In conclusion, simplifying mathematical expressions is a fundamental skill that requires a solid understanding of the order of operations and basic arithmetic principles. By meticulously following the steps outlined in this guide, we successfully simplified the expression (2/15 + 2/5) + (9/10 × 4/3) to 26/15 or 1 11/15. This process not only provided us with the final answer but also reinforced the importance of a systematic approach, attention to detail, and the power of breaking down complex problems into manageable steps. The key takeaways from this exercise include the importance of PEMDAS, finding common denominators when adding fractions, and simplifying fractions to their lowest terms. These skills are not only essential for success in mathematics but also for problem-solving in various aspects of life. The ability to simplify complex information and identify the core components is a valuable asset in any field. As we've seen, simplifying mathematical expressions is not just about finding an answer; it's about developing a deeper understanding of mathematical relationships and honing our critical thinking skills. By mastering these techniques, we empower ourselves to approach challenges with confidence and clarity, knowing that we have the tools to navigate complexity and arrive at meaningful solutions. So, let's continue to practice and refine our simplification skills, and we'll be well-equipped to tackle the mathematical challenges that lie ahead. The journey of mathematical exploration is a rewarding one, and each simplified expression is a testament to our growing understanding and problem-solving abilities.
