Solving 10.20 - [5 × (7 + 2) ÷ 3] A Step-by-Step Guide
In the realm of mathematics, precision and order are paramount. When faced with complex equations, understanding the hierarchy of operations is crucial to arriving at the correct solution. The equation 10.20 - [5 × (7 + 2) ÷ 3] serves as an excellent example to illustrate this principle. This comprehensive exploration will not only dissect the step-by-step solution but also delve into the underlying mathematical concepts and their significance. Understanding the order of operations is not just about getting the right answer; it's about developing a logical and systematic approach to problem-solving, a skill that extends far beyond the classroom. This article aims to provide a clear and detailed explanation, ensuring that readers of all levels can grasp the intricacies of this fundamental mathematical concept.
The Foundation: Order of Operations (PEMDAS/BODMAS)
Before we dive into the specifics of the equation, it’s essential to establish a firm understanding of the order of operations. This set of rules dictates the sequence in which mathematical operations should be performed to ensure consistency and accuracy. The acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) are commonly used to remember this order. Both acronyms represent the same hierarchy, with slight variations in terminology. The key takeaway is that operations within parentheses or brackets are always performed first, followed by exponents or orders (powers and square roots), then multiplication and division (from left to right), and finally addition and subtraction (from left to right). This standardized approach prevents ambiguity and guarantees that everyone arrives at the same solution when solving a mathematical expression.
Why is the order of operations so important? Imagine a world where mathematical equations were interpreted differently by different people. The chaos that would ensue! The order of operations acts as a universal language, a set of rules that ensures everyone understands and solves equations in the same way. Without it, mathematics would be subjective, and the very foundations of science, engineering, and finance would crumble. The consistency provided by PEMDAS/BODMAS is the bedrock upon which all mathematical calculations are built. It’s not just a set of rules to memorize; it’s a fundamental principle that underpins the entire mathematical framework.
Step-by-Step Solution of 10.20 - [5 × (7 + 2) ÷ 3]
Now that we have a solid understanding of the order of operations, let's tackle the equation 10.20 - [5 × (7 + 2) ÷ 3] step by step. By meticulously following the PEMDAS/BODMAS rule, we can unravel the complexities of this equation and arrive at the correct answer. Each step will be clearly explained, highlighting the specific operation being performed and the rationale behind it.
1. Parentheses/Brackets First: (7 + 2)
The first step, according to PEMDAS/BODMAS, is to address the operation within the parentheses. In this case, we have (7 + 2). This is a simple addition problem, and the solution is straightforward:
7 + 2 = 9
Now, our equation looks like this:
10.20 - [5 × 9 ÷ 3]
The parentheses have been resolved, and we move on to the next level of operations. Remember, the goal is to simplify the equation step by step, making it more manageable and less prone to errors. This initial step of addressing the parentheses is crucial in setting the stage for the subsequent calculations.
2. Multiplication and Division (from left to right): 5 × 9 and then ÷ 3
Next, we encounter multiplication and division within the brackets. According to PEMDAS/BODMAS, these operations have equal priority and should be performed from left to right. First, we tackle the multiplication:
5 × 9 = 45
Now our equation becomes:
10.20 - [45 ÷ 3]
We then proceed with the division:
45 ÷ 3 = 15
Now the equation is significantly simplified:
10.20 - 15
It's important to emphasize the left-to-right rule for multiplication and division. If we were to perform the division before the multiplication, we would arrive at an incorrect answer. This highlights the importance of adhering to the order of operations meticulously.
3. Subtraction: 10.20 - 15
Finally, we are left with a simple subtraction problem. This is the last step in our journey to solve the equation:
10.20 - 15 = -4.80
Therefore, the solution to the equation 10.20 - [5 × (7 + 2) ÷ 3] is -4.80. This final step demonstrates how the cumulative effect of each previous operation leads to the ultimate answer. By systematically following the order of operations, we have successfully navigated the complexities of the equation and arrived at the correct solution.
Common Mistakes and How to Avoid Them
Even with a clear understanding of the order of operations, it’s easy to make mistakes if one isn’t careful. Common errors include skipping steps, misinterpreting the order, or simply making arithmetic mistakes. By being aware of these potential pitfalls, we can develop strategies to avoid them and ensure accuracy in our calculations. This section will explore some of the most frequent errors and provide practical tips for preventing them.
One of the most common mistakes is failing to adhere to the left-to-right rule for multiplication and division (or addition and subtraction). For example, in the expression 5 × 9 ÷ 3, some might mistakenly perform the division first, leading to an incorrect result. The correct approach is to multiply 5 by 9 first, then divide the result by 3. Another frequent error is neglecting the importance of parentheses or brackets. Operations within these groupings must always be performed before any other operations outside of them. Failing to do so can drastically alter the outcome of the equation.
To avoid these mistakes, it’s crucial to develop a methodical approach to problem-solving. This includes writing out each step clearly and deliberately, double-checking calculations, and using parentheses or brackets to clarify the order of operations, especially in complex expressions. Another helpful strategy is to use estimation to check the reasonableness of your answer. For example, if you expect an answer to be around 10, and your calculation yields 100, it’s a clear indication that a mistake has been made. By incorporating these strategies into your problem-solving routine, you can significantly reduce the likelihood of errors and increase your confidence in your mathematical abilities.
Real-World Applications of Order of Operations
The order of operations is not just a theoretical concept confined to textbooks and classrooms; it has numerous practical applications in various real-world scenarios. From everyday calculations to complex scientific and engineering problems, understanding and applying the order of operations is essential for accuracy and efficiency. This section will explore some of these real-world applications, highlighting the importance of this fundamental mathematical principle.
In the realm of finance, for example, calculating interest rates, loan payments, or investment returns requires a thorough understanding of the order of operations. Financial formulas often involve multiple operations, and performing them in the wrong sequence can lead to significant errors in the final result. Similarly, in engineering and physics, many formulas involve complex calculations that rely heavily on the correct application of the order of operations. Whether it’s calculating the trajectory of a projectile, designing a bridge, or analyzing electrical circuits, engineers and physicists must adhere to the established mathematical hierarchy to ensure the accuracy and reliability of their calculations.
Even in everyday situations, the order of operations plays a crucial role. For instance, when calculating the total cost of a shopping trip, you might need to apply discounts, sales tax, and other fees. To arrive at the correct total, you must perform these operations in the appropriate order. Similarly, when following a recipe, you need to understand the order in which ingredients are added and mixed to achieve the desired outcome. By recognizing the ubiquitous nature of the order of operations, we can appreciate its importance in both academic and practical contexts. It’s a fundamental tool that empowers us to solve problems accurately and efficiently in a wide range of situations.
Conclusion
Mastering the order of operations is a cornerstone of mathematical proficiency. The equation 10.20 - [5 × (7 + 2) ÷ 3] serves as a compelling illustration of the importance of this principle. By meticulously following the PEMDAS/BODMAS rule, we can navigate complex equations with confidence and accuracy. This exploration has not only provided a step-by-step solution to the equation but also delved into the underlying concepts, common mistakes, and real-world applications of the order of operations. Remember, mathematics is more than just memorizing formulas and procedures; it’s about developing a logical and systematic approach to problem-solving. By embracing the order of operations as a fundamental tool, we can unlock the power of mathematics and apply it to a wide range of challenges, both in and out of the classroom. The journey to mathematical mastery begins with a solid understanding of the basics, and the order of operations is undoubtedly one of the most essential building blocks.
