Solving 9 + 4(3 + 7) - 6 A Step-by-Step Mathematical Guide
Introduction
In this article, we will delve into the process of solving the mathematical expression 9 + 4(3 + 7) - 6. This seemingly simple expression involves several arithmetic operations, including addition, multiplication, and subtraction, along with the crucial use of parentheses. To arrive at the correct answer, we must follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Understanding and applying PEMDAS is fundamental in mathematics to ensure consistent and accurate results. This guide will break down each step in detail, providing clear explanations and justifications along the way. Whether you're a student brushing up on your arithmetic skills or simply someone who enjoys mathematical challenges, this article aims to offer a comprehensive and easily understandable solution to the problem. We will explore the significance of each operation and how the order in which they are performed impacts the final outcome. By the end of this explanation, you will not only know the correct answer but also understand the underlying principles that govern the resolution of such expressions.
Understanding the Order of Operations (PEMDAS)
Before we dive into solving the expression 9 + 4(3 + 7) - 6, it's essential to understand the order of operations, often referred to by the acronym PEMDAS. This mnemonic helps us remember the correct sequence in which to perform mathematical operations: Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Ignoring this order can lead to incorrect results, so adhering to PEMDAS is crucial for accurate calculations. Parentheses come first, as they group operations that should be performed before anything else. This ensures that certain parts of an expression are treated as a single unit. Exponents are next, dealing with powers and roots. Following exponents, we handle Multiplication and Division. It's important to note that multiplication and division have equal priority, so we perform them from left to right in the order they appear. Similarly, Addition and Subtraction have the same priority and are performed from left to right. By following PEMDAS, we ensure consistency and accuracy in our calculations, regardless of the complexity of the expression. This systematic approach provides a clear roadmap for solving mathematical problems, making it easier to break down complex problems into manageable steps. In the context of our expression, 9 + 4(3 + 7) - 6, PEMDAS will guide us to first address the parentheses, then multiplication, and finally addition and subtraction, ensuring we arrive at the correct solution.
Step-by-Step Solution
Let's break down the solution to the expression 9 + 4(3 + 7) - 6 step-by-step, following the order of operations (PEMDAS) to ensure accuracy.
Step 1: Parentheses
First, we focus on the parentheses. Inside the parentheses, we have the operation 3 + 7. Performing this addition gives us:
3 + 7 = 10
So, the expression now becomes:
9 + 4(10) - 6
Step 2: Multiplication
Next, according to PEMDAS, we perform multiplication. We have 4(10), which means 4 multiplied by 10:
4 * 10 = 40
Substituting this result back into the expression, we get:
9 + 40 - 6
Step 3: Addition and Subtraction
Finally, we perform addition and subtraction from left to right. First, we add 9 and 40:
9 + 40 = 49
Now, we subtract 6 from the result:
49 - 6 = 43
Therefore, the final answer to the expression 9 + 4(3 + 7) - 6 is 43. This step-by-step breakdown illustrates how adhering to PEMDAS ensures we arrive at the correct answer. Each operation is performed in the appropriate sequence, leading to a clear and accurate solution. By systematically addressing each part of the expression, we can confidently solve mathematical problems of varying complexity. Understanding the importance of each step and the reasoning behind it is key to mastering arithmetic and algebraic expressions.
Analyzing the Answer Choices
Now that we've meticulously solved the expression 9 + 4(3 + 7) - 6 and arrived at the answer 43, let's analyze the given answer choices to determine which one is correct. The provided options are:
- F 17
- G 13
- H 124
- J 22
- K None of these
Comparing our calculated answer, 43, with the given choices, it's evident that none of the options F, G, H, or J match our result. This indicates that the correct answer is option K, which stands for "None of these." This scenario highlights the importance of not only understanding the mathematical concepts but also being able to critically evaluate the provided options. Sometimes, the correct answer may not be explicitly listed, and recognizing this is a crucial skill in problem-solving. In this case, blindly choosing one of the given options would lead to an incorrect answer. Instead, by carefully following the order of operations and arriving at the accurate solution, we can confidently identify the correct answer as "None of these.” This exercise reinforces the need for thoroughness and attention to detail in mathematical calculations and problem analysis. It's a reminder that sometimes the most accurate answer is recognizing when none of the provided options are correct.
Common Mistakes to Avoid
When solving mathematical expressions like 9 + 4(3 + 7) - 6, it's easy to make mistakes if you're not careful. One common mistake is not following the order of operations (PEMDAS). For instance, someone might add 9 and 4 first, then multiply by the result of (3 + 7), leading to an incorrect answer. Another mistake is misinterpreting the multiplication within the expression. The term 4(3 + 7) means 4 multiplied by the result of (3 + 7), not 4 added to (3 + 7). Failing to recognize this can lead to errors. Additionally, carelessness in basic arithmetic can also lead to mistakes. For example, miscalculating 3 + 7 or 4 * 10 can throw off the entire solution. To avoid these mistakes, it's crucial to write out each step clearly and methodically. Double-checking your calculations can also help catch errors. Remember, PEMDAS is your guide, so always prioritize parentheses, exponents, multiplication and division (from left to right), and then addition and subtraction (from left to right). By being mindful of these common pitfalls and taking a systematic approach, you can increase your accuracy and confidence in solving mathematical expressions. Regular practice and attention to detail are key to mastering these skills and avoiding common mistakes.
Conclusion
In conclusion, solving the mathematical expression 9 + 4(3 + 7) - 6 requires a clear understanding and application of the order of operations (PEMDAS). By meticulously following the steps—first addressing the parentheses, then multiplication, and finally addition and subtraction—we arrive at the correct answer of 43. This detailed process not only provides the solution but also reinforces the importance of a systematic approach to mathematical problems. The analysis of the answer choices further emphasizes the need for critical evaluation and the recognition that sometimes the correct answer might not be explicitly listed among the options. Avoiding common mistakes, such as neglecting PEMDAS or misinterpreting operations, is crucial for accurate calculations. This exercise serves as a valuable reminder of the foundational principles of arithmetic and the significance of precision in mathematical problem-solving. Mastering these skills not only enhances one's ability to solve mathematical expressions but also cultivates a methodical and detail-oriented approach that is beneficial in various aspects of life. Regular practice and a commitment to understanding the underlying concepts are key to building confidence and proficiency in mathematics.
