Solving 6 + (2 + 3) × 5 A Step By Step Guide

Jul 12, 2025 by ADMIN 45 views

Decoding the Expression: 6 + (2 + 3) × 5 - A Comprehensive Guide

In the realm of mathematics, order of operations is paramount. Mastering this concept is crucial for accurately solving mathematical expressions. This article delves into the intricacies of evaluating the expression 6 + (2 + 3) × 5, providing a step-by-step breakdown and comparing it to various alternatives. We aim to equip you with a solid understanding of the order of operations, ensuring you can confidently tackle similar problems. This mathematical journey will not only reveal the correct answer but also illuminate the underlying principles that govern mathematical calculations. Let's embark on this exploration to unravel the solution and solidify your grasp of mathematical operations. Understanding the order of operations is like having a map for navigating the world of numbers, ensuring you arrive at the correct destination every time.

Understanding the Order of Operations: PEMDAS/BODMAS

To effectively evaluate mathematical expressions, we adhere to a specific order of operations, commonly remembered by the acronyms PEMDAS or BODMAS. These acronyms serve as a guide, ensuring consistency and accuracy in our calculations. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Similarly, BODMAS represents Brackets, Orders (powers and square roots, etc.), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). Whether you use PEMDAS or BODMAS, the underlying principle remains the same: certain operations take precedence over others. Parentheses or brackets are always addressed first, as they group terms together and dictate the initial focus of the calculation. Exponents or orders follow, representing repeated multiplication and adding another layer of complexity. Multiplication and division share equal priority and are performed from left to right, ensuring a consistent approach. Finally, addition and subtraction, also of equal importance, are executed from left to right. Mastering this order is akin to learning the grammar of mathematics, allowing you to construct and interpret expressions with precision. Neglecting this order can lead to erroneous results, highlighting the significance of adhering to the established rules. The consistent application of PEMDAS/BODMAS is the cornerstone of accurate mathematical problem-solving, allowing us to navigate complex expressions with confidence and clarity. This framework not only helps in simplifying expressions but also lays the foundation for more advanced mathematical concepts.

Evaluating the Expression: 6 + (2 + 3) × 5

Let's apply the principles of PEMDAS/BODMAS to evaluate the given expression: 6 + (2 + 3) × 5. Following the order of operations, we begin with the parentheses. Inside the parentheses, we have the addition operation: 2 + 3 = 5. This simplifies the expression to 6 + 5 × 5. Now, according to PEMDAS/BODMAS, multiplication takes precedence over addition. Therefore, we perform the multiplication: 5 × 5 = 25. The expression is further simplified to 6 + 25. Finally, we perform the addition: 6 + 25 = 31. Thus, the value of the expression 6 + (2 + 3) × 5 is 31. This step-by-step evaluation demonstrates the importance of adhering to the order of operations. By systematically addressing each operation in the correct sequence, we arrive at the accurate result. Ignoring the order of operations would lead to a different, and incorrect, answer. This meticulous approach ensures clarity and eliminates ambiguity in mathematical calculations. Each step builds upon the previous one, creating a logical progression towards the solution. This process not only yields the correct answer but also reinforces the fundamental principles of mathematical evaluation. Understanding and applying the order of operations is crucial for success in mathematics, enabling us to solve complex problems with confidence and precision.

Analyzing the Answer Choices

Now that we have determined the value of the expression 6 + (2 + 3) × 5 to be 31, let's examine the provided answer choices to identify the one that also equals 31. This process involves applying the same order of operations to each option, ensuring a fair comparison. By meticulously evaluating each choice, we can pinpoint the correct answer and reinforce our understanding of PEMDAS/BODMAS. This exercise not only tests our ability to apply the order of operations but also sharpens our analytical skills. Each option presents a unique challenge, requiring careful attention to detail and a systematic approach. The process of elimination allows us to narrow down the possibilities, ultimately leading us to the correct solution. This comparative analysis highlights the importance of precision and accuracy in mathematical calculations. By thoroughly evaluating each option, we can confidently identify the one that matches the value of the original expression. This step-by-step approach ensures that we arrive at the correct answer with certainty and clarity.

A. (4 × 5) + 3

Let's evaluate option A: (4 × 5) + 3. Following PEMDAS/BODMAS, we first address the parentheses. Inside the parentheses, we have the multiplication operation: 4 × 5 = 20. This simplifies the expression to 20 + 3. Next, we perform the addition: 20 + 3 = 23. Therefore, the value of expression A is 23. Comparing this to our original expression's value of 31, we can see that option A is not the correct answer. This evaluation highlights the importance of accurately applying the order of operations. Each step must be performed meticulously to ensure the correct result. A single error in calculation can lead to an incorrect conclusion. This process of elimination helps us narrow down the possibilities and focus on the remaining options. By systematically evaluating each choice, we increase our chances of identifying the correct answer. This practice also reinforces our understanding of mathematical principles and strengthens our problem-solving skills. The ability to accurately evaluate expressions is crucial for success in mathematics and related fields. Therefore, mastering the order of operations is an essential skill.

B. 1 + 10 × 3

Now, let's evaluate option B: 1 + 10 × 3. According to PEMDAS/BODMAS, multiplication takes precedence over addition. Therefore, we first perform the multiplication: 10 × 3 = 30. This simplifies the expression to 1 + 30. Next, we perform the addition: 1 + 30 = 31. Thus, the value of expression B is 31. Comparing this to our original expression's value of 31, we find that option B matches our result. This indicates that option B is a potential correct answer. However, to ensure we have identified the correct solution, we must evaluate the remaining options as well. This thorough approach guarantees accuracy and eliminates any possibility of overlooking the correct answer. By systematically evaluating each option, we reinforce our understanding of mathematical principles and strengthen our problem-solving skills. This meticulous process ensures that we arrive at the correct conclusion with confidence. The ability to accurately evaluate expressions is crucial for success in mathematics and related fields. Therefore, mastering the order of operations is an essential skill that enables us to solve complex problems with precision.

C. 9 × 5 + 10

Let's proceed to evaluate option C: 9 × 5 + 10. Following the order of operations (PEMDAS/BODMAS), multiplication takes precedence over addition. Therefore, we first perform the multiplication: 9 × 5 = 45. This simplifies the expression to 45 + 10. Next, we perform the addition: 45 + 10 = 55. Thus, the value of expression C is 55. Comparing this to our original expression's value of 31, we can clearly see that option C is not the correct answer. This evaluation further reinforces the importance of adhering to the order of operations. Deviations from this order can lead to significant discrepancies in the final result. The systematic evaluation of each option allows us to eliminate incorrect choices and narrow down the possibilities. This process of elimination is a valuable problem-solving strategy that enhances our ability to identify the correct solution. By meticulously analyzing each option, we strengthen our understanding of mathematical principles and improve our analytical skills. This approach ensures that we arrive at the correct answer with confidence and accuracy. The ability to accurately evaluate expressions is crucial for success in mathematics and related fields, making the mastery of the order of operations an essential skill.

D. 5 + 4 × (5 - 6)

Finally, let's evaluate option D: 5 + 4 × (5 - 6). Following the order of operations (PEMDAS/BODMAS), we begin with the parentheses. Inside the parentheses, we have the subtraction operation: 5 - 6 = -1. This simplifies the expression to 5 + 4 × (-1). Next, multiplication takes precedence over addition, so we perform the multiplication: 4 × (-1) = -4. The expression is now simplified to 5 + (-4). Finally, we perform the addition: 5 + (-4) = 1. Thus, the value of expression D is 1. Comparing this to our original expression's value of 31, it is evident that option D is not the correct answer. This comprehensive evaluation of all options highlights the importance of a systematic approach to problem-solving. By meticulously applying the order of operations to each choice, we can confidently identify the correct answer and eliminate incorrect ones. This process reinforces our understanding of mathematical principles and strengthens our analytical skills. The ability to accurately evaluate expressions is crucial for success in mathematics and related fields. Therefore, mastering the order of operations is an essential skill that enables us to solve complex problems with precision and confidence. This thorough analysis ensures that we arrive at the correct conclusion with certainty.

Conclusion: The Correct Expression

After meticulously evaluating each option using the order of operations (PEMDAS/BODMAS), we have determined that option B, 1 + 10 × 3, is the only expression equal to the original expression, 6 + (2 + 3) × 5, which both evaluate to 31. This exercise underscores the critical importance of adhering to the correct order of operations in mathematical calculations. A seemingly simple expression can yield different results if the operations are performed in the wrong sequence. Mastering PEMDAS/BODMAS is therefore fundamental to achieving accuracy and consistency in mathematical problem-solving. This skill not only helps in solving equations but also forms the basis for more advanced mathematical concepts. By consistently applying the order of operations, we can confidently tackle complex problems and arrive at the correct solutions. This understanding is crucial for success in mathematics and related fields, where precision and accuracy are paramount. The ability to correctly evaluate expressions is a valuable asset that empowers us to navigate the world of numbers with confidence and clarity.