Solving Mathematical Expressions A Step-by-Step Guide To 60-5-12 X 5+20

Navigating the realm of mathematical expressions often involves encountering a seemingly complex string of numbers and operations. Understanding the order of operations is paramount to deciphering these expressions accurately. In this comprehensive exploration, we will dissect the expression $60-5-12 \times 5+20$, meticulously applying the order of operations to arrive at the correct solution. This journey will not only provide the answer but also illuminate the fundamental principles that govern mathematical computations. We will delve into the significance of each operation, emphasizing the critical role they play in achieving precision and clarity in mathematics. By the end of this discussion, you will have a solid grasp of how to approach similar mathematical problems with confidence and accuracy. This understanding is crucial for various mathematical disciplines and real-world applications, making it an invaluable skill for anyone engaging with numerical problems.

The Foundation: Order of Operations (PEMDAS/BODMAS)

To effectively solve $60-5-12 \times 5+20$, we must adhere to the established order of operations, often remembered by the acronyms PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction). These acronyms serve as a roadmap, guiding us through the sequence in which operations must be performed to ensure accuracy. The underlying principle is to resolve operations within parentheses or brackets first, followed by exponents or orders (powers and roots), then multiplication and division (from left to right), and finally addition and subtraction (also from left to right). Ignoring this order can lead to vastly different and incorrect answers, highlighting its importance in mathematical problem-solving. This standardized approach ensures that mathematical expressions are interpreted and solved consistently across different individuals and contexts, forming the bedrock of mathematical communication and understanding. Let's explore the nuances of each step to fully appreciate the rationale behind this order.

Multiplication Takes Precedence

In our expression, $60-5-12 \times 5+20$, the first operation we encounter according to the order of operations is multiplication. The term $12 \times 5$ requires our immediate attention. Performing this multiplication yields $12 \times 5 = 60$. This step is crucial because multiplication holds a higher precedence than both addition and subtraction. By prioritizing multiplication, we ensure that the multiplicative relationship between 12 and 5 is accurately accounted for before any other operations are performed. This adherence to the order of operations is not arbitrary; it's based on the mathematical structure of expressions, ensuring that we correctly capture the relationships between the numbers and operations involved. The result of this multiplication, 60, becomes a key component in the subsequent steps of our calculation.

Simplifying the Expression

After performing the multiplication, our expression transforms from $60-5-12 \times 5+20$ to $60-5-60+20$. This simplified form allows us to focus on the remaining operations: subtraction and addition. It's important to note that subtraction and addition have equal precedence in the order of operations. This means that we perform these operations from left to right, in the order they appear in the expression. This left-to-right approach is essential for maintaining accuracy when dealing with a series of additions and subtractions. By systematically working through the expression, we avoid any ambiguity and ensure that each operation is performed in the correct sequence. This step-by-step simplification is a cornerstone of mathematical problem-solving, breaking down complex expressions into manageable components.

Addition and Subtraction: A Left-to-Right Approach

With the expression now simplified to $60-5-60+20$, we proceed with the remaining operations. Starting from the left, we first encounter the subtraction $60-5$, which results in $55$. Our expression now reads $55-60+20$. Next, we perform the subtraction $55-60$, yielding $-5$. The expression is further simplified to $-5+20$. Finally, we perform the addition $-5+20$, which gives us $15$. This left-to-right execution of addition and subtraction is critical for obtaining the correct result. By adhering to this principle, we ensure that the operations are performed in the intended order, reflecting the true mathematical relationship between the numbers. This systematic approach eliminates the possibility of misinterpreting the expression and ensures a precise solution.

The Final Answer: 15

Through careful application of the order of operations, we have successfully navigated the expression $60-5-12 \times 5+20$ to arrive at the final answer: 15. This journey highlights the importance of adhering to established mathematical principles to achieve accurate results. Each step, from prioritizing multiplication to performing addition and subtraction from left to right, played a crucial role in the solution. The order of operations is not merely a set of rules to be memorized; it's a framework for understanding the structure of mathematical expressions and ensuring consistent interpretation. By mastering this framework, we empower ourselves to tackle a wide range of mathematical problems with confidence and precision. This ability is invaluable in various fields, from science and engineering to finance and everyday life, demonstrating the far-reaching implications of understanding fundamental mathematical principles.

Conclusion: Mastering the Order of Operations

The process of solving $60-5-12 \times 5+20$ exemplifies the power and importance of the order of operations. This fundamental concept in mathematics provides a clear and consistent framework for evaluating expressions, ensuring that we arrive at accurate solutions. By understanding and applying PEMDAS/BODMAS, we can confidently navigate complex mathematical problems, breaking them down into manageable steps. This skill is not only essential for academic success but also for practical applications in various fields. The ability to correctly interpret and solve mathematical expressions is a valuable asset in a world increasingly driven by data and quantitative analysis. The journey to mastering the order of operations is an investment in mathematical literacy, opening doors to deeper understanding and greater proficiency in quantitative reasoning. As we continue to explore the world of mathematics, the principles we have discussed here will serve as a solid foundation for future learning and problem-solving.