Solving (47 - 17) + 10 × 3 A Step-by-Step Mathematical Explanation

Introduction: Deciphering the Arithmetic Expression

In the realm of mathematics, we often encounter expressions that seem daunting at first glance. These expressions, with their mix of numbers, operators, and parentheses, can appear like intricate puzzles waiting to be solved. However, with a systematic approach and a clear understanding of the order of operations, these puzzles can be unraveled with ease. Let's delve into the mathematical puzzle (47 - 17) + 10 × 3 and dissect it step-by-step to arrive at the solution.

This particular expression involves a combination of subtraction, addition, and multiplication, which necessitates a careful adherence to the order of operations. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which mathematical operations should be performed. Ignoring this order can lead to incorrect results, so it's crucial to follow it diligently. Understanding the order of operations is not just a mathematical necessity; it's a fundamental skill that enhances problem-solving abilities in various aspects of life. It teaches us to approach complex tasks methodically, breaking them down into manageable steps and executing them in the correct sequence. This structured approach is invaluable not only in mathematics but also in everyday decision-making and project management. By mastering the order of operations, we develop a critical thinking mindset that empowers us to tackle challenges with confidence and precision.

Step 1: Tackling the Parentheses – The First Priority

The first step in unraveling the expression (47 - 17) + 10 × 3 is to address the parentheses. According to the order of operations, any operation enclosed within parentheses must be performed before any other operations. In this case, we have the subtraction operation 47 - 17 nestled within the parentheses. Performing this subtraction, we get 30. This simplification within the parentheses is a crucial step, as it reduces the complexity of the expression and allows us to proceed with the remaining operations in a clear and organized manner. The emphasis on parentheses in the order of operations highlights the importance of grouping and prioritizing certain calculations. Parentheses act as containers that isolate specific parts of an expression, ensuring that they are evaluated independently before being combined with other elements. This concept of prioritizing operations within parentheses is not limited to simple arithmetic; it extends to more advanced mathematical concepts such as algebraic expressions and complex equations. By understanding the role of parentheses, we gain a deeper appreciation for the structure and logic of mathematical notation. Moreover, the principle of prioritizing tasks within parentheses can be applied to various real-world scenarios. In project management, for instance, certain tasks may need to be completed before others can begin, effectively creating a set of prioritized operations analogous to those within parentheses.

Step 2: Multiplication – The Next in Line

With the parentheses resolved, our expression now transforms into 30 + 10 × 3. The next operation we encounter is multiplication. According to the order of operations, multiplication and division take precedence over addition and subtraction. Therefore, we must perform the multiplication operation 10 × 3 before we proceed with the addition. Multiplying 10 by 3 yields 30. This step further simplifies our expression, bringing us closer to the final solution. The importance of multiplication in the order of operations stems from its nature as a repeated addition. Multiplying two numbers is essentially a shorthand way of adding the first number to itself the number of times indicated by the second number. This fundamental relationship between multiplication and addition underscores its priority in mathematical calculations. In various fields, multiplication plays a vital role, from calculating areas and volumes in geometry to determining financial growth in economics. Understanding the principles of multiplication is essential for both mathematical proficiency and practical applications. Moreover, the concept of precedence in multiplication extends to more complex mathematical operations such as matrix multiplication and vector operations. These operations, which are fundamental to fields like computer graphics and data analysis, rely on a clear understanding of the order in which multiplications are performed. By mastering the principles of multiplication, we equip ourselves with a powerful tool for solving a wide range of mathematical and real-world problems.

Step 3: Addition – The Grand Finale

Having tackled the parentheses and multiplication, we arrive at the final step: addition. Our expression now stands as 30 + 30. This is a straightforward addition operation, and adding 30 to 30 gives us 60. This final calculation completes the unraveling of our mathematical puzzle, revealing the solution. Addition, the fundamental operation of combining quantities, is the cornerstone of arithmetic and a building block for more advanced mathematical concepts. Its simplicity belies its power, as it forms the basis for counting, measuring, and a vast array of calculations in everyday life. From calculating grocery bills to determining travel distances, addition is an indispensable tool. In the context of the order of operations, addition and subtraction are typically performed last, after parentheses, exponents, multiplication, and division have been addressed. This positioning reflects the additive nature of these operations, as they serve to combine the results of previous calculations. However, it's crucial to remember that addition and subtraction are performed from left to right, ensuring that the operations are carried out in the correct sequence. The principles of addition extend far beyond basic arithmetic. In algebra, addition is used to combine like terms and simplify expressions. In calculus, addition is used to find the sum of infinite series. In statistics, addition is used to calculate averages and other descriptive measures. By mastering the art of addition, we unlock a gateway to a deeper understanding of mathematics and its applications.

Conclusion: The Solution Unveiled

Through a meticulous step-by-step process, we have successfully solved the mathematical expression (47 - 17) + 10 × 3. By adhering to the order of operations, we first addressed the parentheses, then performed the multiplication, and finally completed the addition. This systematic approach led us to the solution: 60. This exercise not only demonstrates the importance of following the order of operations but also highlights the power of breaking down complex problems into manageable steps. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), serves as a guiding principle for evaluating mathematical expressions. It ensures that calculations are performed in a consistent and logical manner, leading to accurate results. Mastering the order of operations is not just a mathematical necessity; it's a fundamental skill that enhances problem-solving abilities in various aspects of life. It teaches us to approach complex tasks methodically, breaking them down into manageable steps and executing them in the correct sequence. This structured approach is invaluable not only in mathematics but also in everyday decision-making and project management. By understanding and applying the order of operations, we empower ourselves to tackle challenges with confidence and precision. The solution to this particular expression, 60, represents the culmination of these efforts, a testament to the power of mathematical reasoning and the importance of a systematic approach. As we continue our exploration of mathematics, the principles we have learned here will serve as a solid foundation for tackling more complex problems and expanding our mathematical horizons.