Evaluate 500 - [15 X 10 + (2 X 95)] A Step-by-Step Guide
Evaluating mathematical expressions requires a strong understanding of the order of operations, often remembered by the acronym PEMDAS/BODMAS, which stands for Parentheses/Brackets, Exponents/Orders, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). This systematic approach ensures that we arrive at the correct solution by performing operations in the proper sequence. In this article, we will meticulously evaluate the given expression, 500 - [15 x 10 + (2 x 95)], step-by-step, to illustrate the application of these rules. Understanding and applying the order of operations is a fundamental skill in mathematics, crucial for solving more complex problems and equations. Let’s dive into the detailed solution to enhance your comprehension and mathematical proficiency.
Breaking Down the Expression
The expression we need to evaluate is: 500 - [15 x 10 + (2 x 95)]. To solve this, we must follow the order of operations (PEMDAS/BODMAS), which dictates that we first address the operations within parentheses or brackets. In this case, we have both parentheses and brackets, so we start with the innermost parentheses. This structured approach ensures that we tackle the expression systematically, minimizing the chance of errors and leading us to the correct answer. Remember, the key to accurately evaluating expressions is to adhere to the established rules and proceed step-by-step. This methodical approach not only helps in solving the current problem but also builds a strong foundation for tackling more complex mathematical challenges in the future. Let's begin the step-by-step evaluation to demonstrate how each operation is performed according to the order of operations.
Step 1: Innermost Parentheses
Our expression contains nested parentheses, so we start with the innermost one: (2 x 95). This multiplication operation is the first step according to PEMDAS/BODMAS. Performing this multiplication gives us 2 multiplied by 95, which equals 190. This result now replaces the original parenthetical expression in the larger equation. It's crucial to handle these inner operations first to simplify the expression and move towards the final solution. Remember, each step in the order of operations serves to reduce complexity and make the overall calculation more manageable. By focusing on the innermost parentheses first, we are setting the stage for the subsequent steps, ensuring that we follow the correct sequence and maintain accuracy throughout the evaluation process.
- 2 x 95 = 190
Step 2: Brackets
Now, we substitute the result back into the expression, which becomes: 500 - [15 x 10 + 190]. The next step is to address the operations within the brackets. Inside the brackets, we have both multiplication and addition. According to the order of operations, multiplication takes precedence over addition. Therefore, we first perform the multiplication: 15 multiplied by 10. This calculation results in 150. Now, the expression inside the brackets simplifies to an addition problem: 150 + 190. By following this order, we ensure that we are adhering to the established mathematical rules and progressing towards an accurate solution. This step-by-step approach is vital for maintaining clarity and preventing errors in complex expressions.
- 15 x 10 = 150
Step 3: Addition within Brackets
Continuing with the expression inside the brackets, we now have 150 + 190. This is a straightforward addition operation. Adding these two numbers together gives us a sum of 340. This result replaces the entire expression within the brackets. Remember, each operation simplifies the expression, bringing us closer to the final answer. By performing the addition within the brackets, we have effectively reduced the complexity of the original expression, making it easier to manage the remaining operations. This methodical approach is key to solving mathematical problems accurately and efficiently.
- 150 + 190 = 340
Step 4: Final Subtraction
After simplifying the expression within the brackets, our equation now looks like this: 500 - 340. The final step is a simple subtraction. Subtracting 340 from 500 gives us the final result. Performing this subtraction, we find that 500 minus 340 equals 160. This is the final value of the expression. By systematically following the order of operations, we have successfully evaluated the expression and arrived at the correct answer. This final step demonstrates the importance of adhering to the rules of PEMDAS/BODMAS throughout the entire process to ensure accuracy and precision in mathematical calculations.
- 500 - 340 = 160
The Final Result
Therefore, the final result of the expression 500 - [15 x 10 + (2 x 95)] is 160. This solution is the culmination of a step-by-step process, carefully adhering to the order of operations (PEMDAS/BODMAS). We started by addressing the innermost parentheses, then moved to the brackets, performing multiplication and addition in the correct sequence, and finally completed the subtraction. This systematic approach is not just a way to solve this particular problem but a fundamental skill in mathematics. Understanding and applying the order of operations correctly is crucial for tackling more complex equations and problems in the future. It ensures accuracy and efficiency in mathematical calculations, making it an indispensable tool for anyone working with numbers and expressions.
Importance of Order of Operations
The importance of the order of operations (PEMDAS/BODMAS) cannot be overstated in mathematics. This set of rules provides a standardized approach to evaluating expressions, ensuring that everyone arrives at the same correct answer. Without a consistent order, mathematical expressions could be interpreted in multiple ways, leading to confusion and incorrect results. PEMDAS/BODMAS acts as a universal guideline, dictating the sequence in which operations must be performed: Parentheses/Brackets first, then Exponents/Orders, followed by Multiplication and Division (from left to right), and finally Addition and Subtraction (from left to right). This structured approach is crucial for maintaining accuracy and clarity in mathematical calculations. By adhering to these rules, we eliminate ambiguity and ensure that mathematical expressions are evaluated consistently, regardless of who is solving them. This consistency is essential for effective communication and collaboration in mathematics and related fields.
Avoiding Common Mistakes
A common mistake in evaluating mathematical expressions is failing to follow the correct order of operations. For instance, someone might mistakenly perform addition before multiplication, leading to an incorrect result. Another frequent error is neglecting parentheses or brackets, which can significantly alter the outcome of an expression. To avoid these pitfalls, it is essential to always adhere to the PEMDAS/BODMAS rules. Begin by simplifying expressions within parentheses or brackets, then address exponents, followed by multiplication and division (from left to right), and finally addition and subtraction (from left to right). Consistent application of these rules will minimize errors and ensure accurate evaluations. Additionally, double-checking each step and paying close attention to the placement of parentheses and operators can further enhance accuracy and prevent common mistakes in mathematical calculations.
Practice Problems
To reinforce your understanding of the order of operations, solving practice problems is invaluable. Consider the following expressions:
- 100 + [25 x 2 - (10 ÷ 5)]
- 3 x (20 - 5) + 12 ÷ 4
- 450 - [5 x 10 + (3 x 35)]
Work through each problem step-by-step, carefully applying the PEMDAS/BODMAS rules. Start with parentheses or brackets, then handle exponents, followed by multiplication and division, and finally addition and subtraction. After solving each expression, check your answers to ensure accuracy. Practice helps solidify your understanding of the order of operations and improves your ability to apply these rules confidently. The more you practice, the more proficient you will become in evaluating mathematical expressions correctly and efficiently. Consistent practice is the key to mastering this fundamental skill in mathematics.
Conclusion
In conclusion, evaluating the expression 500 - [15 x 10 + (2 x 95)] demonstrates the critical importance of following the order of operations (PEMDAS/BODMAS). By systematically addressing the innermost parentheses, then the brackets, and performing multiplication, addition, and subtraction in the correct sequence, we arrived at the solution of 160. This step-by-step approach not only ensures accuracy but also simplifies complex expressions into manageable steps. Understanding and applying the order of operations is a fundamental skill in mathematics, essential for solving a wide range of problems. Consistent practice and attention to detail are key to mastering this concept and avoiding common errors. Whether you are a student learning the basics or someone working with complex mathematical equations, a solid grasp of the order of operations is indispensable for success in mathematics and related fields.