Solving 12 + 3 - 15 ÷ (-2 - 1) + 30 - 64 ÷ 8 - 2 + 10 - 72 ÷ (-8) A Step-by-Step Guide

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In this comprehensive guide, we will meticulously walk through the process of solving the mathematical expression: 12 + 3 - 15 ÷ (-2 - 1) + 30 - 64 ÷ 8 - 2 + 10 - 72 ÷ (-8). This expression involves a mix of arithmetic operations, including addition, subtraction, division, and negative numbers. To ensure accuracy, we will adhere to the order of operations, commonly remembered by the acronym PEMDAS/BODMAS (Parentheses/Brackets, Exponents/Orders, Multiplication and Division, Addition and Subtraction). Understanding and applying PEMDAS/BODMAS is crucial for correctly solving any mathematical expression of this nature. This step-by-step approach aims to demystify the process and provide a clear understanding of how each operation is performed in the correct sequence, ultimately leading to the solution.

Understanding the Order of Operations (PEMDAS/BODMAS)

Before we dive into the problem, it's essential to understand the order of operations. This is a set of rules that dictate the sequence in which mathematical operations should be performed. The acronyms PEMDAS and BODMAS serve as memory aids to recall this order:

  • Parentheses / Brackets
  • Exponents / Orders
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

The order of operations is the backbone of simplifying complex mathematical expressions. Adhering to this order ensures that we arrive at the correct answer. When an expression involves multiple operations, PEMDAS/BODMAS provides the roadmap for how to proceed. For instance, division and multiplication have higher precedence than addition and subtraction, meaning we tackle them first. Similarly, operations within parentheses or brackets take precedence over all others. This structured approach eliminates ambiguity and ensures consistency in mathematical calculations.

Now, let’s break down the expression step by step, applying the order of operations diligently. By following each step methodically, we can unravel the complexity of the expression and arrive at the final solution. Understanding PEMDAS/BODMAS isn't just about solving this particular problem; it's a foundational skill that applies to a wide range of mathematical challenges. In the following sections, we'll see exactly how this order is applied to each component of our expression, ensuring clarity and accuracy at every stage.

Step 1: Simplify within Parentheses

The first step in solving the expression 12 + 3 - 15 ÷ (-2 - 1) + 30 - 64 ÷ 8 - 2 + 10 - 72 ÷ (-8) is to address the operation within the parentheses: (-2 - 1). This is the highest priority according to the order of operations (PEMDAS/BODMAS).

  • -2 - 1 = -3

Now, we substitute this result back into the original expression, which gives us:

  • 12 + 3 - 15 ÷ (-3) + 30 - 64 ÷ 8 - 2 + 10 - 72 ÷ (-8)

Simplifying expressions within parentheses first is a critical step because it isolates and resolves a specific part of the equation, making the subsequent steps clearer and more manageable. In this case, by simplifying (-2 - 1) to -3, we've reduced the complexity of the overall expression. This initial step is a cornerstone of the PEMDAS/BODMAS methodology, highlighting the importance of tackling grouped operations before moving on to divisions, multiplications, additions, or subtractions.

This simple subtraction within parentheses sets the stage for the next phase of solving the problem, where we'll focus on division operations. By handling the parentheses first, we ensure that all subsequent calculations are based on accurate intermediate results. This methodical approach is what makes PEMDAS/BODMAS such a valuable tool in mathematics, allowing us to break down seemingly complex problems into a series of smaller, solvable steps. Understanding the significance of this initial simplification is key to mastering the order of operations and tackling more intricate mathematical challenges in the future.

Step 2: Perform Division Operations

Having simplified the parentheses, the next step according to PEMDAS/BODMAS is to address the division operations. In our expression, 12 + 3 - 15 ÷ (-3) + 30 - 64 ÷ 8 - 2 + 10 - 72 ÷ (-8), there are three division operations to perform. We will proceed from left to right:

  1. 15 ÷ (-3): A positive number divided by a negative number results in a negative number. 15 divided by 3 is 5, so 15 ÷ (-3) = -5.
  2. 64 ÷ 8: This is a straightforward division. 64 divided by 8 is 8.
  3. 72 ÷ (-8): Similar to the first division, a positive number divided by a negative number results in a negative number. 72 divided by 8 is 9, so 72 ÷ (-8) = -9.

Now, we replace these divisions with their results in the expression:

  • 12 + 3 - (-5) + 30 - 8 - 2 + 10 - (-9)

Addressing division operations before addition and subtraction is crucial for adhering to the correct order of operations. Division, along with multiplication, holds a higher precedence in PEMDAS/BODMAS, which means these operations must be carried out before any additions or subtractions are performed. This step is essential for maintaining the integrity of the mathematical equation and ensuring that the final answer is accurate.

Each division operation we've performed here has simplified the expression, bringing us closer to the final solution. By methodically working through each division from left to right, we avoid any potential confusion or errors. Furthermore, the handling of negative numbers in these divisions is a key aspect of this step, demonstrating the importance of understanding the rules of signs in arithmetic. With the divisions now completed, the expression is significantly streamlined, setting the stage for the final additions and subtractions that will lead us to the answer. The careful and methodical approach taken in this step is a testament to the effectiveness of PEMDAS/BODMAS in breaking down complex problems.

Step 3: Handle Subtraction of Negative Numbers

Before we proceed with addition and subtraction, it's important to address the subtraction of negative numbers in the expression: 12 + 3 - (-5) + 30 - 8 - 2 + 10 - (-9). Subtracting a negative number is the same as adding its positive counterpart. This is a fundamental rule in arithmetic and is crucial for simplifying expressions correctly.

  1. - (-5) becomes +5
  2. - (-9) becomes +9

So, the expression now becomes:

  • 12 + 3 + 5 + 30 - 8 - 2 + 10 + 9

Transforming subtractions of negative numbers into additions is a pivotal step in simplifying mathematical expressions. This transformation not only makes the expression easier to read and understand but also reduces the risk of making errors in subsequent calculations. The principle behind this conversion—that subtracting a negative is equivalent to adding a positive—is a cornerstone of basic arithmetic and algebra.

By handling these subtractions of negative numbers, we've streamlined the expression into a series of additions and subtractions, which can now be tackled in a straightforward manner. This step underscores the importance of attending to the nuances of mathematical notation and applying fundamental rules to ensure accuracy. The ability to confidently convert between subtracting negatives and adding positives is a key skill in mathematics, simplifying complex expressions and paving the way for accurate solutions. With the expression now in this simplified form, we are perfectly positioned to complete the final additions and subtractions, bringing us closer to the ultimate answer.

Step 4: Perform Addition and Subtraction from Left to Right

Now that we've simplified the expression to 12 + 3 + 5 + 30 - 8 - 2 + 10 + 9, we proceed with the final step: performing addition and subtraction from left to right. According to PEMDAS/BODMAS, addition and subtraction have equal priority, so we execute them in the order they appear.

  1. 12 + 3 = 15
  2. 15 + 5 = 20
  3. 20 + 30 = 50
  4. 50 - 8 = 42
  5. 42 - 2 = 40
  6. 40 + 10 = 50
  7. 50 + 9 = 59

Therefore, the final result of the expression is 59.

Executing addition and subtraction from left to right is the concluding step in adhering to the order of operations. This approach ensures that each operation is performed in its proper sequence, preventing errors that could arise from tackling the problem out of order. The step-by-step method not only guarantees accuracy but also simplifies the process, making it easier to manage and understand.

By systematically adding and subtracting the numbers as they appear, we maintain a clear and logical progression towards the solution. This method reinforces the importance of following established mathematical conventions, which are designed to provide consistency and clarity in calculations. With each operation, the expression is further reduced until we arrive at the single, final answer. The meticulous application of this left-to-right approach in the final step showcases the power and effectiveness of PEMDAS/BODMAS in resolving complex mathematical problems. Completing these calculations in the correct order solidifies our understanding of arithmetic operations and their application in problem-solving scenarios.

Conclusion

In conclusion, by meticulously following the order of operations (PEMDAS/BODMAS), we have successfully solved the complex expression 12 + 3 - 15 ÷ (-2 - 1) + 30 - 64 ÷ 8 - 2 + 10 - 72 ÷ (-8). The step-by-step approach involved simplifying parentheses first, followed by division operations, handling subtraction of negative numbers, and finally, performing addition and subtraction from left to right. Each step was crucial in simplifying the expression and ensuring an accurate final result.

The final answer to the expression is 59. This process underscores the importance of adhering to the correct order of operations in mathematics. PEMDAS/BODMAS serves as a valuable tool for breaking down complex problems into manageable steps, reducing the likelihood of errors and promoting a clear understanding of mathematical principles.

Mastering the order of operations is not just about solving specific problems; it's a fundamental skill that applies to a wide range of mathematical contexts. The systematic approach we've employed here can be used to tackle more complex expressions and equations in the future. By understanding and applying PEMDAS/BODMAS, anyone can confidently navigate the world of mathematics, ensuring accuracy and efficiency in their calculations. This step-by-step guide has demonstrated how even seemingly daunting expressions can be simplified through a logical and methodical approach, highlighting the power of mathematical principles in problem-solving.