Solving 169 ÷ 18 × 9 ÷ 2 ÷ 3 - 132 ÷ 6 A Step-by-Step Mathematical Exploration

Mathematics can often present us with complex-looking expressions, but with a systematic approach, we can break them down into manageable steps and arrive at the correct answer. In this article, we will dissect the expression 169 ÷ 18 × 9 ÷ 2 ÷ 3 - 132 ÷ 6, meticulously solving each operation while highlighting the importance of the order of operations. The book suggests the answer is 174, and we'll explore each step to confirm or challenge this solution.

Understanding the Order of Operations

Before diving into the calculation, it's crucial to remember the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This order dictates the sequence in which we perform mathematical operations to ensure consistent and accurate results. In our expression, we have division, multiplication, and subtraction. Following PEMDAS, we perform division and multiplication from left to right before moving on to subtraction.

In tackling any mathematical expression, the order of operations serves as our compass, guiding us through the maze of numbers and symbols to the correct destination. PEMDAS, or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction) as it's known in some regions, is the universally accepted convention that dictates the sequence in which operations must be performed. This ensures that everyone, regardless of their location or background, arrives at the same answer when evaluating an expression. Without this standard, the world of mathematics would descend into chaos, with each individual potentially interpreting expressions differently.

Our expression, 169 ÷ 18 × 9 ÷ 2 ÷ 3 - 132 ÷ 6, presents a mix of division, multiplication, and subtraction. At first glance, it might seem daunting, a jumble of numbers and symbols demanding to be deciphered. However, with PEMDAS as our guide, we can transform this seemingly complex problem into a series of manageable steps. We'll begin by addressing the division and multiplication operations, working our way from left to right, carefully untangling the numerical web. Then, with the results of these operations in hand, we'll tackle the subtraction, bringing us closer to the final answer. Think of it as peeling an onion, layer by layer, until we reach the core – the solution that lies at the heart of the expression. Each step is a deliberate act, a careful application of mathematical principles, ensuring that we remain on the path to accuracy. Understanding the order of operations is not just about following rules; it's about embracing the logical structure that underpins mathematics, the framework that allows us to navigate complex problems with confidence and precision. It's about transforming chaos into order, confusion into clarity, and ultimately, arriving at the truth.

Step-by-Step Calculation

1. 169 ÷ 18: This division yields approximately 9.3889. This is the first step, dividing 169 by 18, which sets the stage for the subsequent operations. The result, approximately 9.3889, becomes a crucial intermediate value, a stepping stone towards the final answer. We carry this value forward, ensuring that it plays its part in the unfolding calculation. Each digit matters, each decimal place contributes to the overall precision of our solution. This initial division is not just an isolated act; it's the foundation upon which the rest of the calculation is built. It's the first piece of the puzzle, carefully placed, setting the pattern for the pieces that will follow. The accuracy of this step is paramount, as any error here will cascade through the remaining operations, potentially leading us astray. We approach this division with care, with diligence, understanding that it holds the key to unlocking the mystery of the expression.

2. 9. 3889 × 9: Multiplying the result by 9 gives us approximately 84.5. This multiplication builds upon the previous division, taking the quotient and scaling it up by a factor of nine. The result, approximately 84.5, is another intermediate value, a landmark along the path to the final answer. Each multiplication is a step forward, a refinement of our understanding of the expression's value. We are not just crunching numbers; we are constructing a solution, brick by brick. This step is a testament to the interconnectedness of mathematical operations, how each one depends on the ones that came before. The accuracy we maintained in the first step now pays dividends, ensuring the reliability of this multiplication. We proceed with confidence, knowing that we are building on a solid foundation. This multiplication is not just a calculation; it's a step in a journey, a leg of the race towards the solution.

3. 84. 5 ÷ 2: Dividing 84.5 by 2 results in 42.25. The division by two acts as a halving operation, scaling the result down to a more manageable size. The value 42.25 emerges as yet another intermediate value, a milestone passed on our way to the final destination. Each division is a carving away, a reduction of the value, bringing us closer to the true essence of the expression. We are not just manipulating numbers; we are sculpting a solution, carefully shaping it with each operation. This step is a demonstration of the power of division, its ability to break down large quantities into smaller, more comprehensible parts. The precision we have maintained throughout the previous steps is crucial here, ensuring that the halving is accurate and true. We move forward with determination, knowing that we are honing in on the answer with each passing operation. This division is not just a mathematical act; it's a process of refinement, a step towards clarity.

4. 42. 25 ÷ 3: Dividing 42.25 by 3 yields approximately 14.0833. This division further refines our result, reducing the value to approximately 14.0833. This intermediate value is a crucial marker, guiding us closer to the final answer. Each division is a further step in the unraveling of the expression, peeling back the layers to reveal the solution within. We are not simply performing calculations; we are engaged in a process of discovery, uncovering the hidden truth of the equation. This step highlights the importance of continued precision, as even small discrepancies can accumulate and affect the final result. We press on with diligence, knowing that each operation brings us closer to our goal. This division is not just a numerical manipulation; it's an act of unveiling, a step towards understanding.

5. 132 ÷ 6: Separately, we divide 132 by 6, which equals 22. This division is a separate calculation within the larger expression, a tributary feeding into the main stream. The result, 22, is a distinct value, waiting to be incorporated into the final subtraction. Each independent calculation is a piece of the puzzle, a part of the whole that must be correctly assembled. We are not just solving a single problem; we are tackling a multi-faceted challenge, addressing each component with care and attention. This step underscores the importance of recognizing separate operations within a larger expression, treating each with the respect it deserves. The accuracy of this division is paramount, as it will directly impact the final outcome. We proceed with focus, knowing that each calculation contributes to the overall solution. This division is not just an isolated act; it's an integral part of the larger process, a key element in the final equation.

6. 14. 0833 - 22: Subtracting 22 from 14.0833 gives us approximately -7.9167. This final subtraction brings together the results of the previous operations, revealing the overall value of the expression. The result, approximately -7.9167, is the culmination of our efforts, the answer we have been seeking. Each subtraction is a comparison, a determination of the difference between two values, bringing us closer to a complete understanding. We are not just performing an arithmetic operation; we are synthesizing the results of previous calculations, drawing them together to form a coherent whole. This step underscores the importance of the order of operations, as the final subtraction relies on the accuracy of all the preceding steps. We arrive at the final answer with confidence, knowing that we have followed the correct path. This subtraction is not just a mathematical act; it's a synthesis of knowledge, a culmination of effort, the final piece of the puzzle falling into place.

Comparing the Result with the Book's Answer

Our calculation yields approximately -7.9167, which is significantly different from the book's answer of 174. This discrepancy suggests a potential error in the book's answer or a misunderstanding of the expression. It's essential to double-check the original expression and each step of the calculation to identify any mistakes.

The vast difference between our calculated result of approximately -7.9167 and the book's answer of 174 raises a significant red flag. Such a discrepancy demands a thorough investigation, a forensic examination of every step taken in the calculation. It's not simply a matter of dismissing one answer as incorrect; it's an opportunity to delve deeper into the problem, to understand the potential sources of error, and to reinforce the importance of meticulousness in mathematical problem-solving.

First, we must revisit the original expression, ensuring that we have transcribed it correctly. A single misplaced digit or symbol can throw the entire calculation off course. Then, we must meticulously retrace our steps, examining each operation, each intermediate value, for any potential errors in arithmetic. Did we perform the divisions and multiplications in the correct order? Did we carry the decimal places accurately? Did we make any mistakes in the final subtraction?

This process of error detection is not just about finding the mistake; it's about understanding why the mistake occurred. Did we misunderstand a mathematical principle? Did we rush through a step, sacrificing accuracy for speed? Did we overlook a crucial detail? By identifying the root cause of the error, we can learn from our mistakes and prevent them from recurring in the future.

Furthermore, the discrepancy highlights the importance of critical thinking in mathematics. It's not enough to simply accept an answer, even if it comes from a trusted source like a textbook. We must always question, always verify, always strive for a deeper understanding. Mathematics is not just about memorizing formulas and procedures; it's about developing a logical and analytical mindset, the ability to dissect problems, identify patterns, and arrive at solutions with confidence.

In this case, the significant difference between the two answers suggests a fundamental error somewhere in the process. It's a challenge, an invitation to engage with the problem at a deeper level, to hone our mathematical skills, and to emerge with a more robust understanding of the concepts involved. The pursuit of accuracy is not just about getting the right answer; it's about cultivating a mindset of precision, a commitment to excellence, and a relentless pursuit of the truth.

Possible Sources of Error

1. Misinterpretation of the Expression: It's possible that the expression was misinterpreted, leading to a different order of operations. Double-checking the original problem is crucial.

2. Arithmetic Errors: Mistakes in basic arithmetic can easily occur, especially with multiple operations. Reviewing each calculation step is necessary.

3. Book Error: It's not uncommon for textbooks to contain errors. If all steps are verified and the discrepancy remains, the book's answer may be incorrect.

Delving into the potential sources of error is a critical step in the problem-solving process. It's a shift from simply performing calculations to adopting a more analytical and investigative approach. We're no longer just crunching numbers; we're becoming detectives, searching for clues that might lead us to the truth.

One of the most common pitfalls in mathematical problem-solving is the misinterpretation of the expression itself. A misplaced parenthesis, a wrongly transcribed digit, or a misunderstanding of the order of operations can all lead to drastically different results. Therefore, the first step in our investigation must be a meticulous re-examination of the original expression. We must compare it against our working notes, ensuring that we have captured every detail accurately. It's like checking the blueprint before starting construction; any error at this stage can have cascading consequences.

Even if the expression is correctly transcribed, arithmetic errors can still creep in. The human mind is not a perfect calculating machine, and mistakes can happen, especially when dealing with multiple operations and decimal places. Therefore, we must methodically review each calculation step, scrutinizing every addition, subtraction, multiplication, and division. It's like proofreading a document, looking for typos and grammatical errors; attention to detail is paramount.

But what if we've checked and rechecked our work, and we're still confident in our answer? It's important to consider the possibility that the error might lie not with us, but with the source material. Textbooks, while generally reliable, are not immune to mistakes. Typos, misprints, and even conceptual errors can occur. Therefore, we should not be afraid to challenge the given answer, especially if our calculations consistently lead us to a different result.

Identifying the possible sources of error is not just about finding the mistake; it's about developing a critical and analytical mindset. It's about recognizing that errors are a natural part of the learning process, and that by understanding why they occur, we can improve our problem-solving skills and become more confident mathematicians. It's about embracing the challenge, treating each discrepancy as an opportunity to learn and grow.

Conclusion

In conclusion, by meticulously following the order of operations, we arrived at an answer of approximately -7.9167 for the expression 169 ÷ 18 × 9 ÷ 2 ÷ 3 - 132 ÷ 6. This result differs significantly from the book's answer of 174, suggesting a potential error either in the book or in the interpretation of the problem. It's crucial to always double-check calculations and be aware of potential mistakes.

Ultimately, the journey through this mathematical expression serves as a powerful reminder of the importance of precision, attention to detail, and a willingness to question and verify. Mathematics is not just about arriving at the right answer; it's about the process of logical thinking, problem-solving, and the pursuit of truth.

The discrepancy between our calculated result and the book's answer serves as a valuable learning opportunity, a chance to reinforce the fundamental principles of mathematics and to hone our problem-solving skills. It's a testament to the fact that mathematics is not a passive activity, a mere memorization of formulas and procedures; it's an active engagement with logical reasoning, critical thinking, and the relentless pursuit of accuracy.

By meticulously following the order of operations, we have demonstrated a systematic approach to problem-solving, a framework that can be applied to a wide range of mathematical challenges. We have shown the importance of breaking down complex expressions into manageable steps, carefully performing each operation, and maintaining accuracy throughout the process. This step-by-step approach is not just a technique; it's a mindset, a way of organizing our thoughts and tackling problems with confidence.

Our willingness to question the book's answer highlights the importance of critical thinking in mathematics. It's not enough to simply accept information at face value; we must always be prepared to challenge assumptions, verify results, and seek a deeper understanding. This critical mindset is essential not just in mathematics, but in all areas of life.

The journey through this expression has also underscored the importance of attention to detail. A single misplaced digit or symbol can throw the entire calculation off course. Therefore, we must cultivate a habit of meticulousness, of carefully checking our work, and of being aware of potential sources of error. This attention to detail is a valuable skill, applicable not just to mathematics, but to any task that requires precision and accuracy.

In conclusion, the expression 169 ÷ 18 × 9 ÷ 2 ÷ 3 - 132 ÷ 6 has served as a vehicle for learning, a platform for developing our mathematical skills, and a reminder of the importance of precision, critical thinking, and the relentless pursuit of accuracy. The true value of mathematics lies not just in the answers we obtain, but in the intellectual journey we undertake to arrive at those answers. It's a journey that fosters logical reasoning, sharpens our problem-solving abilities, and equips us with the tools to navigate the complexities of the world around us.