Simplifying Complex Mathematical Expressions A Step-by-Step Guide

Understanding the Order of Operations

In the realm of mathematics, simplifying complex expressions requires a systematic approach. The expression we're tackling today, 26 - (6 - 12/3) - 1 2/5 - [3 8/3 - (2/5 - 1 1/6)], might seem daunting at first glance, but by breaking it down step by step and adhering to the order of operations – often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) – we can simplify it effectively. This principle ensures that mathematical expressions are evaluated consistently, leading to a unique and correct solution. Understanding PEMDAS is crucial not just for this problem but for a wide range of mathematical calculations, making it a fundamental concept in arithmetic and algebra. Mastering this order allows for the accurate simplification of expressions, paving the way for solving more complex equations and problems. By following this structured approach, even the most intricate expressions can be untangled and solved with confidence. This method reduces errors and provides a clear path to the final answer, highlighting the importance of procedural accuracy in mathematics. The careful application of these rules transforms a complex problem into a series of manageable steps, showcasing the elegance and logic inherent in mathematical operations. Each step, when correctly executed, brings us closer to the simplified form of the original expression, illustrating the power of methodical problem-solving.

Step-by-Step Simplification

Let’s embark on this journey of simplification, breaking down the expression into manageable parts. Our starting point is the innermost parentheses: (6 - 12/3). Within this, division takes precedence, so we first compute 12/3, which equals 4. Now our expression within the parentheses becomes (6 - 4), which simplifies to 2. Moving on, we have the mixed number 1 2/5, which needs conversion to an improper fraction for easier handling. To do this, we multiply the whole number (1) by the denominator (5) and add the numerator (2), resulting in 7, which we place over the original denominator, giving us 7/5. Next, let's convert 3 8/3 into an improper fraction: multiply 3 by 3 and add 8, resulting in 17, placed over the denominator 3, yielding 17/3. Similarly, 1 1/6 converts to 7/6. Now we address the expression inside the brackets: [17/3 - (2/5 - 7/6)]. Before subtracting 2/5 and 7/6, we need a common denominator, which is 30. Converting the fractions, we get 12/30 - 35/30, which equals -23/30. Thus, our bracketed expression becomes [17/3 - (-23/30)], which simplifies to 17/3 + 23/30. To add these fractions, we again need a common denominator, which is 30. This leads to 170/30 + 23/30, which sums up to 193/30. With these intermediate simplifications in hand, we can now rewrite our original expression with these simplified components. This methodical breakdown not only makes the problem less intimidating but also reduces the likelihood of errors, allowing for a clearer understanding of the solution process. Each simplification acts as a stepping stone, guiding us towards the final, simplified answer.

Calculation Breakdown

Now, let's piece together these simplified components back into the main expression. Our initial expression was 26 - (6 - 12/3) - 1 2/5 - [3 8/3 - (2/5 - 1 1/6)]. After our initial simplifications, this transforms into 26 - 2 - 7/5 - 193/30. To proceed, we need to combine the whole numbers and then deal with the fractions. Subtracting 2 from 26 gives us 24, so our expression now reads 24 - 7/5 - 193/30. To subtract the fractions, we need a common denominator, which is 30. Converting 7/5 to an equivalent fraction with a denominator of 30, we multiply both the numerator and the denominator by 6, giving us 42/30. Thus, the expression becomes 24 - 42/30 - 193/30. Now we can combine the fractions: -42/30 - 193/30 equals -235/30. So, our expression is now 24 - 235/30. To subtract this fraction from 24, we first convert 24 into a fraction with a denominator of 30. Multiplying 24 by 30 gives us 720, so we have 720/30 - 235/30. Subtracting the numerators, we get 720 - 235 = 485. Therefore, our expression simplifies to 485/30. This fraction can be further simplified by finding the greatest common divisor (GCD) of 485 and 30, which is 5. Dividing both the numerator and the denominator by 5, we get 97/6. Finally, we can convert this improper fraction back into a mixed number. Dividing 97 by 6, we get 16 with a remainder of 1. Thus, the simplified form of the expression is 16 1/6. This step-by-step calculation demonstrates how a complex mathematical expression can be methodically simplified to arrive at a concise and understandable answer. Each step builds upon the previous one, showcasing the logical progression of mathematical operations.

Final Simplified Answer

After meticulously working through the expression 26 - (6 - 12/3) - 1 2/5 - [3 8/3 - (2/5 - 1 1/6)], we arrive at the final simplified form: 16 1/6. This result encapsulates the power of step-by-step simplification, adhering to the order of operations (PEMDAS), and converting between mixed numbers and improper fractions. The journey from the complex initial expression to this concise answer underscores the importance of accuracy and methodical problem-solving in mathematics. Each step, from handling parentheses to finding common denominators, played a crucial role in reaching the final solution. This process not only simplifies the expression but also enhances our understanding of mathematical principles. The ability to break down complex problems into smaller, manageable steps is a valuable skill that extends beyond mathematics, applicable in various aspects of life and problem-solving. The final answer, 16 1/6, stands as a testament to the clarity and precision that mathematical simplification can bring. It highlights the elegance of mathematical solutions and the satisfaction derived from a methodical approach to complex problems. This result is not just a number; it represents a journey of simplification, a demonstration of mathematical skill, and an embodiment of the power of logical reasoning.