Evaluating The Expression -4 × (0.1 + -0.2) - 0.5
In the realm of mathematics, evaluating expressions is a fundamental skill. It's the process of simplifying a mathematical statement to a single numerical value. This involves following the order of operations, which ensures that calculations are performed in the correct sequence, leading to an accurate result. In this comprehensive guide, we will dissect the expression -4 × (0.1 + -0.2) - 0.5, step by step, to arrive at the final answer. This will provide a clear understanding of how to handle various operations, including multiplication, addition, and subtraction, especially when dealing with decimals and negative numbers. Understanding the nuances of these operations is crucial for success in algebra, calculus, and various other branches of mathematics. So, let's embark on this journey of mathematical exploration and enhance our ability to evaluate expressions with confidence.
Understanding the Order of Operations (PEMDAS/BODMAS)
The bedrock of evaluating any mathematical expression lies in adhering to the order of operations, often remembered by the acronyms PEMDAS or BODMAS. These acronyms serve as a roadmap, guiding us through the sequence in which operations must be performed. PEMDAS stands for Parentheses, Exponents, Multiplication and Division (from left to right), and Addition and Subtraction (from left to right). Similarly, BODMAS represents Brackets, Orders (exponents and roots), Division and Multiplication (from left to right), and Addition and Subtraction (from left to right). The core principle is to prioritize calculations within parentheses or brackets first, then handle exponents or orders, followed by multiplication and division, and finally, perform addition and subtraction. Ignoring this order can lead to drastically different and incorrect results. For instance, in our expression, the operation within the parentheses must be resolved before we proceed with multiplication or subtraction. Mastering the order of operations is not just about following a rule; it's about developing a logical approach to problem-solving, a skill that transcends mathematics and extends to various aspects of life.
Step-by-Step Evaluation of the Expression
Now, let's apply the order of operations to our expression: -4 × (0.1 + -0.2) - 0.5. The initial step involves simplifying the expression within the parentheses. We have (0.1 + -0.2), which is essentially adding a positive and a negative decimal. When adding numbers with different signs, we find the difference between their absolute values and assign the sign of the number with the larger absolute value. In this case, |0.1| = 0.1 and |-0.2| = 0.2. The difference is 0.2 - 0.1 = 0.1. Since -0.2 has a larger absolute value, the result is -0.1. So, the expression within the parentheses simplifies to -0.1. Our expression now becomes: -4 × (-0.1) - 0.5. The next operation, according to PEMDAS/BODMAS, is multiplication. We need to multiply -4 by -0.1. A negative number multiplied by a negative number yields a positive result. Therefore, -4 × -0.1 = 0.4. The expression is now: 0.4 - 0.5. Finally, we perform the subtraction. Subtracting 0.5 from 0.4 is the same as adding -0.5 to 0.4. Again, we are adding numbers with different signs. The difference between their absolute values is |0.4| = 0.4 and |-0.5| = 0.5, so 0.5 - 0.4 = 0.1. Since -0.5 has a larger absolute value, the result is -0.1. Thus, the final answer is -0.1.
Step 1: Simplify Inside the Parentheses
Our initial focus is on simplifying the expression enclosed within the parentheses: (0.1 + -0.2). This step is paramount as it sets the stage for the subsequent operations. Here, we encounter the addition of two decimal numbers with differing signs. To navigate this, we apply the principle of adding numbers with opposite signs, which involves finding the difference between their absolute values. The absolute value of a number is its distance from zero, regardless of its sign. Thus, the absolute value of 0.1 is 0.1, and the absolute value of -0.2 is 0.2. We then subtract the smaller absolute value from the larger one: 0.2 - 0.1 = 0.1. The sign of the result is determined by the number with the greater absolute value. In this instance, -0.2 has a larger absolute value than 0.1, so our result will be negative. Therefore, 0.1 + -0.2 simplifies to -0.1. This seemingly simple step is crucial, as it transforms the complexity of the original expression into a more manageable form, allowing us to proceed with the next operations with greater clarity and precision.
Step 2: Perform the Multiplication
Having simplified the expression within the parentheses, we now turn our attention to the multiplication operation: -4 × (-0.1). This step is significant as it involves the multiplication of two negative numbers, a concept that often requires careful attention to signs. The fundamental rule here is that the product of two negative numbers is always a positive number. This rule stems from the basic axioms of arithmetic and is crucial for maintaining consistency in mathematical calculations. To perform the multiplication, we first ignore the signs and multiply the absolute values of the numbers: 4 × 0.1 = 0.4. Since both numbers are negative, the result will be positive, as dictated by the rule of signs. Thus, -4 × -0.1 equals 0.4. This step not only reduces the expression further but also highlights the importance of understanding and applying the rules of signs in mathematical operations. Mastering these rules is essential for avoiding common errors and ensuring accuracy in more complex calculations.
Step 3: Perform the Subtraction
The final step in our evaluation process involves the subtraction operation: 0.4 - 0.5. This step requires us to subtract a larger decimal number from a smaller one, which is akin to adding a negative number. To approach this, we can rewrite the subtraction as the addition of a negative number: 0.4 + (-0.5). Now, we are faced with adding two numbers with different signs. As we discussed earlier, when adding numbers with different signs, we find the difference between their absolute values. The absolute value of 0.4 is 0.4, and the absolute value of -0.5 is 0.5. Subtracting the smaller absolute value from the larger one, we get 0.5 - 0.4 = 0.1. The sign of the result is determined by the number with the larger absolute value, which in this case is -0.5, so the result will be negative. Therefore, 0.4 - 0.5 equals -0.1. This final step underscores the interconnectedness of mathematical operations and the importance of a solid understanding of both addition and subtraction principles. The ability to seamlessly transition between these operations is a hallmark of mathematical fluency.
Final Answer: -0.1
After meticulously following the order of operations and performing each step with precision, we arrive at the final answer: -0.1. This result represents the simplified value of the original expression, -4 × (0.1 + -0.2) - 0.5. The process of evaluating this expression has not only yielded a numerical answer but has also served as a valuable exercise in applying fundamental mathematical principles. We have reinforced the importance of the order of operations, the rules of signs, and the techniques for handling decimal numbers. Each step, from simplifying the expression within parentheses to performing multiplication and subtraction, has contributed to our understanding of how to approach mathematical problems systematically. This comprehensive evaluation serves as a testament to the power of structured problem-solving and the satisfaction of arriving at a correct and definitive answer. The ability to confidently evaluate expressions is a cornerstone of mathematical proficiency, paving the way for tackling more complex challenges in the future.
In conclusion, the journey of evaluating the expression -4 × (0.1 + -0.2) - 0.5 has been a testament to the power of mathematical principles and the importance of meticulous execution. We have navigated the intricacies of the order of operations, the nuances of handling negative numbers and decimals, and the seamless integration of addition, subtraction, and multiplication. The final answer, -0.1, is not merely a numerical result; it is a symbol of our understanding and mastery of these fundamental concepts. This exercise has reinforced the idea that mathematics is not just about memorizing formulas, but about developing a logical and systematic approach to problem-solving. The ability to break down a complex expression into manageable steps, to apply the correct rules and principles, and to arrive at a definitive answer is a skill that extends far beyond the classroom. It is a skill that fosters critical thinking, analytical reasoning, and the confidence to tackle challenges in any field. As we continue our mathematical journey, let us remember the lessons learned in this evaluation and apply them to more complex problems, always striving for clarity, accuracy, and a deep understanding of the underlying principles.
