Step-by-Step Solution For -237 - (-328) + (-205) - 6 + 89
This article provides a detailed, step-by-step solution to the mathematical expression -237 - (-328) + (-205) - 6 + 89. Understanding the order of operations and how to handle negative numbers is crucial in solving such problems. We will break down the expression into smaller, manageable parts to ensure clarity and accuracy. This guide is perfect for students, educators, or anyone looking to refresh their arithmetic skills. By following each step carefully, you'll gain confidence in your ability to tackle similar mathematical challenges.
Understanding the Basics: Order of Operations and Negative Numbers
Before we dive into the solution, it's essential to grasp the fundamental concepts involved. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which we perform mathematical operations. In this case, we primarily deal with addition and subtraction, which are performed from left to right. Additionally, understanding how to manipulate negative numbers is crucial. Subtracting a negative number is the same as adding its positive counterpart, and adding a negative number is the same as subtracting its positive counterpart. These rules are vital for accurately solving the expression.
The Significance of Order of Operations
The order of operations is not just a mathematical convention; it's the backbone of consistent and accurate calculations. Without a standardized order, the same expression could yield multiple different answers, leading to confusion and errors. For our problem, following the order of operations ensures that we handle the subtractions and additions in the correct sequence, ultimately arriving at the correct result. This methodical approach is not just applicable to simple arithmetic but extends to more complex mathematical fields like algebra, calculus, and beyond. Mastering PEMDAS is, therefore, a foundational step in any mathematical journey.
Navigating Negative Numbers
Negative numbers often pose a challenge for those new to arithmetic, but understanding their behavior is crucial. A negative number is a real number that is less than zero. On a number line, they are represented to the left of zero. The concept of negative numbers is fundamental to understanding debts, temperatures below zero, and various other real-world scenarios. In our expression, we encounter both subtracting a negative number and adding a negative number. It's vital to remember that subtracting a negative number is equivalent to adding its positive counterpart (e.g., -(-328) becomes +328), and adding a negative number is the same as subtracting its positive counterpart (e.g., +(-205) becomes -205). These transformations are key to simplifying and solving the expression correctly.
Real-World Applications of These Concepts
The principles of order of operations and negative number manipulation aren't confined to the classroom. They permeate various aspects of daily life and professional fields. For instance, in finance, calculating account balances, understanding debts, and managing investments all require a firm grasp of these concepts. In science and engineering, dealing with temperatures, forces, and electrical circuits often involves negative numbers and complex calculations. Even in everyday situations like planning a budget or figuring out travel times, these mathematical foundations play a crucial role. By mastering these basics, you're not just solving mathematical problems; you're enhancing your problem-solving skills in a broader context.
Step 1: Addressing the First Subtraction -237 - (-328)
In this first step, we tackle the initial subtraction: -237 - (-328). As we discussed earlier, subtracting a negative number is equivalent to adding its positive counterpart. Therefore, we can rewrite the expression as -237 + 328. This transformation simplifies the problem, making it easier to compute. Now, we are dealing with the addition of two numbers with different signs. To solve this, we find the difference between their absolute values and assign the sign of the number with the larger absolute value.
Rewriting the Expression
The ability to rewrite mathematical expressions is a critical skill in problem-solving. In this case, transforming -237 - (-328) into -237 + 328 is not merely a cosmetic change; it fundamentally alters the operation we're performing. This transformation stems from the basic principle that subtracting a negative is the inverse operation of adding, effectively canceling out the negative sign. This rewriting step is crucial for accurate calculation and helps to avoid common errors associated with negative numbers.
Calculating the Difference
When adding numbers with different signs, we focus on the difference between their absolute values. The absolute value of a number is its distance from zero, irrespective of its sign. In our case, the absolute value of -237 is 237, and the absolute value of 328 is 328. The difference between these values is 328 - 237, which equals 91. This difference represents the magnitude of the result, but we still need to determine the correct sign.
Determining the Sign
To assign the correct sign to our result, we consider which number has the larger absolute value. In the expression -237 + 328, 328 has a larger absolute value than -237. Since 328 is positive, the result will also be positive. Therefore, -237 + 328 = 91. This step highlights the importance of not only calculating the numerical difference but also paying attention to the signs involved. A common mistake is to overlook the sign, leading to an incorrect answer despite the correct numerical value.
Visualizing on a Number Line
A helpful way to understand operations with negative numbers is to visualize them on a number line. Starting at -237 and moving 328 units to the right (in the positive direction) will indeed land you at 91. This visual representation can solidify the concept and provide a check for your calculations. Number lines are a valuable tool for students learning about negative numbers and can help build intuition around these concepts.
Step 2: Incorporating the Next Term + (-205)
Now that we have simplified the initial part of the expression to 91, we incorporate the next term: + (-205). Adding a negative number is the same as subtracting its positive counterpart. Thus, we rewrite the expression as 91 - 205. This step again involves dealing with numbers with different signs. We will find the difference between their absolute values and assign the sign of the number with the larger absolute value, similar to Step 1.
Simplifying the Addition of a Negative Number
The step of converting + (-205) to - 205 is crucial for simplifying the expression and making it easier to solve. This transformation reinforces the fundamental rule that adding a negative number is equivalent to subtraction. Understanding and applying this rule is key to accurately handling negative numbers in arithmetic. It eliminates an extra operation and allows us to proceed with a straightforward subtraction.
Calculating the Difference and Determining the Sign
We now have the expression 91 - 205. The absolute value of 91 is 91, and the absolute value of 205 is 205. The difference between these values is 205 - 91, which equals 114. Since 205 has a larger absolute value than 91 and is negative, the result will be negative. Therefore, 91 - 205 = -114. This step illustrates the importance of correctly identifying the sign of the result based on the numbers involved.
Common Mistakes to Avoid
A frequent error when dealing with numbers of different signs is to incorrectly determine the sign of the result. It's essential to always compare the absolute values and assign the sign of the number with the larger absolute value. Another common mistake is to perform the subtraction in the wrong order (e.g., 91 - 205 instead of 205 - 91). Double-checking the order of subtraction and the sign of the result can significantly reduce errors.
Practical Examples
Consider a scenario where you have $91 and you owe $205. After paying your $91, you would still owe $114, which can be represented as -$114. This real-world analogy helps to contextualize the mathematical operation and make it more relatable. Similarly, if the temperature was 91 degrees and then dropped by 205 degrees, the new temperature would be -114 degrees. These examples underscore the practical relevance of understanding negative number operations.
Step 3: Subtracting 6 from the Result -114 - 6
Now, we move on to the next operation: -114 - 6. This step involves subtracting a positive number from a negative number. When subtracting a positive number from a negative number, we move further into the negative direction on the number line. We can think of this as adding the negative of 6 to -114, which gives us -114 + (-6). This transformation helps to clarify the operation and makes it easier to compute.
Understanding Subtraction as Addition of a Negative
The concept of rewriting subtraction as the addition of a negative number is a cornerstone of arithmetic. It simplifies calculations and helps to avoid confusion, especially when dealing with negative numbers. Transforming -114 - 6 into -114 + (-6) clarifies that we are moving further into the negative range on the number line. This understanding is crucial for building a strong foundation in mathematical operations.
Performing the Addition of Negative Numbers
When adding two negative numbers, we simply add their absolute values and keep the negative sign. The absolute value of -114 is 114, and the absolute value of -6 is 6. Adding these values gives us 114 + 6 = 120. Since both numbers are negative, the result is also negative. Therefore, -114 + (-6) = -120. This process is straightforward and consistent with the rules of negative number arithmetic.
Visual Aids for Understanding
Imagine a number line where you start at -114. Subtracting 6 means moving 6 units to the left (the negative direction). This visualization reinforces the idea that the result will be further away from zero in the negative direction. Number lines are an excellent tool for building intuition about negative number operations and can help to verify calculations.
Real-Life Scenarios
Think of owing $114 and then incurring an additional debt of $6. Your total debt would be $120, which can be represented as -$120. Another example could be a temperature of -114 degrees dropping by another 6 degrees, resulting in a temperature of -120 degrees. Relating mathematical operations to real-world situations makes the concepts more understandable and memorable.
Step 4: Adding 89 to the Result -120 + 89
Finally, we add 89 to our current result: -120 + 89. This step involves adding a positive number to a negative number. As in Step 1 and Step 2, we will find the difference between their absolute values and assign the sign of the number with the larger absolute value. This is a recurring theme in solving expressions with both positive and negative numbers.
The Process of Adding Numbers with Different Signs
Adding numbers with different signs requires careful attention to both the numerical value and the sign. The fundamental principle is to find the difference between the absolute values and then assign the sign of the number with the larger absolute value. This process ensures that the result accurately reflects the relationship between the two numbers.
Calculating the Difference and Determining the Sign (Again!)
The absolute value of -120 is 120, and the absolute value of 89 is 89. The difference between these values is 120 - 89, which equals 31. Since 120 has a larger absolute value than 89 and is negative, the result will be negative. Therefore, -120 + 89 = -31. This calculation is a straightforward application of the rules we've established in previous steps.
Checking for Consistency
It's always a good practice to check your work, especially when dealing with multiple steps. In this case, we can verify that adding 89 to -120 indeed results in -31. This check reinforces our confidence in the correctness of the solution. Consistency checks are an integral part of problem-solving and help to minimize errors.
Connecting to Everyday Scenarios
Imagine you owe $120 but then receive $89. Your remaining debt would be $31, represented as -$31. This scenario provides a tangible context for the mathematical operation and helps to solidify the concept. Similarly, if the temperature was -120 degrees and then increased by 89 degrees, the new temperature would be -31 degrees. These real-world connections highlight the practical significance of mastering these arithmetic skills.
Final Answer: -31
After carefully following each step, we have arrived at the final answer: -31. We started with the expression -237 - (-328) + (-205) - 6 + 89 and systematically simplified it by addressing each operation in the correct order. We successfully handled negative numbers, subtractions, and additions, ultimately arriving at the solution. This step-by-step process demonstrates the power of breaking down complex problems into smaller, manageable parts.
Reviewing the Entire Process
Let's recap the steps we took to solve the expression:
- We started with -237 - (-328) + (-205) - 6 + 89.
- We simplified -237 - (-328) to -237 + 328, which equals 91.
- We incorporated the next term, + (-205), rewriting it as 91 - 205, which equals -114.
- We subtracted 6 from -114, resulting in -120.
- Finally, we added 89 to -120, which equals -31.
This review highlights the importance of methodical problem-solving and the consistency of applying arithmetic rules.
The Importance of Showing Your Work
Showing each step in your solution is crucial for several reasons. First, it allows you to track your progress and identify any potential errors along the way. Second, it makes your solution easier for others to understand and follow. Third, in an educational setting, showing your work often earns you partial credit even if the final answer is incorrect. Detailed step-by-step solutions are a hallmark of good mathematical practice.
Further Practice and Resources
If you found this guide helpful, consider practicing similar problems to reinforce your understanding. There are numerous online resources, textbooks, and worksheets available that offer a variety of arithmetic exercises. Additionally, seeking help from teachers, tutors, or classmates can provide valuable support and clarification. Consistent practice and a willingness to ask questions are key to mastering mathematical concepts.
By understanding the principles of order of operations and negative number manipulation, you can confidently tackle a wide range of mathematical challenges. This step-by-step solution serves as a valuable guide for anyone looking to improve their arithmetic skills and build a solid foundation in mathematics.
