Step-by-Step Evaluation Of Arithmetic Expressions

In mathematics, evaluating expressions is a fundamental skill. It involves simplifying a given mathematical statement to obtain a single numerical value. This often requires understanding the order of operations (PEMDAS/BODMAS) and the rules for dealing with positive and negative numbers. This article will walk you through the step-by-step evaluation of several expressions involving addition, subtraction, and both positive and negative integers. A solid understanding of these concepts is crucial for more advanced mathematical topics, making it essential for students and anyone working with numerical data.

When evaluating mathematical expressions, accuracy and precision are paramount. A small error in the initial steps can lead to a completely incorrect final answer. Therefore, it is essential to approach each problem systematically, ensuring that every operation is performed correctly. This includes paying close attention to the signs (positive or negative) of the numbers involved, as well as the order in which the operations are carried out. In this comprehensive guide, we will explore various expressions, breaking down each step to illustrate the correct methodology. By the end of this article, you will be equipped with the knowledge and confidence to tackle a wide range of mathematical evaluations.

Let's dive deeper into the process of evaluating expressions. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) or BODMAS (Brackets, Orders, Division and Multiplication, Addition and Subtraction), is the backbone of simplifying complex expressions. This set of rules dictates the sequence in which operations must be performed to arrive at the correct answer. For instance, any calculations within parentheses or brackets must be done first, followed by exponents or orders (powers and square roots), then multiplication and division (from left to right), and finally addition and subtraction (also from left to right). Ignoring this order can lead to significant errors in the final result. Therefore, always double-check your steps and ensure that you adhere to PEMDAS/BODMAS throughout the evaluation process.

a. Evaluate: 814 - (-45) + (-10)

To evaluate the expression 814 - (-45) + (-10), we need to follow the order of operations, which in this case involves dealing with subtraction and addition. The first step is to address the subtraction of a negative number. Subtracting a negative number is the same as adding its positive counterpart. Thus, 814 - (-45) becomes 814 + 45. Performing this addition, we get 859. Now, the expression simplifies to 859 + (-10). Adding a negative number is the same as subtracting its absolute value. So, 859 + (-10) is equivalent to 859 - 10. Subtracting 10 from 859 gives us the final result.

When evaluating 814 - (-45) + (-10), the initial transformation of 814 - (-45) to 814 + 45 is a crucial step. This transformation relies on the fundamental principle that subtracting a negative number is identical to adding the positive version of that number. This principle is a cornerstone of arithmetic and is essential for simplifying expressions involving both positive and negative integers. By correctly applying this rule, we can move forward with the evaluation process without making any common sign-related errors. After performing the addition 814 + 45, we arrive at the intermediate result of 859. This intermediate result serves as a stepping stone towards the final answer, and it highlights the importance of performing each operation in the correct sequence to maintain accuracy.

The subsequent step in evaluating the expression is to address the addition of a negative number, specifically 859 + (-10). Similar to the subtraction rule, adding a negative number is equivalent to subtracting its absolute value. Therefore, 859 + (-10) can be rewritten as 859 - 10. This transformation simplifies the expression further and brings us closer to the final solution. By correctly converting the addition of a negative number into a subtraction operation, we are ensuring that the evaluation process remains accurate and consistent with mathematical rules. Performing the subtraction 859 - 10 provides us with the final result, completing the evaluation of the original expression.

Solution:

814 - (-45) + (-10)
= 814 + 45 + (-10)
= 859 + (-10)
= 859 - 10
= 849

b. Evaluate: (-1308) + (-178) - 700

To evaluate (-1308) + (-178) - 700, we begin by adding the two negative numbers, (-1308) and (-178). When adding two negative numbers, we add their absolute values and keep the negative sign. Thus, (-1308) + (-178) is the same as -(1308 + 178). Performing this addition, we get 1486, so the sum is -1486. Now, the expression becomes -1486 - 700. Subtracting a positive number from a negative number is equivalent to adding the absolute values and keeping the negative sign. Therefore, -1486 - 700 is the same as -(1486 + 700). Adding 1486 and 700 gives us the final result.

When evaluating the expression (-1308) + (-178) - 700, the first critical step is to correctly add the two negative numbers, (-1308) and (-178). The fundamental rule for adding two negative numbers dictates that we add their absolute values and then affix a negative sign to the result. This rule ensures that we are accurately representing the cumulative effect of these negative quantities. In this instance, adding the absolute values of -1308 and -178, which are 1308 and 178 respectively, yields 1486. Consequently, the sum of the two original negative numbers is -1486. This intermediate result is essential for proceeding with the evaluation, and it underscores the importance of adhering to the rules of sign manipulation in arithmetic.

The next step in evaluating the expression involves subtracting 700 from the negative number -1486. This operation, -1486 - 700, can be simplified by recognizing that subtracting a positive number from a negative number is equivalent to adding their absolute values and maintaining the negative sign. This rule is a direct extension of the principles governing operations with negative numbers, and it helps to streamline the evaluation process. Therefore, -1486 - 700 can be interpreted as adding the absolute values of -1486 and -700, which are 1486 and 700 respectively, and then applying a negative sign to the sum. This transformation makes the calculation more straightforward and ensures that the final result accurately reflects the combined effect of the negative quantities.

Solution:

(-1308) + (-178) - 700
= -1486 - 700
= -2186

c. Evaluate: 902 - (-510) + 162 + (-72)

To evaluate 902 - (-510) + 162 + (-72), we first address the subtraction of the negative number. Subtracting a negative is the same as adding the positive, so 902 - (-510) becomes 902 + 510. Adding these numbers gives us 1412. The expression now simplifies to 1412 + 162 + (-72). Next, we add 1412 and 162, which results in 1574. The expression further simplifies to 1574 + (-72). Adding a negative number is the same as subtracting its absolute value, so 1574 + (-72) is equivalent to 1574 - 72. Subtracting 72 from 1574 gives us the final result.

When evaluating the expression 902 - (-510) + 162 + (-72), the initial step is to simplify the subtraction of a negative number. Specifically, we address the term 902 - (-510). A fundamental rule in arithmetic states that subtracting a negative number is equivalent to adding its positive counterpart. Therefore, 902 - (-510) transforms into 902 + 510. This transformation is crucial as it eliminates the double negative, making the expression easier to manage. By correctly applying this rule, we ensure that the subsequent calculations are based on accurate values, thereby minimizing the risk of errors. Performing the addition 902 + 510 yields 1412, which becomes a pivotal intermediate result in our evaluation process.

The next phase in evaluating the expression involves adding the intermediate result, 1412, to 162. This operation, 1412 + 162, is a straightforward addition that combines two positive numbers. Performing this addition gives us a new intermediate result of 1574. This step is important because it progressively simplifies the expression by consolidating terms. By performing each addition in the correct sequence, we are systematically reducing the complexity of the expression, making it more manageable and less prone to errors. The cumulative addition of positive numbers ensures that we are moving closer to the final solution with each step.

The subsequent step in the evaluating process addresses the addition of a negative number, specifically 1574 + (-72). As previously established, adding a negative number is equivalent to subtracting its absolute value. Thus, the expression 1574 + (-72) can be rewritten as 1574 - 72. This transformation simplifies the calculation and aligns with the rules governing operations with signed numbers. By correctly converting the addition of a negative number into a subtraction, we maintain the integrity of the expression and ensure that the final result accurately reflects the operations performed. Completing the subtraction 1574 - 72 will provide the final answer, thus concluding the evaluation of the original expression.

Solution:

902 - (-510) + 162 + (-72)
= 902 + 510 + 162 + (-72)
= 1412 + 162 + (-72)
= 1574 + (-72)
= 1574 - 72
= 1502

d. Evaluate: (-2000) + 1400 - (-230) + (-316)

To evaluate (-2000) + 1400 - (-230) + (-316), we start by adding -2000 and 1400. Since we are adding numbers with different signs, we subtract their absolute values and take the sign of the number with the larger absolute value. So, (-2000) + 1400 is the same as -(2000 - 1400), which equals -600. The expression now becomes -600 - (-230) + (-316). Next, we address the subtraction of the negative number. Subtracting a negative is the same as adding the positive, so -600 - (-230) becomes -600 + 230. Adding these numbers with different signs, we get -(600 - 230), which is -370. The expression now simplifies to -370 + (-316). Adding two negative numbers, we add their absolute values and keep the negative sign. Thus, -370 + (-316) is the same as -(370 + 316). Adding 370 and 316 gives us the final result.

When evaluating the expression (-2000) + 1400 - (-230) + (-316), the first crucial step is to add -2000 and 1400. When adding numbers with different signs, it is essential to subtract their absolute values and then assign the sign of the number with the greater absolute value to the result. This rule ensures that the outcome accurately reflects the combined effect of the positive and negative quantities. In this case, the absolute values are 2000 and 1400, and their difference is 600. Since -2000 has a larger absolute value, the result of the addition is -600. This intermediate result is critical as it simplifies the expression and sets the stage for the subsequent calculations. By correctly applying the rule for adding numbers with different signs, we are maintaining the integrity of the evaluation process.

The next step in evaluating the expression involves addressing the subtraction of a negative number, specifically -600 - (-230). As previously established, subtracting a negative number is equivalent to adding its positive counterpart. Therefore, the expression -600 - (-230) transforms into -600 + 230. This transformation is crucial as it simplifies the expression and eliminates the double negative, making it easier to manage. By correctly applying this rule, we ensure that the subsequent calculations are based on accurate values, thereby minimizing the risk of errors. Adding -600 and 230 requires us to subtract their absolute values and take the sign of the number with the larger absolute value, resulting in -370.

The subsequent step in the evaluating process addresses the addition of two negative numbers, specifically -370 + (-316). When adding two negative numbers, the rule is to add their absolute values and affix a negative sign to the result. This rule ensures that the outcome accurately reflects the cumulative effect of the negative quantities. Thus, we add the absolute values of -370 and -316, which are 370 and 316 respectively, resulting in 686. Since both numbers are negative, the final result is -686. This final calculation completes the evaluation of the original expression, providing the solution that combines all the performed operations while adhering to the rules of signed number arithmetic.

Solution:

(-2000) + 1400 - (-230) + (-316)
= -600 - (-230) + (-316)
= -600 + 230 + (-316)
= -370 + (-316)
= -686

In conclusion, evaluating mathematical expressions involving positive and negative numbers requires a strong understanding of the order of operations and the rules for dealing with signed numbers. By carefully applying these principles, we can simplify complex expressions and arrive at accurate solutions. Remember to always double-check your work and pay close attention to the signs of the numbers involved.