Evaluate The Expression -0.2 × 3 - 0.9 ÷ -1

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Understanding the Order of Operations

In mathematics, accurately evaluating expressions requires a clear understanding and application of the order of operations. This principle ensures that we perform calculations in a consistent and logical sequence, avoiding ambiguity and arriving at the correct result. The universally recognized mnemonic for the order of operations is PEMDAS, which stands for:

  • Parentheses (or brackets)
  • Exponents (or powers and roots)
  • Multiplication and Division (from left to right)
  • Addition and Subtraction (from left to right)

This order dictates that any expressions within parentheses should be simplified first, followed by exponents. Multiplication and division are then performed from left to right, and finally, addition and subtraction are carried out, also from left to right. Ignoring this order can lead to drastically different and incorrect answers.

For instance, consider the expression 2 + 3 × 4. If we perform the addition first, we get 5 × 4 = 20. However, according to PEMDAS, multiplication should precede addition, so the correct evaluation is 2 + (3 × 4) = 2 + 12 = 14. This simple example illustrates the critical importance of adhering to the order of operations. In more complex expressions involving multiple operations and nested parentheses, the consequences of disregarding PEMDAS can be even more significant.

Mathematical expressions often involve a mix of operations, making a systematic approach essential. By following the order of operations diligently, we ensure that each calculation is performed in the correct sequence, leading to accurate and reliable results. This consistency is vital not only in basic arithmetic but also in more advanced mathematical fields like algebra, calculus, and beyond. Therefore, mastering the order of operations is a fundamental skill in mathematics.

Step-by-Step Evaluation of -0.2 × 3 - 0.9 ÷ -1

To accurately evaluate the given expression, -0.2 × 3 - 0.9 ÷ -1, we must meticulously follow the order of operations, commonly remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). This ensures that we perform the calculations in the correct sequence, avoiding errors and arriving at the precise answer.

In our expression, there are no parentheses or exponents, so we move directly to multiplication and division. According to PEMDAS, these operations should be performed from left to right. The first operation we encounter is multiplication: -0.2 × 3. Multiplying these two numbers yields -0.6. It's crucial to pay attention to the signs of the numbers; in this case, a negative number multiplied by a positive number results in a negative product.

Next, we perform the division: -0.9 ÷ -1. Dividing -0.9 by -1 gives us 0.9. Here, a negative number divided by another negative number results in a positive quotient. This step is critical, as misinterpreting the signs would lead to an incorrect final result. Now our expression looks like this: -0.6 - 0.9 ÷ -1 = -0.6 + 0.9.

Having completed the multiplication and division, we proceed to the final operation: subtraction (which can also be viewed as adding a negative number). The expression now reads -0.6 - (-0.9). This is equivalent to -0.6 + 0.9. Adding these two numbers, -0.6 and 0.9, gives us 0.3. Therefore, the final result of the expression is 0.3.

By systematically applying the order of operations, we have successfully evaluated the expression. Each step, from multiplication and division to addition and subtraction, was performed in the correct sequence, ensuring the accuracy of the result. This methodical approach is essential for handling more complex expressions and is a cornerstone of mathematical problem-solving.

Detailed Breakdown of the Calculation

To provide a comprehensive understanding of the evaluation process, let’s break down each step of the calculation in greater detail. The expression we are evaluating is -0.2 × 3 - 0.9 ÷ -1. As we established earlier, the order of operations dictates that we handle multiplication and division before addition and subtraction.

  1. Multiplication: The first operation we encounter is -0.2 × 3. To perform this multiplication, we multiply the absolute values of the numbers and then apply the appropriate sign. The absolute value of -0.2 is 0.2, and multiplying this by 3 gives us 0.6. Since we are multiplying a negative number by a positive number, the result is negative. Therefore, -0.2 × 3 = -0.6.

  2. Division: Next, we address the division operation: -0.9 ÷ -1. When dividing, we again consider the absolute values and the signs. The absolute value of -0.9 is 0.9, and dividing this by the absolute value of -1, which is 1, gives us 0.9. Since we are dividing a negative number by another negative number, the result is positive. Thus, -0.9 ÷ -1 = 0.9.

  3. Rewriting the Expression: After performing the multiplication and division, our expression now looks like this: -0.6 - (-0.9 ÷ -1) = -0.6 + 0.9

  4. Addition: We are left with the subtraction operation, which can be rewritten as adding the negative: -0.6 - (-0.9). This is equivalent to -0.6 + 0.9. To add these numbers, we can think of it as finding the difference between their absolute values and using the sign of the number with the larger absolute value. The absolute value of -0.6 is 0.6, and the absolute value of 0.9 is 0.9. The difference between 0.9 and 0.6 is 0.3. Since 0.9 has a larger absolute value and is positive, the result is positive. Therefore, -0.6 + 0.9 = 0.3.

By meticulously breaking down each step, we have demonstrated how the expression is evaluated according to the order of operations. This detailed explanation highlights the importance of paying close attention to signs and the sequence of operations to arrive at the correct answer.

Common Mistakes to Avoid

When evaluating mathematical expressions, several common mistakes can lead to incorrect results. Being aware of these pitfalls and taking steps to avoid them is crucial for achieving accuracy. Here, we will discuss some frequent errors and how to prevent them.

  1. Ignoring the Order of Operations: As emphasized earlier, the order of operations (PEMDAS) is paramount. A common mistake is performing operations in the wrong sequence. For instance, adding before multiplying or subtracting before dividing. To avoid this, always remember PEMDAS and methodically work through the expression, addressing parentheses, exponents, multiplication and division (from left to right), and finally, addition and subtraction (from left to right).

  2. Incorrectly Handling Negative Signs: Negative signs can be tricky, especially when combined with other operations. For example, subtracting a negative number is equivalent to adding the positive counterpart. Confusing these rules can lead to errors. Pay close attention to the signs and rewrite the expression if necessary to clarify the operations. For instance, -(-x) should be seen as +x.

  3. Sign Errors in Multiplication and Division: When multiplying or dividing numbers with different signs, the result is negative. When the signs are the same, the result is positive. It's easy to make a mistake in these rules, especially when dealing with multiple negative numbers. Double-check your signs to ensure accuracy.

  4. Forgetting the Left-to-Right Rule: When performing multiplication and division or addition and subtraction, the operations should be carried out from left to right. Skipping this rule can lead to incorrect answers. For example, in the expression 6 ÷ 2 × 3, one might incorrectly multiply 2 × 3 first. The correct approach is to divide 6 ÷ 2 first, then multiply by 3.

  5. Misinterpreting Parentheses: Parentheses indicate the operations that should be performed first. Misinterpreting or overlooking parentheses can change the entire outcome of the expression. Always simplify the expressions within parentheses before proceeding with other operations.

By being mindful of these common errors and systematically applying the order of operations, you can significantly improve your accuracy in evaluating mathematical expressions. Practice and careful attention to detail are key to mastering these skills.

Conclusion: The Final Answer

In summary, we have meticulously evaluated the expression -0.2 × 3 - 0.9 ÷ -1 by adhering to the order of operations (PEMDAS). This systematic approach ensures that we perform each calculation in the correct sequence, leading to an accurate and reliable result. Let's recap the steps we took:

  1. Multiplication: We began by performing the multiplication operation: -0.2 × 3. This resulted in -0.6.

  2. Division: Next, we addressed the division operation: -0.9 ÷ -1. Dividing a negative number by another negative number yields a positive result, so this calculation gave us 0.9.

  3. Rewriting the Expression: After the multiplication and division, the expression was simplified to -0.6 + 0.9.

  4. Addition: Finally, we performed the addition: -0.6 + 0.9. Adding these numbers resulted in 0.3.

Therefore, the final answer to the expression -0.2 × 3 - 0.9 ÷ -1 is 0.3. This result underscores the importance of following the order of operations and paying close attention to signs throughout the calculation process.

By systematically applying PEMDAS and meticulously working through each step, we have successfully evaluated the expression. This detailed process not only provides the correct answer but also reinforces the fundamental principles of mathematical operations. Understanding and applying these principles are essential for tackling more complex mathematical problems in the future.

In conclusion, mastering the order of operations and avoiding common mistakes are crucial for accurate mathematical calculations. By following a methodical approach, we can confidently evaluate expressions and arrive at the correct solutions.