Evaluating The Expression 0.3 - (-0.6 + -0.2) × -0.5

In the realm of mathematics, evaluating expressions is a fundamental skill. This article will provide a comprehensive, step-by-step guide to evaluating the expression $0.3 - (-0.6 + -0.2) × -0.5$. We will break down the expression into manageable parts, explaining each operation and the order in which it should be performed. Whether you are a student looking to improve your understanding of arithmetic or simply someone who enjoys solving mathematical puzzles, this guide will help you master the process of evaluating expressions.

Before we dive into the specifics of this expression, it is crucial to understand the order of operations. In mathematics, we follow a specific set of rules to ensure that expressions are evaluated consistently. This order is often remembered by the acronym PEMDAS, which stands for:

• Parentheses
• Exponents
• Multiplication and Division (from left to right)
• Addition and Subtraction (from left to right)

This means that we first deal with any expressions inside parentheses, then exponents, followed by multiplication and division (performed from left to right), and finally addition and subtraction (also performed from left to right). Adhering to this order is paramount to arriving at the correct solution. Let's illustrate this with a simple example before tackling our main expression. Consider the expression $2 + 3 × 4$. If we perform the addition first, we get $5 × 4 = 20$. However, if we follow PEMDAS and perform the multiplication first, we get $2 + 12 = 14$, which is the correct answer. This example underscores the importance of adhering to the established order of operations.

Our expression is $0.3 - (-0.6 + -0.2) × -0.5$. According to PEMDAS, the first step is to simplify the expression within the parentheses: $(-0.6 + -0.2)$.

To add two negative numbers, we add their absolute values and keep the negative sign. In this case, we have:

$-0.6 + -0.2 = -(0.6 + 0.2) = -0.8$

So, the expression within the parentheses simplifies to $-0.8$. This step is a fundamental application of arithmetic rules for adding signed numbers. The ability to correctly handle negative numbers is crucial in many areas of mathematics, and this step provides a solid foundation for the rest of the problem. By focusing on this initial simplification, we reduce the complexity of the overall expression and make it easier to manage. It is important to remember that when adding numbers with the same sign, we simply add their magnitudes and retain the sign.

Now that we have simplified the expression within the parentheses, our expression becomes:

$0.3 - (-0.8) × -0.5$

According to PEMDAS, the next operation we need to perform is multiplication. We have $(-0.8) × -0.5$. When multiplying two negative numbers, the result is a positive number. So, we need to multiply the absolute values of the numbers:

$0.8 × 0.5 = 0.4$

Therefore, $(-0.8) × -0.5 = 0.4$. This step highlights the rules of multiplying signed numbers, a key concept in algebra and beyond. Understanding that the product of two negatives is a positive is essential for correctly evaluating mathematical expressions. The ability to perform decimal multiplication accurately is also vital. In this case, multiplying 0.8 by 0.5 yields 0.4, demonstrating a basic but important arithmetic skill. This multiplication step helps to further simplify the expression, bringing us closer to the final answer.

After performing the multiplication, our expression is now:

$0.3 - 0.4$

The final step is to perform the subtraction. Subtracting a larger positive number from a smaller positive number will result in a negative number. To find the result, we subtract the smaller absolute value from the larger absolute value and take the sign of the number with the larger absolute value:

$0.3 - 0.4 = - (0.4 - 0.3) = -0.1$

Therefore, $0.3 - 0.4 = -0.1$. This subtraction step is the final operation in our expression, and it showcases the rules for subtracting decimals. The ability to subtract decimals accurately is a fundamental arithmetic skill, and this step reinforces its importance. The result, -0.1, is a decimal number, and it represents the final value of the expression. This step completes the evaluation process, demonstrating how following the order of operations leads to a precise and accurate solution.

After following the order of operations and performing each step carefully, we have found that the value of the expression $0.3 - (-0.6 + -0.2) × -0.5$ is:

$-0.1$

This result is a decimal number, as requested in the instructions. We did not need to round the answer, as the result was already a terminating decimal. The entire process of evaluating this expression serves as a practical example of how the PEMDAS rule is applied in mathematics. By breaking down the expression into smaller, manageable steps, we were able to systematically arrive at the correct answer. This exercise demonstrates the importance of accuracy in arithmetic calculations and the value of understanding the fundamental rules of mathematics. The final answer, -0.1, is the culmination of our efforts and a testament to the power of step-by-step problem-solving.

When evaluating mathematical expressions, it's easy to make mistakes if you're not careful. Here are some common pitfalls to watch out for:

1. Ignoring the Order of Operations: This is the most frequent mistake. Always remember PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction) and perform operations in the correct order.
2. Incorrectly Handling Negative Signs: Be extra cautious when dealing with negative numbers. A misplaced negative sign can drastically change the outcome. Remember the rules for adding, subtracting, multiplying, and dividing negative numbers.
3. Arithmetic Errors: Simple addition, subtraction, multiplication, or division errors can throw off your entire calculation. Double-check your work at each step to minimize these mistakes.
4. Forgetting to Distribute: When dealing with parentheses, make sure to distribute any multiplication or division across all terms inside the parentheses.
5. Rushing Through the Problem: Take your time and work through the problem step by step. Rushing can lead to careless errors.

By being mindful of these common mistakes, you can significantly improve your accuracy when evaluating mathematical expressions. Remember, practice makes perfect, so the more you work through these types of problems, the better you'll become at avoiding these errors. It's always a good idea to review your work to catch any potential mistakes before arriving at your final answer.

To solidify your understanding of evaluating expressions, try solving these practice problems. Remember to follow the order of operations (PEMDAS) and be careful with negative signs.

1. $2.5 + (3.2 - 1.7) × 0.4$
2. $-1.8 - (-2.1 + 0.5) ÷ 0.2$
3. $0.7 × (1.9 - 0.4 × 3) + 1.2$
4. $4.6 ÷ (-0.8 - 1.5) - 0.9$
5. $-0.3 + (-0.6 - 0.2) × -0.5$

Work through each problem step by step, showing your work as you go. Once you've found your answers, you can compare them to the solutions provided below.

Here are the solutions to the practice problems listed above:

1. $2.5 + (3.2 - 1.7) × 0.4 = 3.1$
2. $-1.8 - (-2.1 + 0.5) ÷ 0.2 = 6.2$
3. $0.7 × (1.9 - 0.4 × 3) + 1.2 = 1.69$
4. $4.6 ÷ (-0.8 - 1.5) - 0.9 = -2.9$
5. $-0.3 + (-0.6 - 0.2) × -0.5 = 0.1$

Review your work and see if you arrived at the correct answers. If you made any mistakes, try to identify where you went wrong and learn from it. Practice is key to mastering the evaluation of mathematical expressions. The more problems you solve, the more confident you'll become in your abilities. Remember to focus on understanding the underlying principles and techniques, and you'll be well on your way to success.

In conclusion, evaluating mathematical expressions requires a solid understanding of the order of operations (PEMDAS) and careful attention to detail. By breaking down complex expressions into smaller, manageable steps, we can systematically arrive at the correct answer. This article provided a comprehensive guide to evaluating the expression $0.3 - (-0.6 + -0.2) × -0.5$, demonstrating each step in detail. We also discussed common mistakes to avoid and offered practice problems to further enhance your skills. Mastering the evaluation of expressions is a crucial skill in mathematics, and with consistent practice, you can become proficient in this area. Remember to always follow the order of operations, double-check your work, and learn from any mistakes you make. With dedication and effort, you can excel in evaluating mathematical expressions and build a strong foundation for more advanced mathematical concepts.