Evaluate The Expression (-0.2 × 0.25 - 0.09) × -3 A Step-by-Step Guide

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#evaluating-expressions #mathematics #decimal-operations #order-of-operations

Introduction

In this article, we will delve into the process of evaluating a mathematical expression involving decimal numbers and the order of operations. The expression we aim to simplify is (-0.2 × 0.25 - 0.09) × -3. This problem requires a clear understanding of arithmetic operations, particularly multiplication and subtraction with decimals, and the correct application of the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). Let's break down each step to ensure clarity and accuracy in our calculation.

Understanding the Order of Operations (PEMDAS)

The order of operations is a fundamental concept in mathematics that dictates the sequence in which mathematical operations should be performed. Without a standardized order, the same expression could yield different results depending on the sequence of operations. PEMDAS provides this standardization, ensuring consistent and correct evaluations. Here’s what each letter stands for:

  • Parentheses: Operations inside parentheses (or brackets) are performed first.
  • Exponents: Exponentiation (powers and roots) is performed next.
  • Multiplication and Division: These operations are performed from left to right.
  • Addition and Subtraction: These operations are performed from left to right.

Adhering to PEMDAS is crucial for accurately simplifying mathematical expressions. In our case, the expression (-0.2 × 0.25 - 0.09) × -3 contains parentheses, multiplication, and subtraction, making PEMDAS directly applicable.

Step-by-Step Evaluation

Step 1: Evaluate the Expression Within the Parentheses

The first step, according to PEMDAS, is to address the operations inside the parentheses: (-0.2 × 0.25 - 0.09). Within the parentheses, we have both multiplication and subtraction. Following PEMDAS, we perform multiplication before subtraction.

Substep 1.1: Perform the Multiplication

We need to multiply -0.2 by 0.25. When multiplying decimals, it’s often helpful to ignore the decimal points initially and multiply the numbers as if they were whole numbers. Then, we count the total number of decimal places in the original numbers and apply that count to the product.

    1. 2 × 0.25 can be seen as 2 × 25, which equals 50.
  • There are a total of three decimal places (one in -0.2 and two in 0.25), so we apply three decimal places to the result: 0.050.
  • Since one of the numbers is negative (-0.2), the result is negative: -0.050.

So, -0.2 × 0.25 = -0.05.

Substep 1.2: Perform the Subtraction

Now we substitute the result of the multiplication back into the parentheses: (-0.05 - 0.09). This is a subtraction operation between two negative numbers. To subtract 0.09 from -0.05, we can think of it as adding two negative numbers: -0.05 + (-0.09).

  • Adding the absolute values: 0.05 + 0.09 = 0.14.
  • Since both numbers are negative, the result is negative: -0.14.

Thus, (-0.05 - 0.09) = -0.14.

Step 2: Multiply the Result by -3

Now that we have simplified the expression inside the parentheses to -0.14, we can proceed with the final operation, which is multiplying -0.14 by -3.

Substep 2.1: Perform the Multiplication

To multiply -0.14 by -3, we again multiply the numbers as if they were whole numbers and then adjust for the decimal places.

    1. 14 × 3 = 42.
  • There are two decimal places in -0.14, so we apply two decimal places to the result: 0.42.
  • Since we are multiplying two negative numbers, the result is positive.

Therefore, -0.14 × -3 = 0.42.

Final Answer

After performing all the necessary operations according to the correct order, we have successfully evaluated the expression (-0.2 × 0.25 - 0.09) × -3. The final result is 0.42.

Alternative Methods and Verification

While the step-by-step method provides a clear path to the solution, it’s always beneficial to consider alternative approaches and verification methods to ensure accuracy. Here, we'll discuss a few ways to double-check our result.

Method 1: Using a Calculator

The most straightforward way to verify the result is to use a calculator. Input the original expression exactly as it is: (-0.2 × 0.25 - 0.09) × -3. The calculator should return the same result, 0.42, confirming our calculations.

Method 2: Breaking Down the Parentheses Differently

Another way to approach the problem is to redistribute if possible or to convert the decimals into fractions. This method can sometimes simplify the calculations and provide a fresh perspective on the problem.

  1. Convert Decimals to Fractions (If Applicable): While not strictly necessary in this case, converting decimals to fractions can sometimes make calculations easier. However, for this specific problem, it might complicate the process rather than simplify it.

  2. Recheck Each Step: Go through each step of the original method and verify the calculations. This involves ensuring that the multiplication and subtraction within the parentheses were done correctly and that the final multiplication was also accurate.

Method 3: Estimation and Approximation

Estimating the result can help identify if the calculated answer is reasonable. For instance, we can round the numbers and approximate the expression:

  • -0. 2 × 0.25 is approximately -0.2 × 0.3, which is -0.06.
  • Subtracting 0.09 from -0.06 gives approximately -0.15.
  • Multiplying -0.15 by -3 gives approximately 0.45.

Our calculated result of 0.42 is close to the estimated result of 0.45, which increases our confidence in the accuracy of our solution.

Common Mistakes and How to Avoid Them

Evaluating mathematical expressions, especially those involving decimals and multiple operations, can be prone to errors. Understanding common mistakes and learning how to avoid them is crucial for achieving accurate results. Here are some typical pitfalls and strategies to prevent them:

1. Ignoring the Order of Operations

Mistake: Not following PEMDAS can lead to incorrect results. For instance, performing subtraction before multiplication within the parentheses.

How to Avoid: Always remember and apply the order of operations (PEMDAS). Prioritize parentheses, exponents, multiplication and division (from left to right), and finally, addition and subtraction (from left to right). Writing down the steps and clearly marking which operation is being performed can help maintain order.

2. Decimal Placement Errors

Mistake: Misplacing the decimal point when multiplying or dividing decimals.

How to Avoid: When multiplying decimals, count the total number of decimal places in the factors and ensure the product has the same number of decimal places. For division, ensure the decimal point is correctly placed in the quotient by aligning it with the decimal point in the dividend after making the divisor a whole number.

3. Sign Errors

Mistake: Incorrectly applying the rules for multiplying and dividing negative numbers.

How to Avoid: Remember that multiplying or dividing two numbers with the same sign (both positive or both negative) results in a positive number, while multiplying or dividing numbers with different signs results in a negative number. Double-check the signs at each step to avoid errors.

4. Arithmetic Errors

Mistake: Simple calculation mistakes such as addition or subtraction errors.

How to Avoid: Perform calculations carefully and double-check each step. If the numbers are complex, break the problem down into smaller, manageable parts. Using a calculator for complex arithmetic can also help reduce errors.

5. Incorrectly Handling Parentheses

Mistake: Not properly simplifying expressions within parentheses before proceeding with other operations.

How to Avoid: Always evaluate the expression inside parentheses first, following PEMDAS within the parentheses if necessary. Ensure that all operations within the parentheses are completed before moving on to the next step.

6. Rushing Through the Problem

Mistake: Trying to solve the problem too quickly, leading to careless errors.

How to Avoid: Take your time and work through the problem methodically. Rushing increases the likelihood of making mistakes. It’s better to solve one problem accurately than to attempt multiple problems with errors.

7. Not Verifying the Solution

Mistake: Failing to check the answer using an alternative method or estimation.

How to Avoid: Always verify your solution. Use a calculator, re-perform the calculations, or apply estimation techniques to ensure your answer is reasonable. Verification can catch errors that might otherwise go unnoticed.

Conclusion

Evaluating the expression (-0.2 × 0.25 - 0.09) × -3 demonstrates the importance of understanding and applying the correct order of operations (PEMDAS). Through a step-by-step approach, we first tackled the multiplication within the parentheses, then the subtraction, and finally, the multiplication outside the parentheses. Our careful calculations led us to the final answer of 0.42. By breaking down the problem, we ensure that each operation is performed accurately, minimizing the risk of errors.

In conclusion, mastering the evaluation of mathematical expressions not only reinforces fundamental arithmetic skills but also enhances problem-solving abilities. By consistently applying the order of operations and verifying solutions, one can confidently tackle complex mathematical challenges. Remember to take your time, double-check each step, and utilize alternative methods for verification. With practice and attention to detail, you can achieve accuracy and proficiency in evaluating any mathematical expression.