Evaluating Expressions Step-by-Step Guide

In mathematics, evaluating expressions is a fundamental skill. It involves simplifying a mathematical expression to obtain a single numerical value. This process often requires applying the order of operations, which dictates the sequence in which different operations should be performed. In this comprehensive guide, we will dissect the expression $-\frac{1}{2} \div \frac{2}{5} \times(-\frac{1}{3}+-\frac{2}{3})$ step by step, ensuring a clear understanding of each operation and the reasoning behind it. We will walk you through each stage, from handling fractions to applying the order of operations, providing a clear and detailed explanation for every step.

When it comes to evaluating mathematical expressions, understanding the order of operations is very important. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), ensures consistency and accuracy in mathematical calculations. Parentheses are always addressed first, simplifying the expression from the inside out. Exponents follow, then multiplication and division (from left to right), and finally, addition and subtraction (also from left to right). This set of rules acts as a universal guide, providing a structured approach to simplify complex mathematical expressions. Without this order, the same expression could yield different results depending on the sequence of operations, leading to confusion and errors. For instance, if we were to perform multiplication before division in our expression, we would arrive at a different answer. Adhering to PEMDAS guarantees that everyone evaluating the same expression will arrive at the same correct answer, which is crucial in fields like engineering, physics, and finance, where precision is paramount.

Fractions, with their numerators and denominators, may seem daunting at first, but with a clear understanding of the basic operations, they can be easily managed. In our given expression, we encounter fractions in several places, requiring us to perform division, multiplication, and addition involving these fractional values. When dividing fractions, we multiply by the reciprocal of the divisor. For example, $-\frac{1}{2} \div \frac{2}{5}$ becomes $-\frac{1}{2} \times \frac{5}{2}$. This transformation allows us to convert division into multiplication, simplifying the calculation. Multiplying fractions involves multiplying the numerators together and the denominators together. For example, $-\frac{1}{2} \times \frac{5}{2}$ results in $-\frac{5}{4}$. When adding fractions, it is essential to have a common denominator. If the fractions do not have the same denominator, we need to find the least common multiple (LCM) of the denominators and convert each fraction to an equivalent fraction with the LCM as the denominator. Once the denominators are the same, we can add the numerators while keeping the denominator constant. By mastering these fraction operations, we can confidently handle complex expressions involving fractional values.

Step-by-Step Evaluation

Let's break down the expression $-\frac{1}{2} \div \frac{2}{5} \times(-\frac{1}{3}+-\frac{2}{3})$ into manageable steps.

1. Simplify the Parentheses

• Start with the expression inside the parentheses: $(-\frac{1}{3}+-\frac{2}{3})$.
• Since the fractions have the same denominator, we can add the numerators directly: $-\frac{1}{3} + -\frac{2}{3} = \frac{-1 + (-2)}{3}$.
• Simplify the numerator: $\frac{-1 + (-2)}{3} = \frac{-3}{3}$.
• Reduce the fraction: $\frac{-3}{3} = -1$.
• So, the expression inside the parentheses simplifies to $-1$.

This initial step demonstrates the importance of starting with the parentheses in the order of operations. By simplifying the expression within the parentheses first, we reduce the complexity of the overall expression, making it easier to manage in subsequent steps. The addition of negative fractions is a key concept here, and understanding how to combine these fractions is crucial for accurately evaluating the entire expression. This step not only simplifies the expression but also sets the stage for the next operations, ensuring that we proceed in the correct order and maintain accuracy.

2. Rewrite the Expression

• Substitute the simplified value back into the original expression: $-\frac{1}{2} \div \frac{2}{5} \times (-1)$.
• Now, the expression looks cleaner and more manageable.

By replacing the parentheses with its simplified value, we transform the original expression into a more streamlined format. This step is vital in maintaining clarity and preventing errors as we move forward with the evaluation. The substitution highlights the progressive nature of solving mathematical expressions, where each step builds upon the previous one. This simplified expression now contains only division and multiplication operations, which we will address next in the order of operations.

3. Perform Division

• Convert division to multiplication by using the reciprocal: $-\frac{1}{2} \div \frac{2}{5}$ becomes $-\frac{1}{2} \times \frac{5}{2}$.
• Multiply the fractions: $-\frac{1}{2} \times \frac{5}{2} = \frac{-1 \times 5}{2 \times 2}$.
• Simplify: $\frac{-1 \times 5}{2 \times 2} = \frac{-5}{4}$.

In this step, we tackle the division operation by converting it into multiplication, a standard technique when dealing with fractions. By multiplying by the reciprocal, we maintain the mathematical integrity of the expression while making it easier to compute. The multiplication of the fractions involves multiplying the numerators and denominators separately, resulting in a new fraction. This step underscores the importance of understanding fraction operations and how they are applied in the context of more complex expressions. The result, $-\frac{5}{4}$, is a key intermediate value that we will use in the final step.

4. Perform Multiplication

• Multiply the result from the division by $-1$: $\frac{-5}{4} \times (-1)$.
• Remember that multiplying a negative number by $-1$ results in a positive number: $\frac{-5}{4} \times (-1) = \frac{5}{4}$.

Here, we complete the evaluation by performing the final multiplication. Multiplying a negative fraction by -1 changes its sign, resulting in a positive fraction. This step reinforces the rules of multiplying signed numbers, a fundamental concept in algebra. The final result, $\frac{5}{4}$, represents the simplified value of the original expression. This result can also be expressed as a mixed number, providing an alternative way to represent the same value.

5. Express the Answer

• The final result is $\frac{5}{4}$.
• This can also be expressed as the mixed number $1\frac{1}{4}$.

Finally, we present the result in both improper fraction and mixed number forms. The improper fraction $\frac{5}{4}$ clearly shows the numerical value, while the mixed number $1\frac{1}{4}$ provides a more intuitive understanding of the quantity (one whole and one-quarter). This final step emphasizes the flexibility in expressing mathematical results and the importance of choosing the representation that best suits the context.

Conclusion

By following the order of operations and carefully handling fractions, we have successfully evaluated the expression $-\frac{1}{2} \div \frac{2}{5} \times(-\frac{1}{3}+-\frac{2}{3})$. The final result is $\frac{5}{4}$, which is also equivalent to the mixed number $1\frac{1}{4}$. This step-by-step guide illustrates the importance of methodical calculation and attention to detail in mathematics. Each operation, from simplifying within parentheses to multiplying fractions, builds upon the previous one, leading to the final solution. Understanding these steps not only helps in solving similar expressions but also reinforces foundational mathematical principles.

Evaluating mathematical expressions is a critical skill that extends beyond the classroom. In real-world applications, from engineering calculations to financial analysis, the ability to accurately simplify expressions is essential. This guide has provided a detailed walkthrough of the process, highlighting the key concepts and techniques involved. By mastering these skills, individuals can approach complex mathematical problems with confidence and precision. Practice is key to solidifying understanding, and working through various expressions will further enhance proficiency in this area. The goal is not just to arrive at the correct answer but also to understand the underlying logic and principles that govern mathematical operations.

Furthermore, the ability to evaluate expressions accurately is a stepping stone to more advanced mathematical concepts. Algebra, calculus, and other higher-level math courses rely heavily on the foundational skills of simplifying expressions. A strong grasp of these basics ensures a smoother transition to more complex topics. In addition, the problem-solving skills developed through evaluating expressions are transferable to other areas of life. The ability to break down a problem into smaller, manageable steps, follow a logical process, and arrive at a solution is a valuable asset in any field. Thus, mastering the evaluation of expressions is not just about numbers and symbols; it’s about developing a way of thinking and approaching challenges that will serve you well in various aspects of life.

• Evaluating expressions
• Order of operations
• PEMDAS
• Fractions
• Division
• Multiplication
• Addition
• Simplifying expressions
• Mathematical calculations
• Mixed numbers
• Improper fractions
• Reciprocal
• Numerator
• Denominator
• Least common multiple (LCM)
• Step-by-step guide
• Algebra