Evaluating The Expression -3(2/5) - 4(-3/2) * 3 A Step-by-Step Guide

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Introduction

In this article, we will delve into the process of evaluating the numerical expression -3(2/5) - 4(-3/2)3. This expression involves a combination of arithmetic operations, including multiplication, division, and subtraction, along with both fractional and integer values. Understanding the order of operations and how to work with fractions is crucial for accurately solving this type of problem. This mathematical expression might seem daunting at first glance, but with a systematic approach and a clear understanding of the order of operations, it can be simplified step by step. The key is to break down the expression into smaller, manageable parts and to perform the operations in the correct sequence. By carefully applying the rules of arithmetic and paying close attention to detail, we can arrive at the correct answer. The importance of mastering these fundamental mathematical skills cannot be overstated, as they form the building blocks for more advanced concepts and problem-solving techniques. In the subsequent sections, we will guide you through each step of the evaluation process, providing explanations and insights to enhance your understanding. We will also highlight common pitfalls to avoid and offer tips for tackling similar expressions in the future. This comprehensive approach ensures that you not only learn how to solve this particular problem but also develop a broader proficiency in arithmetic operations.

Step-by-Step Evaluation

To accurately evaluate the expression -3(2/5) - 4(-3/2)3, we need to follow the order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction). However, in this case, we don't have any parentheses or exponents, so we can focus on multiplication and division first, followed by addition and subtraction. The order of operations, often represented by the acronym PEMDAS, serves as a roadmap for simplifying complex mathematical expressions. It dictates the sequence in which operations should be performed to ensure consistency and accuracy in the final result. This principle is not merely a convention; it is a fundamental rule of mathematics that ensures everyone arrives at the same answer when presented with the same expression. The initial step in tackling any mathematical expression is to identify the different operations involved and their respective priorities. This involves scanning the expression for parentheses, exponents, multiplication, division, addition, and subtraction. Once the operations are identified, the next step is to apply PEMDAS systematically, starting with the operations enclosed in parentheses (if any) and progressing through the order until all operations have been performed. This meticulous approach minimizes the risk of errors and ensures that the final answer is correct. Ignoring the order of operations can lead to drastically different results, highlighting the critical importance of adhering to this mathematical principle. Let's break down the expression into manageable parts:

  1. First, let's handle the multiplication of -3 and (2/5): -3 * (2/5) = -6/5 When multiplying a whole number by a fraction, we can treat the whole number as a fraction with a denominator of 1. In this case, -3 can be written as -3/1. Then, we multiply the numerators (-3 and 2) and the denominators (1 and 5) separately. This gives us (-3 * 2) / (1 * 5) = -6/5. This step is crucial because it simplifies one part of the expression, making it easier to manage in subsequent steps. It's important to remember that a negative number multiplied by a positive number results in a negative number, which is why the result is -6/5. This result, -6/5, will be used in the next stage of the evaluation process. Understanding how to multiply fractions and integers is a fundamental skill in arithmetic, and this step provides a clear example of its application. By performing this multiplication first, we adhere to the order of operations and ensure that the expression is simplified correctly.

  2. Next, we'll address the term 4(-3/2)3. Here, we have multiplication involving a fraction and a whole number: 4 * (-3/2) = -12/2 -12/2 = -6 -6 * 3 = -18 In this part of the expression, we encounter a sequence of multiplications involving a whole number, a fraction, and another whole number. To simplify this, we first multiply 4 by -3/2. As before, we can treat 4 as a fraction 4/1 and multiply the numerators and denominators: (4 * -3) / (1 * 2) = -12/2. Next, we simplify the fraction -12/2, which equals -6. This simplification is essential for making the subsequent calculation easier. Finally, we multiply -6 by 3, which gives us -18. This series of multiplications demonstrates how to handle mixed operations involving integers and fractions. It also highlights the importance of simplifying fractions whenever possible to reduce the complexity of the calculations. By breaking down the multiplication into smaller steps, we minimize the risk of errors and ensure that we arrive at the correct result. This result, -18, will be used in the final step of the evaluation.

  3. Now, let's combine the results: -6/5 - (-18) -6/5 + 18 To subtract a negative number, we add its positive counterpart. So, - (-18) becomes + 18. This is a fundamental rule of arithmetic that is crucial for correctly simplifying expressions involving subtraction and negative numbers. Adding the opposite is an equivalent operation, but it changes the sign of the number being subtracted, which can lead to confusion if not handled carefully. In this case, subtracting -18 is the same as adding 18, so we rewrite the expression as -6/5 + 18. This transformation makes it easier to proceed with the addition, as we now have a simple addition problem involving a fraction and a whole number. Understanding this rule is essential for accurately solving mathematical expressions and avoiding common errors. The ability to manipulate negative numbers and subtraction correctly is a key skill in algebra and beyond. This step sets the stage for the final calculation, where we will combine the fraction and the whole number to arrive at the final answer.

  4. To add these, we need a common denominator. We can express 18 as a fraction with a denominator of 5: 18 = 18/1 = (18 * 5) / (1 * 5) = 90/5 To add or subtract fractions, they must have a common denominator. In this case, we have -6/5 and 18, which can be written as 18/1. The least common denominator (LCD) for 5 and 1 is 5. Therefore, we need to convert 18/1 into an equivalent fraction with a denominator of 5. To do this, we multiply both the numerator and the denominator of 18/1 by 5. This gives us (18 * 5) / (1 * 5) = 90/5. This process ensures that we are adding equivalent quantities, which is crucial for obtaining the correct result. Finding a common denominator is a fundamental step in fraction arithmetic, and it's essential for simplifying expressions and solving equations. This conversion allows us to combine the fractions easily in the next step. The ability to work with fractions and find common denominators is a key skill in mathematics, and this step provides a clear example of its application.

  5. Now, we can add the fractions: -6/5 + 90/5 = 84/5 Now that we have both fractions with the same denominator, we can add them by adding their numerators and keeping the denominator the same. In this case, we have -6/5 + 90/5. Adding the numerators, we get -6 + 90 = 84. The denominator remains 5. So, the result is 84/5. This step demonstrates the fundamental principle of adding fractions with common denominators. It's a straightforward process once the fractions have been converted to a common denominator. The resulting fraction, 84/5, is the final answer in improper fraction form. We can also convert this improper fraction to a mixed number for a different representation of the same value. This final step completes the evaluation of the expression, providing us with the simplified result. The ability to add fractions correctly is a fundamental skill in mathematics, and this step provides a clear example of its application.

Final Result

Therefore, -3(2/5) - 4(-3/2)3 = 84/5. This can also be expressed as the mixed number 16 4/5. The ability to convert between improper fractions and mixed numbers is a valuable skill in mathematics, as different contexts may call for different representations of the same value. An improper fraction, like 84/5, has a numerator that is greater than or equal to the denominator, while a mixed number combines a whole number and a proper fraction. To convert 84/5 to a mixed number, we divide 84 by 5. The quotient is 16, and the remainder is 4. This means that 84/5 is equal to 16 whole units and 4/5 of another unit, which is written as 16 4/5. Both representations, 84/5 and 16 4/5, are correct and equivalent. Understanding how to convert between these forms allows for flexibility in problem-solving and communication of mathematical results. In some cases, an improper fraction may be preferred for calculations, while a mixed number may be more intuitive for understanding the quantity. The final result, whether expressed as an improper fraction or a mixed number, represents the simplified value of the original expression.

Conclusion

In conclusion, by carefully following the order of operations and applying the rules of arithmetic, we have successfully evaluated the expression -3(2/5) - 4(-3/2)3 and arrived at the final answer of 84/5 or 16 4/5. This process highlights the importance of breaking down complex expressions into smaller, manageable steps. Each step, from multiplying fractions and integers to finding common denominators and adding fractions, requires a clear understanding of mathematical principles. The order of operations, often represented by PEMDAS, serves as a roadmap for simplifying expressions and ensuring accuracy. By adhering to this order, we can avoid common errors and arrive at the correct result. Furthermore, the ability to convert between improper fractions and mixed numbers allows for flexibility in representing mathematical quantities. This skill is particularly useful in real-world applications where different forms may be more appropriate depending on the context. The process of evaluating this expression not only provides a specific answer but also reinforces fundamental arithmetic skills that are essential for more advanced mathematical concepts. Mastering these skills is crucial for success in mathematics and related fields. By practicing and applying these techniques, you can develop confidence in your ability to solve complex mathematical problems.