Evaluating The Expression 4/5 - 3/4 * 1/5 A Step-by-Step Guide

In the realm of mathematics, evaluating expressions is a fundamental skill. It's the process of simplifying a mathematical statement to arrive at a single numerical value. Today, we'll delve into evaluating a specific expression involving fractions, a common topic in arithmetic and algebra. This article provides a detailed, step-by-step explanation of how to solve the expression, ensuring clarity and understanding for anyone looking to improve their math skills. Whether you're a student, a teacher, or simply someone who enjoys math, this guide will help you master the art of evaluating expressions with fractions. Our focus is on providing clear, concise instructions that make complex calculations seem straightforward.

The Expression: A Fraction Challenge

Our mission is to evaluate the expression: 4/5 - 3/4 * 1/5. This expression involves both subtraction and multiplication of fractions, requiring us to carefully follow the order of operations to arrive at the correct answer. Understanding the order of operations is crucial when dealing with mathematical expressions that involve multiple operations. The order of operations, often remembered by the acronym PEMDAS (Parentheses, Exponents, Multiplication and Division, Addition and Subtraction), dictates the sequence in which operations should be performed. This ensures that mathematical expressions are evaluated consistently and accurately. In this case, we'll be focusing on multiplication and subtraction of fractions, but the principles of PEMDAS apply to a wide range of mathematical problems. By mastering these principles, you'll be well-equipped to tackle more complex expressions in the future.

The Order of Operations: Multiplication First

According to the order of operations (PEMDAS/BODMAS), multiplication must be performed before subtraction. This is a crucial rule in mathematics that ensures consistency in calculations. If we were to subtract before multiplying, we would end up with a different, and incorrect, answer. Understanding the order of operations is like understanding the grammar of mathematics; it provides the structure and rules that allow us to communicate mathematical ideas clearly and effectively. By adhering to these rules, we can avoid confusion and ensure that our calculations are accurate. So, before we even think about subtracting, we need to tackle the multiplication part of our expression. This step-by-step approach is essential for solving complex mathematical problems and building a strong foundation in arithmetic.

Multiplying Fractions: Numerators and Denominators

To multiply fractions, we multiply the numerators (the top numbers) together and the denominators (the bottom numbers) together. This is a fundamental rule of fraction multiplication. The numerator represents the number of parts we have, and the denominator represents the total number of parts the whole is divided into. When we multiply fractions, we're essentially finding a fraction of a fraction. This concept is used in many real-world applications, from dividing a recipe to calculating proportions. In our expression, we need to multiply 3/4 by 1/5. To do this, we'll multiply 3 (the numerator of the first fraction) by 1 (the numerator of the second fraction) and 4 (the denominator of the first fraction) by 5 (the denominator of the second fraction). This simple yet powerful rule allows us to combine fractions and simplify complex expressions.

Performing the Multiplication: 3/4 * 1/5

Let's perform the multiplication: 3/4 * 1/5. Multiplying the numerators (3 * 1) gives us 3, and multiplying the denominators (4 * 5) gives us 20. Therefore, 3/4 * 1/5 equals 3/20. This calculation is a straightforward application of the rule we just discussed. It demonstrates how easily we can combine fractions through multiplication. The result, 3/20, represents a smaller portion than either of the original fractions, which makes sense when we consider that we're taking a fraction of a fraction. This step is a critical milestone in solving our original expression. Now that we've handled the multiplication, we can move on to the next operation, which is subtraction. But first, let's pause and appreciate the simplicity and elegance of fraction multiplication.

Subtraction of Fractions: Finding a Common Denominator

Now that we've multiplied the fractions, we're left with the subtraction: 4/5 - 3/20. To subtract fractions, they must have a common denominator. This is because we can only directly add or subtract fractions that represent parts of the same whole. Imagine trying to add apples and oranges – you need a common unit, like "fruit," to combine them meaningfully. Similarly, fractions need a common denominator to be combined. The common denominator is a number that both denominators can divide into evenly. Finding this common denominator is a crucial step in fraction subtraction. It allows us to rewrite the fractions in a way that makes the subtraction possible. Once we have a common denominator, we can simply subtract the numerators and keep the denominator the same. This principle of common denominators is fundamental to fraction arithmetic.

Identifying the Least Common Multiple (LCM)

The easiest way to find a common denominator is to determine the Least Common Multiple (LCM) of the denominators. The LCM is the smallest number that is a multiple of both denominators. In our case, the denominators are 5 and 20. To find the LCM, we can list the multiples of each number and identify the smallest one they share. Multiples of 5 are 5, 10, 15, 20, 25, and so on. Multiples of 20 are 20, 40, 60, and so on. The smallest number that appears in both lists is 20. Therefore, the LCM of 5 and 20 is 20. This means that 20 is the ideal common denominator for our fractions. Using the LCM as the common denominator makes our calculations simpler and avoids the need to reduce the fraction at the end. This method is a reliable and efficient way to find the common denominator in fraction subtraction and addition problems.

Converting Fractions to a Common Denominator

With the LCM identified as 20, we need to convert 4/5 to an equivalent fraction with a denominator of 20. To do this, we ask ourselves, "What do we multiply 5 by to get 20?" The answer is 4. So, we multiply both the numerator and the denominator of 4/5 by 4. This gives us (4 * 4) / (5 * 4), which simplifies to 16/20. Remember, multiplying both the numerator and the denominator by the same number doesn't change the value of the fraction; it simply changes the way it's expressed. Now we have two fractions with the same denominator: 16/20 and 3/20. The fraction 3/20 already has the desired denominator, so we don't need to change it. This conversion is a key step in preparing the fractions for subtraction. By expressing both fractions with a common denominator, we've set the stage for a straightforward subtraction operation.

Performing the Subtraction: 16/20 - 3/20

Now that both fractions have the same denominator, we can perform the subtraction. We subtract the numerators (16 - 3) and keep the denominator the same. So, 16/20 - 3/20 equals 13/20. This calculation is the culmination of our efforts to find a common denominator and convert the fractions. The result, 13/20, represents the difference between the two original fractions. It's a single fraction that expresses the outcome of the subtraction operation. This step highlights the simplicity of fraction subtraction once a common denominator is established. By focusing on the process of finding and using common denominators, we can make fraction subtraction a much more manageable task. This skill is essential for anyone working with fractions, whether in mathematics, science, or everyday life.

The Final Answer: 13/20

Therefore, the expression 4/5 - 3/4 * 1/5 evaluates to 13/20. This is our final answer, expressed as a fraction. We have successfully navigated the complexities of fraction multiplication and subtraction by following the order of operations and applying the principles of common denominators. This journey from the initial expression to the final answer demonstrates the power of step-by-step problem-solving in mathematics. By breaking down a complex problem into smaller, more manageable steps, we can arrive at the solution with confidence. The result, 13/20, represents the simplified form of the original expression, and it's a testament to the effectiveness of our methodical approach. This process not only provides the answer but also reinforces our understanding of fraction arithmetic.

Conclusion: Mastering Fraction Evaluation

In conclusion, we have successfully evaluated the expression 4/5 - 3/4 * 1/5, arriving at the answer 13/20. This exercise has highlighted the importance of the order of operations, the rules of fraction multiplication, and the concept of common denominators in fraction subtraction. By understanding and applying these principles, we can confidently tackle a wide range of mathematical expressions involving fractions. Evaluating expressions is a fundamental skill in mathematics, and mastering it opens the door to more advanced concepts. This step-by-step guide has provided a clear and concise roadmap for solving fraction problems. With practice and a solid understanding of the underlying principles, anyone can become proficient in evaluating expressions with fractions. Remember, mathematics is a journey of discovery, and each problem solved is a step forward on that journey.