Evaluating Algebraic Expressions With Fractions A Step-by-Step Guide

In the realm of mathematics, the ability to evaluate algebraic expressions involving fractions is a fundamental skill. This article delves into a step-by-step approach to solving such expressions, focusing on the expression

$$\frac{4}{5}x - \frac{1}{3}x - \frac{1}{15}x$$

We will explore the underlying principles, techniques, and potential pitfalls to ensure a clear understanding of the process. By mastering these concepts, you'll be well-equipped to tackle more complex algebraic problems.

Understanding the Basics of Fraction Operations

Before diving into the specific expression, it's crucial to have a solid grasp of the basic operations involving fractions. Fractions represent parts of a whole, and understanding how to add, subtract, multiply, and divide them is essential for algebraic manipulation. The core concept we need to know for this article is subtracting fractions. To add or subtract fractions, they must have a common denominator. The common denominator is a common multiple of the denominators of the fractions. The least common multiple (LCM) is usually used, because it is easier to compute with smaller numbers.

Finding the Least Common Denominator (LCD)

The least common denominator (LCD) is the smallest common multiple of the denominators of the fractions. It's the key to combining fractions effectively. To find the LCD, you can use several methods, including listing multiples, prime factorization, or the greatest common factor (GCF). The LCD is the smallest number that each of the denominators can divide into evenly. Identifying the LCD is a crucial first step in simplifying expressions involving fractions. In our example, we have denominators of 5, 3, and 15. Let's find their LCD:

• Multiples of 5: 5, 10, 15, 20, 25...
• Multiples of 3: 3, 6, 9, 12, 15, 18...
• Multiples of 15: 15, 30, 45...

The least common multiple (LCM) is 15, hence LCD is 15.

Rewriting Fractions with the LCD

Once you've found the LCD, the next step is to rewrite each fraction with the LCD as its new denominator. This involves multiplying both the numerator and the denominator of each fraction by a factor that will result in the LCD. Rewriting the fractions ensures that they have the same denominator, making it possible to add or subtract them directly. The key here is to maintain the fraction's value while changing its form. We need to rewrite each fraction in the original expression with a denominator of 15:

• For 4/5: Multiply both numerator and denominator by 3 to get (4 * 3) / (5 * 3) = 12/15
• For 1/3: Multiply both numerator and denominator by 5 to get (1 * 5) / (3 * 5) = 5/15
• For 1/15: Already has the desired denominator, so it remains 1/15

Combining Fractions

After rewriting the fractions with a common denominator, you can combine them by adding or subtracting their numerators while keeping the denominator the same. This is a straightforward process, but it's essential to pay attention to the signs of the numerators. By combining the numerators, you effectively perform the required operation on the fractions. Remember that the denominator remains the same throughout this process, as it represents the size of the parts being added or subtracted. Now that all fractions have the same denominator, we can subtract them by subtracting the numerators:

$$\frac{12}{15}x - \frac{5}{15}x - \frac{1}{15}x = \frac{12 - 5 - 1}{15}x$$

Simplifying the Resulting Fraction

After combining the fractions, it's crucial to simplify the resulting fraction to its lowest terms. This involves finding the greatest common factor (GCF) of the numerator and denominator and dividing both by the GCF. Simplifying ensures that the fraction is expressed in its most concise form, making it easier to work with in further calculations. Always look for opportunities to simplify fractions to ensure accuracy and clarity in your mathematical work. If possible, the final result should be expressed in simplified form. This means reducing the fraction to its lowest terms by dividing both the numerator and denominator by their greatest common divisor (GCD). By simplifying the fraction, we present the answer in its most elegant and easily understandable form.

Step-by-Step Evaluation of the Expression

Now, let's apply these principles to evaluate the expression:

$$\frac{4}{5}x - \frac{1}{3}x - \frac{1}{15}x$$

Step 1: Find the Least Common Denominator (LCD)

As we determined earlier, the LCD of 5, 3, and 15 is 15.

Step 2: Rewrite Fractions with the LCD

We rewrite each fraction with a denominator of 15:

• $$\frac{4}{5}x = \frac{4 * 3}{5 * 3}x = \frac{12}{15}x$$

• $$\frac{1}{3}x = \frac{1 * 5}{3 * 5}x = \frac{5}{15}x$$

• \frac{1}{15}x$ remains the same.

Step 3: Combine Fractions

Now, we can combine the fractions:

$$\frac{12}{15}x - \frac{5}{15}x - \frac{1}{15}x = \frac{12 - 5 - 1}{15}x$$

Step 4: Simplify the Numerator

Simplify the numerator:

$$\frac{12 - 5 - 1}{15}x = \frac{6}{15}x$$

Step 5: Simplify the Fraction

Finally, simplify the fraction by dividing both the numerator and denominator by their greatest common divisor (GCD), which is 3:

$$\frac{6}{15}x = \frac{6 ÷ 3}{15 ÷ 3}x = \frac{2}{5}x$$

Therefore, the simplified expression is:

$$\frac{2}{5}x$$

Common Mistakes to Avoid

When evaluating algebraic expressions with fractions, several common mistakes can occur. Being aware of these pitfalls can help you avoid errors and ensure accurate solutions. Here are some frequent mistakes to watch out for:

• Forgetting to Find a Common Denominator: This is a critical step when adding or subtracting fractions. Failing to find a common denominator will lead to incorrect results. Always ensure that the fractions have the same denominator before combining them.
• Incorrectly Rewriting Fractions: When rewriting fractions with the LCD, it's essential to multiply both the numerator and the denominator by the same factor. Multiplying only one of them will change the value of the fraction.
• Arithmetic Errors: Mistakes in basic arithmetic, such as addition, subtraction, multiplication, or division, can lead to incorrect answers. Double-check your calculations to minimize errors.
• Forgetting to Simplify: Always simplify the final fraction to its lowest terms. Failing to do so will result in an incomplete solution.
• Incorrectly Handling Negative Signs: Pay close attention to negative signs when combining fractions. A misplaced or mishandled negative sign can significantly alter the result.

Practice Problems

To solidify your understanding, try evaluating these similar expressions:

1. $$\frac{3}{4}x + \frac{1}{2}x - \frac{1}{8}x$$

2. $$\frac{5}{6}x - \frac{2}{3}x + \frac{1}{4}x$$

3. $$\frac{2}{5}x + \frac{1}{10}x - \frac{1}{2}x$$

Conclusion

Evaluating algebraic expressions involving fractions requires a systematic approach and a solid understanding of fraction operations. By following the steps outlined in this article – finding the LCD, rewriting fractions, combining fractions, and simplifying the result – you can confidently tackle these types of problems. Remember to avoid common mistakes and practice regularly to hone your skills. With dedication and a clear understanding of the principles, you can master the art of evaluating algebraic expressions with fractions.

This article provided a comprehensive guide to evaluating the algebraic expression:

$$\frac{4}{5}x - \frac{1}{3}x - \frac{1}{15}x$$

By understanding the underlying principles and following the step-by-step approach, you can confidently solve similar problems and enhance your mathematical proficiency.