Simplifying Complex Fractions A Step-by-Step Guide

Understanding Complex Fractions

In the realm of mathematics, complex fractions often appear daunting at first glance. However, with a systematic approach, these expressions can be simplified effectively. A complex fraction, as the name suggests, is a fraction where either the numerator, the denominator, or both contain fractions themselves. These fractions within fractions can seem intimidating, but they are nothing more than a combination of basic arithmetic operations. At their core, complex fractions represent division. Just as a simple fraction like 1/2 means 1 divided by 2, a complex fraction represents one fractional expression divided by another. The key to simplifying these expressions lies in recognizing this fundamental principle and applying the rules of fraction manipulation. This might involve finding common denominators, adding or subtracting fractions, and ultimately dividing one fraction by another. Understanding this underlying structure is the first step towards mastering the art of simplifying complex fractions. Remember, the goal is to transform a seemingly complicated expression into a more manageable and easily understandable form. By breaking down the problem into smaller steps and focusing on the core concepts, anyone can confidently tackle complex fractions. This guide will walk you through the necessary steps and provide a clear methodology for simplifying these expressions. Whether you are a student learning algebra or someone brushing up on your math skills, mastering complex fractions is a valuable skill that opens doors to more advanced mathematical concepts.

The Problem: A Step-by-Step Approach

Let's consider the specific complex fraction Ari is trying to simplify: $\frac{\frac{5}{a-3}-4}{2+\frac{1}{a-3}}$. This expression might look challenging, but by breaking it down into smaller, manageable steps, we can systematically simplify it. Our goal is to transform this complex fraction into a simpler, equivalent expression. The first step in simplifying any complex fraction is to address the numerator and denominator separately. In this case, the numerator is $\frac{5}{a-3}-4$, and the denominator is $2+\frac{1}{a-3}$. Each of these expressions involves fractions and integers, so we'll need to find a common denominator to combine them. This involves rewriting the integer terms (4 and 2) as fractions with the same denominator as the other fractions in their respective expressions. Once we have a common denominator for both the numerator and the denominator, we can perform the addition and subtraction operations. This will result in a single fraction for the numerator and a single fraction for the denominator. The complex fraction will then be reduced to the form of one fraction divided by another fraction. The final step involves dividing the numerator fraction by the denominator fraction. This is achieved by multiplying the numerator fraction by the reciprocal of the denominator fraction. After this multiplication, we will have a single fraction, which should be simplified if possible. Simplification might involve canceling out common factors between the numerator and the denominator. By following these steps, we can systematically simplify the complex fraction and arrive at the final, simplified expression. Each step builds upon the previous one, making the process clear and easy to follow.

Step 1: Simplify the Numerator

The numerator of the complex fraction is $\frac{5}{a-3}-4$. To simplify this, we need to combine the fraction and the integer term. The key to combining these terms is to find a common denominator. In this case, the denominator of the fraction is $(a-3)$, so we need to rewrite the integer 4 as a fraction with the same denominator. We can express 4 as $\frac{4}{1}$, and then multiply both the numerator and the denominator by $(a-3)$ to get a common denominator: $4 = \frac{4}{1} * \frac{a-3}{a-3} = \frac{4(a-3)}{a-3}$. Now we can rewrite the numerator as $\frac{5}{a-3} - \frac{4(a-3)}{a-3}$. With a common denominator, we can subtract the fractions by subtracting the numerators: $\frac{5 - 4(a-3)}{a-3}$. Next, we need to simplify the numerator by distributing the -4 and combining like terms: $5 - 4(a-3) = 5 - 4a + 12 = 17 - 4a$. Therefore, the simplified numerator is $\frac{17-4a}{a-3}$. This process of finding a common denominator, rewriting the terms, and combining them is a fundamental technique in simplifying complex fractions. By carefully applying these steps, we have successfully simplified the numerator into a single fraction. This simplified numerator will be used in the next step when we combine it with the simplified denominator to further reduce the complex fraction. This step-by-step approach ensures accuracy and clarity in the simplification process. Understanding how to manipulate fractions and find common denominators is crucial for success in algebra and beyond.

Step 2: Simplify the Denominator

The denominator of the complex fraction is $2+\frac{1}{a-3}$. Similar to the numerator, we need to combine the integer term and the fraction. To do this, we need to find a common denominator. The denominator of the fraction is $(a-3)$, so we need to rewrite the integer 2 as a fraction with the same denominator. We can express 2 as $\frac{2}{1}$, and then multiply both the numerator and the denominator by $(a-3)$ to get a common denominator: $2 = \frac{2}{1} * \frac{a-3}{a-3} = \frac{2(a-3)}{a-3}$. Now we can rewrite the denominator as $\frac{2(a-3)}{a-3} + \frac{1}{a-3}$. With a common denominator, we can add the fractions by adding the numerators: $\frac{2(a-3) + 1}{a-3}$. Next, we need to simplify the numerator by distributing the 2 and combining like terms: $2(a-3) + 1 = 2a - 6 + 1 = 2a - 5$. Therefore, the simplified denominator is $\frac{2a-5}{a-3}$. This process mirrors the simplification of the numerator, highlighting the importance of finding common denominators when combining fractions and integers. By carefully applying these steps, we have successfully simplified the denominator into a single fraction. This simplified denominator will be used in the next step when we divide the simplified numerator by it. Mastering the manipulation of fractions and the technique of finding common denominators is essential for simplifying complex expressions in algebra. This step-by-step approach ensures accuracy and clarity in the simplification process, making the overall problem more manageable.

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Now that we have simplified both the numerator and the denominator, we can proceed to the final step of simplifying the complex fraction. We have the simplified numerator as $\frac{17-4a}{a-3}$ and the simplified denominator as $\frac{2a-5}{a-3}$. Dividing the numerator by the denominator means performing the operation $\frac{\frac{17-4a}{a-3}}{\frac{2a-5}{a-3}}$. Dividing by a fraction is the same as multiplying by its reciprocal. Therefore, we can rewrite the division as a multiplication: $\frac{17-4a}{a-3} * \frac{a-3}{2a-5}$. Now we have a multiplication of two fractions. To multiply fractions, we multiply the numerators and multiply the denominators: $\frac{(17-4a)(a-3)}{(a-3)(2a-5)}$. Notice that we have a common factor of $(a-3)$ in both the numerator and the denominator. We can cancel out this common factor, which simplifies the expression further: $\frac{17-4a}{2a-5}$. This is the simplified form of the complex fraction. We have successfully transformed the original complex expression into a more manageable and understandable fraction. This final step demonstrates the power of combining the previous simplifications to arrive at the solution. By understanding the relationship between division and multiplication, and by recognizing and canceling out common factors, we can efficiently simplify complex fractions. This process highlights the importance of each step in the simplification process, from finding common denominators to multiplying by reciprocals.

Final Answer: ${\frac{17-4a}{2a-5}}$

In conclusion, by following a systematic approach, Ari can simplify the complex fraction $\frac{\frac{5}{a-3}-4}{2+\frac{1}{a-3}}$ to its final form: $\frac{17-4a}{2a-5}$. This process involves breaking down the problem into smaller, more manageable steps. First, the numerator and denominator are simplified separately by finding common denominators and combining terms. Then, the division of the simplified numerator by the simplified denominator is performed by multiplying the numerator by the reciprocal of the denominator. Finally, common factors are canceled out to arrive at the simplest form of the fraction. This step-by-step method provides a clear and organized way to tackle complex fractions, making them less intimidating and more accessible. Each step builds upon the previous one, ensuring accuracy and understanding throughout the simplification process. Mastering these techniques is crucial for success in algebra and higher-level mathematics, as it provides a solid foundation for manipulating and simplifying more complex expressions. By practicing and applying these principles, anyone can confidently simplify complex fractions and gain a deeper understanding of mathematical concepts. The final answer, $\frac{17-4a}{2a-5}$, represents the culmination of these efforts, showcasing the power of systematic problem-solving in mathematics. This solution not only provides the answer but also illustrates the underlying principles and techniques involved in simplifying complex fractions.

A. Ari can write the numerator and denominator with a common denominator. Then divide the numerator by the denominator. To do this, multiply the numerator by the reciprocal of the denominator.