Simplifying Complex Fractions: A Step-by-Step Guide

by Alex Johnson 52 views

Complex fractions can seem daunting at first glance, but with a systematic approach, they become much easier to handle. This guide will walk you through the process of simplifying complex fractions, using the example of the expression ((-2/x) + (5/y)) / ((3/y) - (2/x)). By the end of this article, you'll be able to tackle similar problems with confidence. Complex fractions are fractions where the numerator, the denominator, or both contain fractions themselves. Simplifying these fractions involves eliminating the inner fractions to obtain a single, simplified fraction. This skill is crucial in various areas of mathematics, including algebra and calculus, where complex expressions often need to be reduced to their simplest forms to facilitate further calculations or analysis. Understanding how to manipulate and simplify complex fractions not only enhances your problem-solving abilities but also provides a foundational understanding for more advanced mathematical concepts. Therefore, mastering the techniques for simplifying complex fractions is an essential step in your mathematical journey, enabling you to tackle more intricate problems with ease and precision.

Understanding the Complex Fraction

Before diving into the solution, let's break down the given complex fraction:

−2x+5y3y−2x\frac{\frac{-2}{x}+\frac{5}{y}}{\frac{3}{y}-\frac{2}{x}}

This fraction has fractions in both its numerator and denominator. Our goal is to simplify it into a single fraction. To begin simplifying this complex fraction, it's crucial to first understand its structure. The fraction consists of a main division bar, separating the numerator and the denominator. Within the numerator, we have the sum of two fractions, -2/x and 5/y, while in the denominator, we have the difference between two fractions, 3/y and -2/x. This layering of fractions within fractions is what makes it complex. Identifying these components is the first step in developing a strategy for simplification. We need to eliminate these nested fractions by finding common denominators and combining terms, which will eventually lead us to a single fraction. This initial assessment allows us to approach the problem methodically, ensuring each step contributes to the overall goal of simplifying the expression. The ability to dissect a complex fraction into its basic parts is a valuable skill that aids not only in simplification but also in understanding the relationships between different parts of the expression.

Step 1: Find a Common Denominator for the Numerator and Denominator

The first step is to find a common denominator for the fractions in the numerator and the fractions in the denominator separately. For both, the common denominator is xy.

Numerator:

−2x+5y=−2yxy+5xxy\frac{-2}{x} + \frac{5}{y} = \frac{-2y}{xy} + \frac{5x}{xy}

Denominator:

3y−2x=3xxy−2yxy\frac{3}{y} - \frac{2}{x} = \frac{3x}{xy} - \frac{2y}{xy}

The first critical step in simplifying this complex fraction involves tackling the numerator and the denominator independently. To do this effectively, we need to identify a common denominator for the fractions within each part. For both the numerator and the denominator in our example, the common denominator is xy. This is because xy is the simplest expression that both x and y can divide into evenly. Once we've identified the common denominator, the next step is to convert each fraction so that it has this denominator. This involves multiplying the numerator and denominator of each fraction by the appropriate factor. For instance, to convert -2/x to an equivalent fraction with a denominator of xy, we multiply both the numerator and the denominator by y, resulting in -2y/xy. Similarly, we convert 5/y to 5x/xy. Applying this process to both the numerator and the denominator sets the stage for combining the fractions and simplifying the expression, making it a pivotal step in the overall simplification strategy.

Step 2: Combine the Fractions

Now that we have a common denominator, we can combine the fractions in the numerator and the denominator:

Numerator:

−2yxy+5xxy=−2y+5xxy\frac{-2y}{xy} + \frac{5x}{xy} = \frac{-2y + 5x}{xy}

Denominator:

3xxy−2yxy=3x−2yxy\frac{3x}{xy} - \frac{2y}{xy} = \frac{3x - 2y}{xy}

Having established the common denominator in both the numerator and the denominator, the next logical step is to combine the individual fractions into single fractions. This consolidation is achieved by performing the addition or subtraction operations as indicated. In the numerator, we add the fractions -2y/xy and 5x/xy. Since they share a common denominator, we can simply add their numerators: -2y + 5x. This results in the single fraction (-2y + 5x)/xy. A similar process is applied to the denominator, where we subtract 2y/xy from 3x/xy. Again, because the denominators are the same, we subtract the numerators: 3x - 2y, yielding the single fraction (3x - 2y)/xy. Combining the fractions in this way is a crucial step because it simplifies the structure of the complex fraction. Instead of dealing with multiple fractions in the numerator and the denominator, we now have a single fraction in each, making the next steps in the simplification process more manageable and straightforward. This step significantly reduces the complexity of the expression, bringing us closer to the final simplified form.

Step 3: Divide the Fractions

Now we have a fraction divided by another fraction. Remember that dividing by a fraction is the same as multiplying by its reciprocal:

−2y+5xxy3x−2yxy=−2y+5xxy⋅xy3x−2y\frac{\frac{-2y + 5x}{xy}}{\frac{3x - 2y}{xy}} = \frac{-2y + 5x}{xy} \cdot \frac{xy}{3x - 2y}

With the numerator and denominator each simplified into single fractions, the next key step is to address the division operation implied by the main fraction bar. Dividing by a fraction is mathematically equivalent to multiplying by its reciprocal. This principle is crucial for simplifying complex fractions. In our case, we have the fraction (-2y + 5x)/xy divided by the fraction (3x - 2y)/xy. To apply the principle, we invert the denominator fraction, which means swapping its numerator and denominator, resulting in xy/(3x - 2y). We then change the division operation to multiplication. Now, the problem transforms into multiplying the fraction (-2y + 5x)/xy by the reciprocal xy/(3x - 2y). This transformation is significant because multiplication is often easier to handle than division, especially when dealing with fractions. By converting the division into multiplication, we set the stage for further simplification, as we can now look for common factors between the numerators and the denominators to cancel out, leading us closer to the simplest form of the complex fraction.

Step 4: Simplify

Now we can cancel out the xy terms:

−2y+5xxy⋅xy3x−2y=−2y+5x3x−2y\frac{-2y + 5x}{xy} \cdot \frac{xy}{3x - 2y} = \frac{-2y + 5x}{3x - 2y}

After converting the division of fractions into multiplication, the most rewarding step in simplifying complex fractions is the cancellation of common factors. This step not only reduces the expression to its simplest form but also showcases the elegance of mathematical simplification. In our example, we are multiplying the fraction (-2y + 5x)/xy by the fraction xy/(3x - 2y). A careful observation reveals that the term xy appears in both the numerator and the denominator. This presence of a common factor allows us to cancel it out. The process of cancellation involves dividing both the numerator and the denominator by the common factor. In this case, we divide both by xy, effectively removing it from both positions. Once xy is canceled, the expression simplifies significantly, leaving us with (-2y + 5x) in the numerator and (3x - 2y) in the denominator. This cancellation is a pivotal moment because it drastically reduces the complexity of the fraction, resulting in a much cleaner and more manageable expression. The ability to identify and cancel common factors is a cornerstone of fraction simplification, making it an indispensable skill in algebra and beyond.

Final Answer

So, the equivalent expression is:

−2y+5x3x−2y\frac{-2y + 5x}{3x - 2y}

This corresponds to option A.

After meticulously working through each step, from identifying common denominators to inverting and multiplying, and finally canceling out common factors, we arrive at the final simplified form of the complex fraction. This resulting expression, (-2y + 5x) / (3x - 2y), represents the complex fraction in its most reduced and understandable state. The journey from the initial complex expression to this simplified form underscores the power of systematic mathematical manipulation. Each step, while seemingly small on its own, contributes to the overall goal of simplification. The ability to arrive at this final answer not only demonstrates a mastery of fraction manipulation but also highlights the importance of precision and attention to detail in mathematical problem-solving. This final answer serves as a testament to the effectiveness of a step-by-step approach in tackling complex problems, reinforcing the idea that even the most daunting expressions can be simplified with the right techniques and a clear understanding of mathematical principles. By successfully simplifying this complex fraction, we've not only solved a specific problem but also honed our skills in algebraic manipulation, skills that are invaluable in a wide range of mathematical contexts.

Conclusion

Simplifying complex fractions involves a series of steps, including finding common denominators, combining fractions, and multiplying by the reciprocal. By following these steps carefully, you can simplify even the most complex-looking fractions. Remember, practice makes perfect! The key to mastering complex fractions lies in understanding the underlying principles and applying them methodically. Each step, from finding common denominators to multiplying by reciprocals and canceling common factors, plays a crucial role in reducing the expression to its simplest form. This systematic approach not only simplifies the problem-solving process but also enhances your understanding of fraction manipulation. As you practice more examples, you'll develop a keen eye for identifying common factors and applying the appropriate steps, making the process feel more intuitive. Remember, the goal is not just to arrive at the correct answer but to understand the 'why' behind each step. This deeper understanding will empower you to tackle a wider range of mathematical problems with confidence. So, keep practicing, and you'll find that even the most daunting complex fractions can be simplified with patience and a clear strategy. For further practice and resources, consider visiting Khan Academy's section on algebraic expressions.