Simplifying Complex Fractions: A Step-by-Step Guide

Complex fractions, those fractions within fractions, can seem daunting at first glance. But don't worry, guys! Simplifying them is totally achievable with the right approach. In this guide, we'll break down the process step-by-step, using the example: $\frac{\frac{6}{x+4}+\frac{9}{x+5}}{\frac{5 x+22}{x^2+9 x+20}}$.

Understanding Complex Fractions

Before we dive into the solution, let's define what complex fractions are. Complex fractions are fractions where the numerator, the denominator, or both contain fractions themselves. They look intimidating, but they are really just a fraction divided by another fraction. Our main goal here is to transform these complex beasts into simpler, more manageable fractions. This involves getting rid of the smaller fractions within the larger one. We often achieve this by finding a common denominator and performing some algebraic manipulations. Keep in mind, guys, that the key to success is understanding the fundamental principles of fraction manipulation and applying them strategically. So, let's get started and demystify these complex fractions together!

Step 1: Simplify the Numerator

The first thing we're going to tackle is simplifying the numerator. Our numerator is $\frac{6}{x+4} + \frac{9}{x+5}$. To add these two fractions, we need to find a common denominator. Remember from basic fraction addition, we can't directly add fractions unless they share the same denominator. So, what’s the common denominator here? It's the product of the two denominators, which is (x + 4)(x + 5). Now, we need to rewrite each fraction with this common denominator.

To get the first fraction, $\frac{6}{x+4}$, to have the denominator (x + 4)(x + 5), we multiply both the numerator and denominator by (x + 5). This gives us $\frac{6(x+5)}{(x+4)(x+5)}$. For the second fraction, $\frac{9}{x+5}$, we multiply both the numerator and denominator by (x + 4), resulting in $\frac{9(x+4)}{(x+4)(x+5)}$. Now that both fractions have the same denominator, we can add them. We add the numerators and keep the common denominator: $\frac{6(x+5) + 9(x+4)}{(x+4)(x+5)}$. Next, we distribute and combine like terms in the numerator. 6 multiplied by (x + 5) is 6x + 30, and 9 multiplied by (x + 4) is 9x + 36. Adding these together, we get 6x + 30 + 9x + 36, which simplifies to 15x + 66. So, the simplified numerator becomes $\frac{15x + 66}{(x+4)(x+5)}$. That's one big step down! We've successfully combined those two fractions in the numerator into a single fraction. Remember, guys, take it one step at a time, and you'll conquer these complex fractions!

Step 2: Simplify the Denominator

Now, let's shift our focus to the denominator. In our complex fraction, the denominator is $\frac{5x + 22}{x^2 + 9x + 20}$. The good news is, it's already a single fraction, which means we don’t need to combine any terms like we did in the numerator. However, we can try to simplify it further by factoring, if possible. Factoring is like reverse-distributing, and it can help us identify common factors that we can cancel out later.

Looking at the denominator $x^2 + 9x + 20$, we need to find two numbers that multiply to 20 and add up to 9. Those numbers are 4 and 5. So, we can factor the quadratic expression as (x + 4)(x + 5). This means our denominator now looks like $\frac{5x + 22}{(x + 4)(x + 5)}$. At this stage, it's essential to keep an eye out for potential simplifications. We've factored the quadratic in the denominator, and now we have a better view of the entire fraction. In the next step, we'll see how this factored form helps us to simplify the entire complex fraction. Factoring is a powerful tool, guys, so make sure you're comfortable with it!

Step 3: Divide the Simplified Numerator by the Simplified Denominator

Okay, we've simplified both the numerator and the denominator. Now comes the exciting part: dividing the simplified numerator by the simplified denominator. Remember, guys, dividing by a fraction is the same as multiplying by its reciprocal. This is a crucial step in simplifying complex fractions.

Our simplified numerator is $\frac{15x + 66}{(x+4)(x+5)}$, and our simplified denominator is $\frac{5x + 22}{(x + 4)(x + 5)}$. So, dividing the numerator by the denominator is the same as multiplying $\frac{15x + 66}{(x+4)(x+5)}$ by the reciprocal of $\frac{5x + 22}{(x + 4)(x + 5)}$, which is $\frac{(x + 4)(x + 5)}{5x + 22}$.

Now, we have the expression: $\frac{15x + 66}{(x+4)(x+5)} \cdot \frac{(x + 4)(x + 5)}{5x + 22}$. Before we multiply, let’s look for opportunities to cancel out common factors. We see that (x + 4)(x + 5) appears in both the numerator and the denominator, so we can cancel them out! This leaves us with $\frac{15x + 66}{5x + 22}$. We are in the final stretch, guys! We've transformed our complex fraction into a much simpler fraction. Now, let's see if we can simplify this even further.

Step 4: Further Simplification by Factoring (If Possible)

We've arrived at the fraction $\frac{15x + 66}{5x + 22}$. To see if we can simplify further, we need to factor both the numerator and the denominator, if possible. Factoring helps us identify common factors that we can cancel out.

Let's start with the numerator, 15x + 66. We look for the greatest common factor (GCF) of 15 and 66. The GCF is 3. Factoring out 3, we get 3(5x + 22). Now let's look at the denominator, 5x + 22. Can we factor this? In this case, there's no common factor other than 1, so it remains as 5x + 22.

Now our fraction looks like $\frac{3(5x + 22)}{5x + 22}$. Do you see any common factors now? Yes! We have (5x + 22) in both the numerator and the denominator. We can cancel these out, leaving us with just 3. So, the simplified form of our complex fraction is 3. Wow, guys, that’s quite a transformation from where we started!

Conclusion

And there you have it! We've successfully simplified the complex fraction $\frac{\frac{6}{x+4}+\frac{9}{x+5}}{\frac{5 x+22}{x^2+9 x+20}}$ to 3. Remember, the key to simplifying complex fractions is to break them down into smaller, manageable steps. First, simplify the numerator and denominator separately. Then, divide the simplified numerator by the simplified denominator (which is the same as multiplying by the reciprocal). Finally, look for opportunities to factor and cancel out common factors. With practice, guys, you'll become complex fraction masters! Keep practicing, and you'll be simplifying complex fractions like a pro in no time.