Division of Complex Fractions

Oct 13, 2025 by miodeya.com 47 views

Introduction

When dealing with complex fractions, it's essential to understand the concept of division and how it applies to these types of expressions. In this article, we will explore the process of dividing two complex fractions, specifically $\frac{y^2-y-6}{6y+66} \div \frac{-y^2-y+12}{12y+132}$ . We will break down the steps involved in this process and provide a clear explanation of the mathematical concepts used.

Understanding Complex Fractions

A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given expression, both the numerator and denominator contain fractions. To simplify this expression, we need to understand the concept of division and how it applies to complex fractions.

Division of Complex Fractions

When dividing two complex fractions, we can use the following steps:

Invert and Multiply: To divide two fractions, we can invert the second fraction (i.e., flip the numerator and denominator) and multiply the two fractions.
Simplify the Expression: After inverting and multiplying, we can simplify the resulting expression by combining like terms and canceling out any common factors.

Step 1: Invert and Multiply

To divide the given complex fractions, we will invert the second fraction and multiply the two fractions.

$\frac{y^2-y-6}{6y+66} \div \frac{-y^2-y+12}{12y+132} = \frac{y^2-y-6}{6y+66} \times \frac{12y+132}{-y^2-y+12}$

Step 2: Simplify the Expression

Now that we have multiplied the two fractions, we can simplify the resulting expression by combining like terms and canceling out any common factors.

$\frac{y^2-y-6}{6y+66} \times \frac{12y+132}{-y^2-y+12} = \frac{(y^2-y-6)(12y+132)}{(6y+66)(-y^2-y+12)}$

Step 3: Factor the Numerator and Denominator

To simplify the expression further, we can factor the numerator and denominator.

$\frac{(y^2-y-6)(12y+132)}{(6y+66)(-y^2-y+12)} = \frac{(y-3)(y+2)(12y+132)}{(6y+66)(-y-3)(y-4)}$

Step 4: Cancel Out Common Factors

Now that we have factored the numerator and denominator, we can cancel out any common factors.

$\frac{(y-3)(y+2)(12y+132)}{(6y+66)(-y-3)(y-4)} = \frac{(y-3)(y+2)(12)(y+11)}{(6)(y+11)(y-4)}$

Step 5: Simplify the Expression

Finally, we can simplify the expression by canceling out any remaining common factors.

$\frac{(y-3)(y+2)(12)(y+11)}{(6)(y+11)(y-4)} = \frac{(y-3)(y+2)(2)}{(y-4)}$

Conclusion

In this article, we have explored the process of dividing two complex fractions, specifically $\frac{y^2-y-6}{6y+66} \div \frac{-y^2-y+12}{12y+132}$ . We have broken down the steps involved in this process and provided a clear explanation of the mathematical concepts used. By following these steps, we can simplify complex fractions and arrive at a final expression.

Final Answer

The final answer to the given problem is $\frac{(y-3)(y+2)(2)}{(y-4)}$ .

Discussion

The given problem involves dividing two complex fractions, which can be a challenging task. However, by following the steps outlined in this article, we can simplify the expression and arrive at a final answer. This problem requires a strong understanding of mathematical concepts, including division, factoring, and simplifying expressions.

References

[1] "Algebra" by Michael Artin
[2] "Calculus" by Michael Spivak
[3] "Mathematics for Computer Science" by Eric Lehman

Note: The references provided are for general information purposes only and are not directly related to the given problem.

Introduction

In our previous article, we explored the process of dividing two complex fractions, specifically $\frac{y^2-y-6}{6y+66} \div \frac{-y^2-y+12}{12y+132}$ . We broke down the steps involved in this process and provided a clear explanation of the mathematical concepts used. In this article, we will answer some of the most frequently asked questions related to this topic.

Q&A

Q: What is the first step in dividing two complex fractions?

A: The first step in dividing two complex fractions is to invert the second fraction (i.e., flip the numerator and denominator) and multiply the two fractions.

Q: How do I simplify the expression after inverting and multiplying?

A: After inverting and multiplying, you can simplify the resulting expression by combining like terms and canceling out any common factors.

Q: What is the difference between inverting and multiplying and simplifying the expression?

A: Inverting and multiplying involves flipping the second fraction and multiplying the two fractions, while simplifying the expression involves combining like terms and canceling out any common factors.

Q: Can I simplify the expression before inverting and multiplying?

A: No, you cannot simplify the expression before inverting and multiplying. The order of operations requires that you invert and multiply before simplifying the expression.

Q: What is the final answer to the given problem?

A: The final answer to the given problem is $\frac{(y-3)(y+2)(2)}{(y-4)}$ .

Q: How do I know if I have simplified the expression correctly?

A: To ensure that you have simplified the expression correctly, you can check your work by plugging in values for the variables and verifying that the expression is true.

Q: Can I use this method to divide any two complex fractions?

A: Yes, you can use this method to divide any two complex fractions. However, you may need to adjust the steps depending on the specific fractions you are working with.

Q: What are some common mistakes to avoid when dividing complex fractions?

A: Some common mistakes to avoid when dividing complex fractions include:

Failing to invert the second fraction
Failing to multiply the two fractions
Failing to simplify the expression
Not canceling out common factors

Conclusion

In this article, we have answered some of the most frequently asked questions related to dividing complex fractions. We have provided a clear explanation of the mathematical concepts used and have highlighted some common mistakes to avoid. By following the steps outlined in this article, you can simplify complex fractions and arrive at a final expression.