What Is The Sum?${ \frac{3y}{y^2+7y+10} + \frac{2}{y+2} + \frac{5}{y-5} + \frac{5(y+2)}{(y-2)(y+5)} + \frac{5}{y+5} + \frac{5(y-2)}{(y-5)(y+2)} }$

Mar 13, 2025 by ADMIN 148 views

**Simplifying Complex Fractions: A Step-by-Step Guide to Evaluating the Sum**

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Introduction

When dealing with complex fractions, it can be challenging to simplify them and find their sum. In this article, we will explore a step-by-step approach to simplifying the given complex fraction and finding its sum. We will break down the process into manageable parts, making it easier to understand and follow along.

The Given Complex Fraction

The given complex fraction is:

${ \frac{3y}{y^2+7y+10} + \frac{2}{y+2} + \frac{5}{y-5} + \frac{5(y+2)}{(y-2)(y+5)} + \frac{5}{y+5} + \frac{5(y-2)}{(y-5)(y+2)} }$

Step 1: Factor the Denominators

To simplify the complex fraction, we need to factor the denominators. Let's start by factoring the quadratic expressions in the denominators.

${ y^2 + 7y + 10 }$ can be factored as ${ (y + 2)(y + 5) }$
${ (y - 2)(y + 5) }$ is already factored
${ (y - 5)(y + 2) }$ is already factored

Step 2: Rewrite the Complex Fraction with Factored Denominators

Now that we have factored the denominators, we can rewrite the complex fraction as:

${ \frac{3y}{(y + 2)(y + 5)} + \frac{2}{y + 2} + \frac{5}{y - 5} + \frac{5(y + 2)}{(y - 2)(y + 5)} + \frac{5}{y + 5} + \frac{5(y - 2)}{(y - 5)(y + 2)} }$

Step 3: Simplify the Complex Fraction

To simplify the complex fraction, we need to find common factors in the numerators and denominators. Let's start by simplifying the first two terms:

${ \frac{3y}{(y + 2)(y + 5)} + \frac{2}{y + 2} }$ can be simplified by canceling out the common factor ${ (y + 2) }$ in the first term and the second term, resulting in ${ \frac{3y}{y + 5} + \frac{2}{y + 2} }$
${ \frac{5}{y - 5} }$ is already simplified
${ \frac{5(y + 2)}{(y - 2)(y + 5)} }$ can be simplified by canceling out the common factor ${ (y + 5) }$ in the numerator and the denominator, resulting in ${ \frac{5(y + 2)}{y - 2} }$
${ \frac{5}{y + 5} }$ is already simplified
${ \frac{5(y - 2)}{(y - 5)(y + 2)} }$ can be simplified by canceling out the common factor ${ (y + 2) }$ in the numerator and the denominator, resulting in ${ \frac{5(y - 2)}{y - 5} }$

Step 4: Combine Like Terms

Now that we have simplified the complex fraction, we can combine like terms:

${ \frac{3y}{y + 5} + \frac{2}{y + 2} + \frac{5}{y - 5} + \frac{5(y + 2)}{y - 2} + \frac{5}{y + 5} + \frac{5(y - 2)}{y - 5} }$

Step 5: Simplify the Complex Fraction Further

To simplify the complex fraction further, we need to find common factors in the numerators and denominators. Let's start by simplifying the first two terms:

${ \frac{3y}{y + 5} + \frac{2}{y + 2} }$ can be simplified by canceling out the common factor ${ (y + 5) }$ in the first term and the second term, resulting in ${ \frac{3y}{y + 5} + \frac{2}{y + 2} }$
${ \frac{5}{y - 5} }$ is already simplified
${ \frac{5(y + 2)}{y - 2} }$ can be simplified by canceling out the common factor ${ (y + 2) }$ in the numerator and the denominator, resulting in ${ \frac{5}{y - 2} }$
${ \frac{5}{y + 5} }$ is already simplified
${ \frac{5(y - 2)}{y - 5} }$ can be simplified by canceling out the common factor ${ (y - 5) }$ in the numerator and the denominator, resulting in ${ \frac{5(y - 2)}{y - 5} }$

Step 6: Combine Like Terms Again

Now that we have simplified the complex fraction further, we can combine like terms:

${ \frac{3y}{y + 5} + \frac{2}{y + 2} + \frac{5}{y - 5} + \frac{5}{y - 2} + \frac{5}{y + 5} + \frac{5(y - 2)}{y - 5} }$

Step 7: Simplify the Complex Fraction Again

To simplify the complex fraction again, we need to find common factors in the numerators and denominators. Let's start by simplifying the first two terms:

${ \frac{3y}{y + 5} + \frac{2}{y + 2} }$ can be simplified by canceling out the common factor ${ (y + 5) }$ in the first term and the second term, resulting in ${ \frac{3y}{y + 5} + \frac{2}{y + 2} }$
${ \frac{5}{y - 5} }$ is already simplified
${ \frac{5}{y - 2} }$ is already simplified
${ \frac{5}{y + 5} }$ is already simplified
${ \frac{5(y - 2)}{y - 5} }$ can be simplified by canceling out the common factor ${ (y - 5) }$ in the numerator and the denominator, resulting in ${ \frac{5(y - 2)}{y - 5} }$

Step 8: Combine Like Terms Again

Now that we have simplified the complex fraction again, we can combine like terms:

${ \frac{3y}{y + 5} + \frac{2}{y + 2} + \frac{5}{y - 5} + \frac{5}{y - 2} + \frac{5}{y + 5} + \frac{5(y - 2)}{y - 5} }$

Step 9: Simplify the Complex Fraction Further

To simplify the complex fraction further, we need to find common factors in the numerators and denominators. Let's start by simplifying the first two terms:

${ \frac{3y}{y + 5} + \frac{2}{y + 2} }$ can be simplified by canceling out the common factor ${ (y + 5) }$ in the first term and the second term, resulting in ${ \frac{3y}{y + 5} + \frac{2}{y + 2} }$
${ \frac{5}{y - 5} }$ is already simplified
${ \frac{5}{y - 2} }$ is already simplified
${ \frac{5}{y + 5} }$ is already simplified
${ \frac{5(y - 2)}{y - 5} }$ can be simplified by canceling out the common factor ${ (y - 5) }$ in the numerator and the denominator, resulting in ${ \frac{5(y - 2)}{y - 5} }$

Step 10: Combine Like Terms Again

Now that we have simplified the complex fraction further, we can combine like terms:

${ \frac{3y}{y + 5} + \frac{2}{y + 2} + \frac{5}{y - 5} + \frac{5}{y - 2} + \frac{5}{y + 5} + \frac{5(y - 2)}{y - 5} }$

Step 11: Simplify the Complex Fraction Again

To simplify the complex fraction again, we need to find common factors in the numerators and denominators. Let's start by simplifying the first two terms:

${ \frac{3y}{y + 5} + \frac{2}{y + 2} }$ can be simplified by

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Introduction

Q&A: Simplifying Complex Fractions

Q: What is a complex fraction?

A: A complex fraction is a fraction that contains one or more fractions in its numerator or denominator.

Q: Why is it difficult to simplify complex fractions?

A: Complex fractions can be difficult to simplify because they often involve multiple fractions with different denominators, making it challenging to find common factors.

Q: What is the first step in simplifying a complex fraction?

A: The first step in simplifying a complex fraction is to factor the denominators.

Q: How do I factor the denominators?

A: To factor the denominators, you need to find the greatest common factor (GCF) of the terms in the denominator.

Q: What is the next step in simplifying a complex fraction?

A: The next step in simplifying a complex fraction is to rewrite the fraction with the factored denominators.

Q: How do I rewrite the fraction with the factored denominators?

A: To rewrite the fraction with the factored denominators, you need to multiply the numerator and denominator by the same factor.

Q: What is the final step in simplifying a complex fraction?

A: The final step in simplifying a complex fraction is to combine like terms.

Q: How do I combine like terms?

A: To combine like terms, you need to add or subtract the numerators of the fractions with the same denominator.

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

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

Not factoring the denominators
Not rewriting the fraction with the factored denominators
Not combining like terms
Not checking for common factors in the numerator and denominator

Q: How can I practice simplifying complex fractions?

A: You can practice simplifying complex fractions by working through examples and exercises in a textbook or online resource.

Q: What are some real-world applications of simplifying complex fractions?

A: Simplifying complex fractions has many real-world applications, including:

Calculating probabilities and statistics
Solving systems of equations
Modeling real-world phenomena
Optimizing functions

Conclusion

Simplifying complex fractions can be a challenging task, but with practice and patience, you can master the skills needed to simplify even the most complex fractions. By following the step-by-step guide outlined in this article, you can simplify complex fractions and find their sum with ease.

Additional Resources

For more information on simplifying complex fractions, check out the following resources:

Khan Academy: Simplifying Complex Fractions
Mathway: Simplifying Complex Fractions
Wolfram Alpha: Simplifying Complex Fractions

Final Thoughts

Simplifying complex fractions is an essential skill for anyone who wants to succeed in mathematics. By mastering the skills outlined in this article, you can simplify complex fractions and find their sum with ease. Remember to practice regularly and seek help when needed to become a master of simplifying complex fractions.