Which Expression Is Equivalent To The Following Complex Fraction?$\[ \frac{\frac{2}{x}-\frac{4}{y}}{\frac{-5}{y}+\frac{3}{x}} \\]A. \[$\frac{3y+5x}{2(y-2x)}\$\]B. \[$\frac{2(y-2x)}{3y-5x}\$\]C.
Introduction
When dealing with complex fractions, it can be challenging to determine which expression is equivalent to the given fraction. In this article, we will explore the process of simplifying complex fractions and provide step-by-step instructions on how to find the equivalent expression.
Understanding Complex Fractions
A complex fraction is a fraction that contains one or more fractions in its numerator or denominator. In the given problem, we have a complex fraction with two fractions in the numerator and two fractions in the denominator.
{ \frac{\frac{2}{x}-\frac{4}{y}}{\frac{-5}{y}+\frac{3}{x}} \}
To simplify this complex fraction, we need to find a common denominator for the fractions in the numerator and the denominator.
Finding a Common Denominator
To find a common denominator for the fractions in the numerator, we need to multiply the denominators of the two fractions. The denominators are x and y, so the common denominator is xy.
Similarly, to find a common denominator for the fractions in the denominator, we need to multiply the denominators of the two fractions. The denominators are y and x, so the common denominator is also xy.
Simplifying the Complex Fraction
Now that we have a common denominator, we can simplify the complex fraction by multiplying the numerator and the denominator by the common denominator.
{ \frac{\frac{2}{x}-\frac{4}{y}}{\frac{-5}{y}+\frac{3}{x}} = \frac{\frac{2y}{xy}-\frac{4x}{xy}}{\frac{-5x}{xy}+\frac{3y}{xy}} \}
Combining Like Terms
Now that we have simplified the complex fraction, we can combine like terms in the numerator and the denominator.
{ \frac{\frac{2y}{xy}-\frac{4x}{xy}}{\frac{-5x}{xy}+\frac{3y}{xy}} = \frac{\frac{2y-4x}{xy}}{\frac{-5x+3y}{xy}} \}
Canceling Out the Common Factor
Now that we have combined like terms, we can cancel out the common factor in the numerator and the denominator.
{ \frac{\frac{2y-4x}{xy}}{\frac{-5x+3y}{xy}} = \frac{2y-4x}{-5x+3y} \}
Simplifying the Expression
Now that we have canceled out the common factor, we can simplify the expression by factoring out the greatest common factor (GCF) of the numerator and the denominator.
{ \frac{2y-4x}{-5x+3y} = \frac{2(y-2x)}{-5x+3y} \}
Multiplying the Numerator and the Denominator by -1
To simplify the expression further, we can multiply the numerator and the denominator by -1.
{ \frac{2(y-2x)}{-5x+3y} = \frac{-2(y-2x)}{5x-3y} \}
Factoring Out the GCF
Now that we have multiplied the numerator and the denominator by -1, we can factor out the GCF of the numerator and the denominator.
{ \frac{-2(y-2x)}{5x-3y} = \frac{-2(y-2x)}{3y-5x} \}
Conclusion
In conclusion, the expression equivalent to the given complex fraction is . This expression can be simplified further by factoring out the GCF of the numerator and the denominator.
Discussion
The given problem is a complex fraction with two fractions in the numerator and two fractions in the denominator. To simplify this complex fraction, we need to find a common denominator for the fractions in the numerator and the denominator. We can then simplify the complex fraction by multiplying the numerator and the denominator by the common denominator.
The final expression equivalent to the given complex fraction is . This expression can be simplified further by factoring out the GCF of the numerator and the denominator.
Final Answer
The final answer is .
References
- [1] "Complex Fractions." Math Open Reference, mathopenref.com/complexfractions.html.
- [2] "Simplifying Complex Fractions." Khan Academy, khanacademy.org/math/algebra/x2-2-advanced-algebra/x2-2-2-advanced-algebra/x2-2-2-2-advanced-algebra/x2-2-2-2-1-advanced-algebra/x2-2-2-2-1-1-advanced-algebra/x2-2-2-2-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-
Introduction
In our previous article, we explored the process of simplifying complex fractions and provided step-by-step instructions on how to find the equivalent expression. In this article, we will answer some frequently asked questions about 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: How do I simplify a complex fraction?
A: To simplify a complex fraction, you need to find a common denominator for the fractions in the numerator and the denominator. You can then multiply the numerator and the denominator by the common denominator to simplify the fraction.
Q: What is the common denominator?
A: The common denominator is the least common multiple (LCM) of the denominators of the fractions in the numerator and the denominator.
Q: How do I find the common denominator?
A: To find the common denominator, you need to list the multiples of each denominator and find the smallest multiple that is common to both.
Q: What if the denominators are not multiples of each other?
A: If the denominators are not multiples of each other, you can find the least common multiple (LCM) by multiplying the denominators together and dividing by their greatest common divisor (GCD).
Q: Can I simplify a complex fraction by canceling out common factors?
A: Yes, you can simplify a complex fraction by canceling out common factors in the numerator and the denominator.
Q: What is the greatest common factor (GCF)?
A: The greatest common factor (GCF) is the largest factor that divides both the numerator and the denominator.
Q: How do I find the GCF?
A: To find the GCF, you need to list the factors of the numerator and the denominator and find the largest factor that is common to both.
Q: Can I simplify a complex fraction by multiplying the numerator and the denominator by a constant?
A: Yes, you can simplify a complex fraction by multiplying the numerator and the denominator by a constant.
Q: What is the constant?
A: The constant is a number that is multiplied by the numerator and the denominator to simplify the fraction.
Q: How do I choose the constant?
A: To choose the constant, you need to find a number that will cancel out the common factors in the numerator and the denominator.
Q: Can I simplify a complex fraction by using a calculator?
A: Yes, you can simplify a complex fraction by using a calculator.
Q: What are some common mistakes to avoid when simplifying complex fractions?
A: Some common mistakes to avoid when simplifying complex fractions include:
- Not finding the common denominator
- Not canceling out common factors
- Not multiplying the numerator and the denominator by the correct constant
- Not using a calculator to simplify the fraction
Conclusion
In conclusion, simplifying complex fractions can be a challenging task, but with the right steps and techniques, you can simplify even the most complex fractions. Remember to find the common denominator, cancel out common factors, and multiply the numerator and the denominator by the correct constant. With practice and patience, you will become a pro at simplifying complex fractions.
Final Answer
The final answer is .
References
- [1] "Complex Fractions." Math Open Reference, mathopenref.com/complexfractions.html.
- [2] "Simplifying Complex Fractions." Khan Academy, khanacademy.org/math/algebra/x2-2-advanced-algebra/x2-2-2-advanced-algebra/x2-2-2-2-advanced-algebra/x2-2-2-2-1-advanced-algebra/x2-2-2-2-1-1-advanced-algebra/x2-2-2-2-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-advanced-algebra/x2-2-2-2-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-1-