Which expression is equivalent to the following complex fraction?
[tex]$\frac{\frac{2}{x}-\frac{4}{y}}{\frac{-5}{y}+\frac{3}{x}}$[/tex]
A. [tex]$\frac{3 y+5 x}{2(y-2 x)}$[/tex]
B. [tex]$\frac{2(y-2 x)}{3 y-5 x}$[/tex]
C. [tex]$\frac{2(y-2 x)(3 y-5 x)}{x^2 y^2}$[/tex]
D. [tex]$\frac{x^2 y^2}{2(y-2 x)(3 y-5 x)}$[/tex]
Answer (2)
Multiply the numerator and denominator by x y to eliminate the inner fractions.
Simplify the expression: x y ( y − 5 + x 3 ) x y ( x 2 − y 4 ) = − 5 x + 3 y 2 y − 4 x .
Factor out a 2 from the numerator: 3 y − 5 x 2 ( y − 2 x ) .
The equivalent expression is 3 y − 5 x 2 ( y − 2 x ) .
Explanation
Understanding the Problem We are given the complex fraction y − 5 + x 3 x 2 − y 4 and we need to find an equivalent expression from the given options.
Simplifying the Fraction To simplify the complex fraction, we multiply both the numerator and the denominator by x y :
y − 5 + x 3 x 2 − y 4 = x y ( y − 5 + x 3 ) x y ( x 2 − y 4 ) = x y ⋅ y − 5 + x y ⋅ x 3 x y ⋅ x 2 − x y ⋅ y 4 This simplifies to − 5 x + 3 y 2 y − 4 x = 3 y − 5 x 2 ( y − 2 x )
Finding the Equivalent Expression Comparing the simplified expression with the given options, we find that the equivalent expression is 3 y − 5 x 2 ( y − 2 x )
Final Answer Therefore, the expression equivalent to the given complex fraction is 3 y − 5 x 2 ( y − 2 x ) .
Examples
Complex fractions might seem abstract, but they appear in various real-world scenarios. For example, when calculating the combined resistance of parallel circuits in electronics, you often encounter complex fractions. If you have two resistors with resistances that are expressed as fractions involving variables, the total resistance of the parallel circuit can be represented by a complex fraction. Simplifying this complex fraction helps you determine the overall resistance more easily, which is crucial for designing and analyzing electronic circuits effectively. Understanding how to manipulate these fractions allows engineers and technicians to work with circuit parameters more efficiently.
The expression equivalent to the given complex fraction is 3 y − 5 x 2 ( y − 2 x ) , which corresponds to option B. The process involved multiplying by the common denominator to eliminate inner fractions and then simplifying. Finally, we factored the numerator to match the answer choices.
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