Simplify the complex fraction.
$\frac{\frac{4}{x+4}+2}{\frac{10}{x+4}-2}$
Answer (1)
Multiply the numerator and denominator by ( x + 4 ) .
Distribute and simplify the numerator and denominator: 10 − 2 x − 8 4 + 2 x + 8 = − 2 x + 2 2 x + 12 .
Factor out common factors: 2 ( − x + 1 ) 2 ( x + 6 ) = − x + 1 x + 6 .
Simplify the fraction: − x − 1 x + 6 .
The simplified complex fraction is − x − 1 x + 6 .
Explanation
Understanding the Problem We are asked to simplify the complex fraction x + 4 10 − 2 x + 4 4 + 2 . A complex fraction is a fraction where the numerator, the denominator, or both contain fractions themselves. To simplify this, we will multiply both the numerator and the denominator by the least common denominator (LCD) of the fractions within the complex fraction. In this case, the LCD is ( x + 4 ) .
Multiplying by the LCD Multiply the numerator and denominator by ( x + 4 ) : x + 4 10 − 2 x + 4 4 + 2 × x + 4 x + 4 = ( x + 4 10 − 2 ) ( x + 4 ) ( x + 4 4 + 2 ) ( x + 4 )
Distributing Distribute ( x + 4 ) in the numerator and the denominator: x + 4 10 ( x + 4 ) − 2 ( x + 4 ) x + 4 4 ( x + 4 ) + 2 ( x + 4 ) = 10 − 2 ( x + 4 ) 4 + 2 ( x + 4 )
Simplifying Simplify the numerator and the denominator by distributing the constants: 10 − 2 x − 8 4 + 2 x + 8 = − 2 x + 2 2 x + 12
Factoring Factor out the common factors in the numerator and the denominator: 2 ( − x + 1 ) 2 ( x + 6 ) = − x + 1 x + 6
Rewriting the fraction We can rewrite the denominator to have a positive leading coefficient: − ( x − 1 ) x + 6 = − x − 1 x + 6
Examples
Complex fractions can appear when calculating rates of change in various contexts, such as determining the average speed of a vehicle over a journey with varying speeds or analyzing financial ratios in business. For instance, if you're calculating the overall return on investment where the initial investment and the profit are themselves expressed as fractions, you'll encounter a complex fraction. Simplifying these fractions makes it easier to understand and compare the underlying rates or ratios, providing clearer insights for decision-making.
