Question

Simplify the complex rational expression by using the LCD:

$\frac{\frac{4y}{y+5}+\frac{2}{y+6}}{\frac{3y}{y^2+11y+30}}$

Short answer

The simplified expression is$\frac{\left(2y^2+13y+5\right)}{3y}$.

Step-by-step solution

Step 1. Given Information.

We have:

$\frac{\frac{4y}{y+5}+\frac{2}{y+6}}{\frac{3y}{y^2+11y+30}}$

Step 2. Find the LCD of all fractions in complex rational expression. 

The least common divisor is $\left(y+5\right)\left(y+6\right)$.

Multiply the numerator and denominator by the LCD:

$=\frac{\left(y+5\right)\left(y+6\right)\left(\frac{4y}{y+5}+\frac{2}{y+6}\right)}{\left(y+5\right)\left(y+6\right)\left(\frac{3y}{y^2+11y+30}\right)}$

Step 3. Simplify the expression.

Further, calculate:

$$\begin{gathered} =\frac{\left(y+5\right)\left(y+6\right)\left(\frac{4y}{\left(y+5\right)}+\frac{2}{\left(y+6\right)}\right)}{\left(y+5\right)\left(y+6\right)\left(\frac{3y}{y^2+5y+6y+30}\right)} \\ =\frac{\left(y+5\right)\left(y+6\right)\left(\frac{4y\left(y+6\right)+2\left(y+5\right)}{\left(y+5\right)\left(y+6\right)}\right)}{\left(y+5\right)\left(y+6\right)\left(\frac{3y}{y\left(y+5\right)+6\left(y+5\right)}\right)} \\ =\frac{\bcancel{\left(y+5\right)\left(y+6\right)}\left(\frac{4y^2+24y+4y+10}{\bcancel{\left(y+5\right)\left(y+6\right)}}\right)}{\bcancel{\left(y+5\right)\left(y+6\right)}\left(\frac{3y}{\bcancel{\left(y+5\right)\left(y+6\right)}}\right)} \\ =\frac{4y^2+26y+10}{3y} \\ =\frac{2\left(2y^2+13y+5\right)}{3y} \end{gathered}$$