Simplify rational expression How do I simplfy this expression?
$$\dfrac{\frac{x}{2}+\frac{y}{3}}{6x+4y}$$
I tried to use the following rule $\dfrac{\frac{a}{b}}{\frac{c}{d}}=\frac{a}{b}\cdot \frac{d}{c}$
But I did not get the right result.
Thanks!!
 A: $$\frac{\frac{x}{2}+\frac{y}{3}}{6x+4y}=\frac{6 \cdot \left ( \frac{x}{2}+\frac{y}{3}\right ) }{6 \cdot (6x+4y)}=\frac{3x+2y}{6 \cdot 2 \cdot (3x+2y)}=\frac{1}{12}$$
A: $$\frac{\frac{x}{2}+\frac{y}{3}}{6x+4y}$$
Start by simplifying the numerator. Specifically, add the two fractions.
$$\frac{\frac{x}{2}+\frac{y}{3}}{6x+4y}=\frac{\frac{3x}{6}+\frac{2y}{6}}{6x+4y}=\frac{\frac{3x+2y}{6}}{6x+4y}$$
Then, since the fraction bar means division, you have:
$$\frac{\frac{3x+2y}{6}}{6x+4y}=\frac{3x+2y}{6}\div(6x+4y)$$
And the rest is just the division of two fractions.
$$\frac{3x+2y}{6}\div(6x+4y)=\frac{3x+2y}{6}\times\frac{1}{6x+4y}=\frac{3x+2y}{36x+24y}$$
However, we're not done. We need to factor the numerator and denominator and simplify.
$$\frac{3x+2y}{36x+24y}=\frac{3x+2y}{12(3x+2y)}=\frac{1}{12}$$
A: $$\begin{align}
\frac{\frac{x}{2} + \frac{y}{3}}{6x+4y} &=  \frac{\frac{x}{2} + \frac{y}{3}}{\frac{6x+4y}{1}}\\
&= \frac{\frac{x}{2} + \frac{y}{3}}{\frac{6x+4y}{1}}\cdot\frac{\frac{1}{6x+4y}}{\frac{1}{6x+4y}}\\
&= \frac{\bigg(\frac{x}{2}+\frac{y}{3}\bigg)\cdot\frac{1}{6x+4y}}{\frac{6x+4y}{6x+4y}}\\
&= \frac{\bigg(\frac{x}{2}+\frac{y}{3}\bigg)\cdot\frac{1}{6x+4y}}{1}\\
&= \bigg(\frac{x}{2}+\frac{y}{3}\bigg)\cdot\frac{1}{6x+4y}\\
&= \bigg(\frac{12}{12}\cdot\frac{x}{2}+\frac{12}{12}\cdot\frac{y}{3}\bigg)\cdot\frac{1}{6x+4y}\\
&= \bigg(\frac{1}{12}\cdot\frac{12x}{2}+\frac{1}{12}\cdot\frac{12y}{3}\bigg)\cdot\frac{1}{6x+4y}\\
&= \bigg(\frac{1}{12}\cdot6x+\frac{1}{12}\cdot4x\bigg)\cdot\frac{1}{6x+4y}\\
&= \frac{1}{12}(6x+4y)\cdot\frac{1}{6x+4y}\\
&=\frac{1}{12}\cdot\frac{6x+4y}{6x+4y} \\
&= \frac{1}{12}\cdot 1 \\
&= \frac{1}{12}
\end{align}$$
