Simplify the complex fraction to eliminate fractions from the numerators and denominators. Assume there are no zero denominators. $$\frac { x + \frac { x } { 5 } } { 5 - \frac { x } { 2 } }$$
A) $\frac { 12 x } { 5 ( 15 - x ) }$
B) $\frac { 13 x } { 5 ( 10 - x ) }$
C) $\frac { 12 x } { 5 ( 10 - x ) }$
D) $\frac { 13 x } { 5 ( 15 - x ) }$
E) $\frac { 8 x } { 5 ( 10 - x ) }$

Question

Quizplus · Accepted Answer

We first need to simplify the numerator and denominator separately. 
For the numerator, we can rewrite it as a common fraction:
\begin{align*}
\frac{x+\frac{x}{5}}{5-\frac{x}{2}} &= \frac{\frac{5x}{5}+\frac{x}{5}}{5-\frac{x}{2}} \
&= \frac{\frac{6x}{5}}{5-\frac{x}{2}} \
&= \frac{\frac{12x}{10}}{5-\frac{x}{2}} \
&= \frac{\frac{12x}{10}}{\frac{10}{2}-\frac{x}{2}} \
&= \frac{\frac{12x}{10}}{\frac{10-x}{2}} \
&= \frac{\frac{6x}{5}}{\frac{10-x}{2}} \
&= \frac{6x}{5} \cdot \frac{2}{10-x} \
&= \frac{12x}{(10-x)5}
\end{align*}
For the denominator, we simply multiply by the conjugate:
\begin{align*}
5-\frac{x}{2} &= \frac{10}{2}-\frac{x}{2} \
&= \frac{10-x}{2}
\end{align*}
Therefore, the original expression simplifies to:
$$\frac{x+\frac{x}{5}}{5-\frac{x}{2}} = \frac{12x}{(10-x)5} = \boxed{\textbf{(C)}}$$

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