Recently, Frank took a one-hundred question aptitude test where each correct answer scored $5$ points, each incorrect answer scored $-2$ points, and each question not answered scored zero points. Frank answered $80$ questions and scored $232$ points. How many questions did he answer correctly?
Let the number of questions Frank answered correctly be $a$ and the number of questions he answered incorrectly be $b$. We have the two equations \begin{align*}
a+b&=80\\
5a-2b&=232
\end{align*} From the first equation, we have $b=80-a$. Substituting this into the second equation to eliminate $b$, we have $5a-2(80-a)=232\Rightarrow a=56$. Thus, Frank answered $\boxed{56}$ questions correctly.