Jay,

Thanks for your solution. Appreciate it.

vinni.k wrote:In a school election, Joan and Peter were the only candidates for class president. Only students in the junior and senior class were allowed to vote and all of them voted for exactly one of the two candidates. Joan received 390 of the votes cast by seniors and Peter received 336 of the votes cast by juniors. How many votes did Joan receive?

(1) Joan received 40% of the votes cast by seniors

(2) Peter received 60% of the votes cast by juniors

\[\left. \begin{gathered}
{\text{seniors}}\,\,\,\left\{ \begin{gathered}
\,390\,\,\,\, \to \,\,\,\,{\text{Joan}} \hfill \\
S - 390\,\,\,\, \to \,\,\,\,{\text{Peter}} \hfill \\
\end{gathered} \right. \hfill \\
{\text{juniors}}\,\,\,\left\{ \begin{gathered}
\,J - 336\,\,\,\, \to \,\,{\text{Joan}} \hfill \\
\,336\,\,\,\, \to \,\,{\text{Peter}} \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\,\,\,\,\, \hfill \\
\end{gathered} \right\}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,? = 390 + \left( {J - 336} \right)\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{\,\,\,\,? = J\,\,}\]
\[\left( 1 \right)\,\,\,\,S - 390 = \frac{6}{{10}}\left( S \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,S\,\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,J\,\,{\text{bifurcates}}\,\,{\text{trivially}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.\]
\[\left( 2 \right)\,\,\,J - 336 = \frac{4}{{10}}\left( J \right)\,\,\,\, \Rightarrow \,\,\,\,\,J\,\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.\]

This solution follows the notations and rationale taught in the GMATH method.

Regards,
Fabio.