At Perry High School, the ratio of students who participate

AAPL wrote:Princeton Review

At Perry High School, the ratio of students who participate in either the band program or the choral program to students who participate in neither program is 3 to 8. If 220 students attend Perry High School, how many of them do NOT participate in either program?

A. 40
B. 60
C. 100
D. 160
E. 180

Trivial application of the k technique and Venn diagrams ("overlapping sets"):

\[\frac{{B \cup C}}{{{\text{none}}}} = \frac{3}{8}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \begin{gathered}
\,B \cup C\,\,{\text{ = }}\,\,{\text{3}}k \hfill \\
\,{\text{none}}\,\, = \,\,8k \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,(k > 0)\,\,\,\,\,\,\,\,\,\left( * \right)\]
\[? = {\text{none}}\,\, = \,\,8k\]
\[220 = {\text{Total}}\,\,{\text{ = }}\,\,B \cup C\,\,{\text{ + }}\,\,{\text{none}}\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,220\,\, = \,\,\,11k\]
\[?\,\,\, = \,\,\,\left( {\frac{8}{{11}}} \right) \cdot 11k\,\,\, = \,\,\,\left( {\frac{8}{{11}}} \right) \cdot 220\,\,\, = \,\,\,160\]

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

Regards,
fskilnik.