Problem Solving

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

OA D.

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.