AAPL wrote:Princeton Review
Mr. Jones spends $25 on movie tickets for a party of adults and children. How many children's tickets did he buy?
1) Adult movie tickets cost $3 each and children's tickets cost $2 each.
2) Mr. Jones buys a total of 11 tickets.
OA C
$$\left\{ \matrix{
\,A\,\,{\rm{adults}}\,\,{\rm{,}}\,\,\,{{\rm{p}}_{\rm{A}}} = {\rm{cost}}\,\,\left( {\rm{\$ }} \right)\,\,{\rm{per}}\,\,{\rm{ticket}} \hfill \cr
\,C\,\,{\rm{children}}\,\,{\rm{,}}\,\,\,{{\rm{p}}_{\rm{C}}} = {\rm{cost}}\,\,\left( {\rm{\$ }} \right)\,\,{\rm{per}}\,\,{\rm{ticket}} \hfill \cr} \right.\,\,\,\,\, \Rightarrow \,\,\,\,A \cdot {p_A} + C \cdot {p_C} = 25\,\,\,\,\left( * \right)$$
$$? = C$$
$$\left( 1 \right)\,\,\left( {{p_A},{p_C}} \right) = \left( {3,2} \right)\,\,\,\,:\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {A,C} \right) = \left( {7,2} \right)\,\,\,\, \Rightarrow \,\,\,? = 2\,\,\,\,\,\,\,\left[ {\left( * \right)\,\,7 \cdot 3 + 2 \cdot 2 = 25} \right] \hfill \cr
\,{\rm{Take}}\,\,\left( {A,C} \right) = \left( {5,5} \right)\,\,\,\, \Rightarrow \,\,\,? = 5\,\,\,\,\,\,\,\left[ {\left( * \right)\,\,5 \cdot 3 + 5 \cdot 2 = 25} \right] \hfill \cr} \right.$$
$$\left( 2 \right)\,\,A + C = 11\,\,\,\,:\,\,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {A,C} \right) = \left( {1,10} \right)\,\,{\rm{and}}\,\,\left( {{p_A},{p_C}} \right) = \left( {5,2} \right)\,\,\,\, \Rightarrow \,\,\,? = 10\,\,\,\,\,\,\,\,\left[ {\left( * \right)\,\,1 \cdot 5 + 10 \cdot 2 = 25} \right] \hfill \cr
\,{\rm{Take}}\,\,\left( {A,C} \right) = \left( {2,9} \right)\,\,{\rm{and}}\,\,\left( {{p_A},{p_C}} \right) = \left( {8,1} \right)\,\,\,\, \Rightarrow \,\,\,? = 9\,\,\,\,\,\,\,\,\left[ {\left( * \right)\,\,2 \cdot 8 + 9 \cdot 1 = 25} \right] \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\left\{ \matrix{
\,3A + 2C = 25\,\,\,\,\left( * \right) \hfill \cr
\,A + C = 11 \hfill \cr} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,3\left( {11 - C} \right) + 2C = 25\,\,\,\,\,\, \Rightarrow \,\,\,\,\,C\,\,{\rm{unique}}$$
The correct answer is therefore (C).
We follow the notations and rationale taught in the GMATH method.
Regards,
Fabio.
