Bunuel wrote:
Albert spent $100 in total on four different types of scratch tickets, and he did not spend more than $40 on any one type of scratch ticket. How much did he spend on the scratch ticket on which he spent the least?
(1) Albert spent twice as much on C-type tickets as he did on G-type tickets.
(2) Albert spent at least $20 on each type of ticket.
\(B,C,D,G\,\, \ge 0\,\,\,(\$ \,\,{\rm{spent}}\,\,{\rm{in}}\,\,{\rm{each}}\,\,{\rm{type}})\)
\(B + C + D + G = 100\)
\(B,C,D,G\,\,\, \le \,\,40\)
\(? = \min \left\{ {B,C,D,G} \right\}\)
\(\left( 1 \right)\,C = 2G\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {B,C,D,G} \right) = \left( {10,40,30,20} \right)\,\,\, \Rightarrow \,\,\,? = 10 \hfill \cr \\
\,{\rm{Take}}\,\,\left( {B,C,D,G} \right) = \left( {5,40,35,20} \right)\,\,\, \Rightarrow \,\,\,? = 5 \hfill \cr} \right.\)
\(\left( 2 \right)\,\,B,C,D,G\,\,\, \ge \,\,20\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,\left( {B,C,D,G} \right) = \left( {20,25,25,30} \right)\,\,\, \Rightarrow \,\,\,? = 20 \hfill \cr \\
\,{\rm{Take}}\,\,\left( {B,C,D,G} \right) = \left( {21,22,23,34} \right)\,\,\, \Rightarrow \,\,\,? = 21 \hfill \cr} \right.\)
\(\left( {1 + 2} \right)\,\,\left\{ \matrix{\\
G \ge 20\,\,\, \Rightarrow \,\,\,C \ge 40 \hfill \cr \\
B \ge 20 \hfill \cr} \right.\,\,\,\, \Rightarrow \,\,\,D \le 100 - \left( {20 + 20 + 40} \right) = 20\)
\({\rm{Hence}}:\,\,\left( {B,C,D,G} \right) = \left( {20,40,20,20} \right)\,\,\,\,\,\, \Rightarrow \,\,\,? = 20\)
The correct answer is therefore (C).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.