BTGmoderatorLU wrote:Source: Official Guide
At a certain picnic, each of the guests was served either a single scoop or a double scoop ice-cream. How many of the guests were served a double scoop ice-cream?
1. At the picnic, 60 percent of the guests were served a double scoop ice-cream.
2. A total of 120 scoops of ice-cream were served to all the guests at the picnic.
$$5\,G\,\,{\rm{guests}}\,\,\,\left\{ \matrix{
\,x\,\,{\rm{guests}}\,\,\,::\,\,{\rm{one}}\,\,{\rm{2 - balls}}\,\,{\rm{ice - cream}}\,\,{\rm{each}} \hfill \cr
\,\left( {5G - x} \right)\,\,{\rm{guests}}\,\,\,::\,\,{\rm{one}}\,\,1{\rm{ - ball}}\,\,{\rm{ice - cream}}\,\,{\rm{each}} \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\,\,\,\,\,;\,\,\,\,\,\,\,\,\,\,\,? = x\,\,$$
$$\left( 1 \right)\,\,\,\,\,{x \over {5G}} = \,{3 \over 5}\,\,\,\,\, \Rightarrow \,\,\,\,x = 3G\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,G} \right) = \left( {3,1} \right)\,\,\,\,\,\left[ {5G = 5} \right]\,\,\,\,\,\, \Rightarrow \,\,\,\,? = 3\,\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,G} \right) = \left( {6,2} \right)\,\,\,\,\,\left[ {5G = 10} \right]\,\,\,\,\,\, \Rightarrow \,\,\,\,? = 6\,\, \hfill \cr} \right.$$
$$\left( 2 \right)\,\,x \cdot 2 + \left( {5G - x} \right) \cdot 1 = 120\,\,\,\,\left[ {{\rm{balls}}\,\,{\rm{of}}\,\,{\rm{ice - cream}}} \right]$$
$$\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {x,G} \right) = \left( {10,22} \right)\,\,\,\, \Rightarrow \,\,\,\,? = 10\,\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {x,G} \right) = \left( {20,20} \right)\,\,\,\, \Rightarrow \,\,\,\,? = 20\,\, \hfill \cr} \right.$$
$$\left( {1 + 2} \right)\,\,\,\left\{ \matrix{
\,x = 3G \hfill \cr
\,x \cdot 2 + \left( {5G - x} \right) \cdot 1 = 120 \hfill \cr} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,G\,\,\,{\rm{unique}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,x\,\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}{\rm{.}}$$
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
