Bunuel wrote:

At a certain picnic, each of the guests was served either a single scoop or a double scoop of ice cream. How many of the guests were served a double scoop of ice cream?

(1) At the picnic, 60 percent of the guests were served a double scoop of 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.

_________________

Fabio Skilnik :: https://www.GMATH.net (Math for the GMAT)

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