Nevernevergiveup wrote:
In the first year of a pyramid scheme, John convinced y of his friends to pay 30 dollars each to join a particular website that he created. Each of those y friends then convinced another y people to pay 15 dollars each to join the same website. If no one else joined the website that year and each person joined only once, what was the value of y?
A. The revenue for the website that year was $36,000.
B. The first y friends accounted for 1/25 of the total revenue for the website that year.
\({\rm{Total}}\,\,\,{\rm{ = }}\,\,\,30y + 15{y^2}\,\,\,\,\,\left( \$ \right)\)
\(? = y\,\,\,\,\,\left( {y \ge 1\,\,{\mathop{\rm int}} } \right)\)
\(\left( 1 \right)\,\,\,\,15{y^2} + 30y - 36000 = 0\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{roots}}} \,\,\,\,\,{y_1} \cdot {y_2} = - {{36000} \over {15}} < 0\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,y\,\,\left( {{\rm{root}}} \right)\,\,\, > 0\,\,\,\,{\rm{unique}}\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,{\rm{SUFF}}.\,\)
\(\left( 2 \right)\,\,\,25 \cdot 30y\,\, = \,\,30y + 15{y^2}\,\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,15} \,\,\;\,\,\,25 \cdot 2y = {y^2} + 2y\,\,\,\,\,\, \Rightarrow \,\,\,y\left( {y - 48} \right) = 0\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,y\,\,\left( {{\rm{root}}} \right)\,\,\, > 0\,\,\,\,{\rm{unique}}\,\,\,\,\left( { = 48} \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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Fabio Skilnik :: GMATH method creator (Math for the GMAT)