seofah wrote:

Mary persuaded n friends to donate $500 each to her election campaign, and then each of these n friends persuaded n more people to donate $500 each to Mary’s campaign. If no one donated more than once and if there were no other donations, what was the value of n?

(1) The first n people donated 1/16 of the total amount donated.

(2) The total amount donated was $120,000.

\({\rm{Total}}\,\, = \,\,500 \cdot n + 500 \cdot n \cdot n\,\,\,\,\,\,\left[ \$ \right]\)

\(? = n\)

\(\left( 1 \right)\,\,\,500 \cdot n = {1 \over {16}} \cdot 500 \cdot n \cdot \left( {1 + n} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,\,\left( {500\,n} \right)\,\,\,\left[ {\,n\, \ne \,0\,} \right]} \,\,\,1 = {1 \over {16}} \cdot \left( {1 + n} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,n\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)

\(\left( 2 \right)\,\,\,500 \cdot n\left( {1 + n} \right) = 120000\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,500} \,\,\,\,n\left( {1 + n} \right) = 240\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,n\,\, > 0\,\,\,\,{\rm{unique}}\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}.\)

\(\left( * \right)\,\,15 \cdot 16 = 240\,\,\, \Rightarrow \,\,\,\left\{ \matrix{

\,n\left( {n + 1} \right) < 240\,\,{\rm{for}}\,\,0 < n < 15 \hfill \cr

\,n\left( {n + 1} \right) > 240\,\,{\rm{for}}\,\,n \ge 16 \hfill \cr} \right.\,\,\,\,\,\,\left( {{\rm{Now}}\,\,{\rm{rethink}}\,\,{\rm{without}}\,\,{\rm{knowing}}\,\,{\rm{that}}\,\,n = 15...} \right)\)

This solution follows the notations and rationale taught in the GMATH method.

Regards,

Fabio.

_________________

Fabio Skilnik :: https://GMATH.net (Math for the GMAT) or GMATH.com.br (Portuguese version)

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