Robin split a total of $24,000 between two investments, X and Y. If in

\(\left. \matrix{\\
X:\,\,\,24 \cdot k\,\,\,\,\,\,\,;\,\,\,\,\,\,{i \over {100}}\left( {24k} \right)\,\,\,\,\,{\rm{annual}}\,\,{\rm{simple}}\,\,{\rm{interest}}\,\,\left( * \right) \hfill \cr \\
Y:\,\,\,24 \cdot \left( {1 - k} \right)\,\,\,\,\,;\,\,\,\,\,\,{7 \over {100}}\left[ {24\left( {1 - k} \right)} \right]\,\,\,\,{\rm{annual}}\,\,{\rm{simple}}\,\,{\rm{interest}}\,\,\, \hfill \cr} \right\}\,\,\,\,\,\,\,\,\left[ {{\rm{unit}}:\,\,\,\$ \,\,{{10}^3}} \right]\)

\(\left( * \right)\,\,{\rm{just}}\,\,{\rm{a}}\,\,{\rm{initial}}\,\,{\rm{assumption}}\,\)



\(? = 24\left( {1 - k} \right)\,\,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,\,\,\,\,\,\boxed{? = k}\,\,\,\,\,\,\left( {0 < k < 1} \right)\,\)


\(\left( 1 \right)\,\,\,\frac{i}{{100}}\left( {24k} \right) = \frac{7}{{100}}\left[ {24\left( {1 - k} \right)} \right]\,\,\,\,\,\, \Rightarrow \,\,\,\,\,ik = 7\left( {1 - k} \right)\,\,\,\,\,\left( {**} \right)\)

\(\left\{ \matrix{\\
\,{\rm{Take}}\,\,i = 7\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,? = k = 0.5\,\,\,\, \hfill \cr \\
\,{\rm{Take}}\,\,i = 0.0001\,\,\,\left( { \cong 0} \right)\,\,\,\mathop \Rightarrow \limits^{\left( {**} \right)} \,\,\,? = k \cong 1\,\,\,\, \hfill \cr} \right.\)


\(\left( 2 \right)\,\,i = 5\,\,\,\left( {{\rm{trivial}}\,\,{\rm{bifurcation}}} \right)\,\,\,\left\{ \matrix{\\
\,{\rm{Take}}\,\,k = 0.5\,\,\,\, \hfill \cr \\
\,{\rm{Take}}\,\,k = 0.6\,\,\,\, \hfill \cr} \right.\,\,\)


\(\left( {1 + 2} \right)\,\,\,\left\{ \matrix{\\
\,\,\left( {**} \right)\,\,\,ik = 7\left( {1 - k} \right)\,\,\,\, \hfill \cr \\
\,\,\,i = 5 \hfill \cr} \right.\,\,\, \Rightarrow \,\,\,\,k\,\,{\rm{unique}}\,\,\, \Rightarrow \,\,\,\,{\rm{SUFF}}.\,\,\,\,\,\)



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

Regards,
Fabio.