BTGmoderatorDC wrote:Julie opened a lemonade stand and sold lemonade in two different sizes, a 52-cent (12oz) and a 58-cent (16oz) size. How many 52-cent (12oz) lemonade drinks did Julie sell?
(1) Julie sold a total of 9 lemonades
(2) The total value of the lemonade drinks Julie sold was $4.92
Source: Veritas Prep
$$\left\{ \matrix{
\,m \ge 1\,\,{\mathop{\rm int}} \,\,\,12{\rm{oz - units}}\,\,,\,\,52\,{\rm{cents}}\,{\rm{each}} \hfill \cr
\,n \ge 1\,\,{\mathop{\rm int}} \,\,\,\,16{\rm{oz - units}}\,\,,\,\,58\,{\rm{cents}}\,{\rm{each}}\,\,\, \hfill \cr} \right.\,\,\,\,\left( * \right)$$
$$? = m$$
$$\left( 1 \right)\,\,m + n = 9\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {1,8} \right)\,\,\,\, \Rightarrow \,\,\,? = 1\,\, \hfill \cr
\,{\rm{Take}}\,\,\left( {m,n} \right) = \left( {2,7} \right)\,\,\,\, \Rightarrow \,\,\,? = 2\,\, \hfill \cr} \right.$$
Money unit will be CENTS. (
All amounts in cents are integers!)
$$\left( 2 \right)\,\,52m + 58n = 492\,\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,2} \,\,\,\,\,26m + 29n = 246\,\,\,$$
$$\left[ {29n\,\,\mathop = \limits^{\left( * \right)} \,} \right]\,\,{\rm{positive}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,29\,\, = \,\,\,246 - 26m = 2\left( {123 - 13m} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{GCF}}\,\left( {2,29} \right)\,\, = \,\,1} \,\,\,123 - 13m\,\,{\rm{is}}\,\,{\rm{a}}\,\,{\rm{positive}}\,\,{\rm{multiple}}\,\,{\rm{of}}\,\,29$$
\[\left. \begin{gathered}
m = 1\,\,\,\, \Rightarrow \,\,\,123 - 13 = 110\,\,\,\,\left( {{\text{NO}}} \right) \hfill \\
m = 2\,\,\,\, \Rightarrow \,\,\,123 - 26 = 97\,\,\left[ { = 110 - 13} \right]\,\,\,\,\,\left( {{\text{NO}}} \right) \hfill \\
m = 3\,\,\,\, \Rightarrow \,\,\,123 - 39 = 84\,\,\left[ { = 97 - 13} \right]\,\,\,\,\,\,\left( {{\text{NO}}} \right)\,\,\,\, \hfill \\
m = 4\,\,\,\, \Rightarrow \,\,\,84 - 13 = 71\,\,\,\,\,\,\left( {{\text{NO}}} \right) \hfill \\
\boxed{m = 5}\,\,\,\, \Rightarrow \,\,\,71 - 13 = 58 = 2 \cdot 29\,\,\,\,\,\left( {{\text{YES}}} \right)\,\,\,\,\,\,\,\,\,\,\, \hfill \\
m = 6\,\,\,\, \Rightarrow \,\,\,58 - 13 = 45\,\,\,\,\,\left( {{\text{NO}}} \right) \hfill \\
m = 7\,\,\,\, \Rightarrow \,\,\,45 - 13 = 32\,\,\,\,\,\left( {{\text{NO}}} \right) \hfill \\
m = 8\,\,\,\, \Rightarrow \,\,\,32 - 13 = 19\,\,\,\,\,\left( {{\text{NO}}} \right) \hfill \\
\end{gathered} \right\}\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = 5\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
