tim415 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

\(\left\{ \begin{gathered}

\,m \geqslant 1\,\,\operatorname{int} \,\,\,12{\text{oz - units}}\,\,,\,\,52\,{\text{cents}}\,{\text{each}} \hfill \\

\,n \geqslant 1\,\,\operatorname{int} \,\,\,\,16{\text{oz - units}}\,\,,\,\,58\,{\text{cents}}\,{\text{each}}\,\,\, \hfill \\

\end{gathered} \right.\,\,\,\,\left( * \right)\)

\(? = m\)

\(\left( 1 \right)\,\,m + n = 9\,\,\,\left\{ \begin{gathered}

\,{\text{Take}}\,\,\left( {m,n} \right) = \left( {1,8} \right)\,\,\,\, \Rightarrow \,\,\,? = 1\,\, \hfill \\

\,{\text{Take}}\,\,\left( {m,n} \right) = \left( {2,7} \right)\,\,\,\, \Rightarrow \,\,\,? = 2\,\, \hfill \\

\end{gathered} \right.\)

Money unit will be CENTS. (

All amounts in cents are integers!)

\(\left( 2 \right)\,\,52m + 58n = 492\,\,\,\,\,\,\mathop \Rightarrow \limits^{:\,\,2} \,\,\,\,\,26m + 29n = 246\,\,\,\)

\(\left[ {29n\,\,\mathop = \limits^{\left( * \right)} \,} \right]\,\,{\text{positive}}\,\,{\text{multiple}}\,\,{\text{of}}\,\,29\,\, = \,\,\,246 - 26m = 2\left( {123 - 13m} \right)\,\,\,\,\,\mathop \Rightarrow \limits^{{\text{GCF}}\,\left( {2,29} \right)\,\, = \,\,1} \,\,\,123 - 13m\,\,{\text{is}}\,\,{\text{a}}\,\,{\text{positive}}\,\,{\text{multiple}}\,\,{\text{of}}\,\,29\)

\(\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.

_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)

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