CAMANISHPARMAR wrote:

A casino pays players with chips that are either turquoise- or violet- colored. If each turquoise- colored chip is worth t dollars, and each violet- colored chip is worth v dollars, where t and v are integers, what is the combined value of four turquoise- colored chips and two violet- colored chips?

(1) The combined value of six turquoise- colored chips and three violet- colored chips is 42 dollars.

(2) The combined value of five turquoise- colored chips and seven violet- colored chips is 53 dollars.

\({\text{chips}}\,\left\{ \begin{gathered}

\,{\text{turqu}}:\$ \,t\,\,{\text{each}} \hfill \\

\,{\text{violet}}:\$ \,v\,\,{\text{each}} \hfill \\

\end{gathered} \right.\,\,\,\,\,\,\left( {t,v \geqslant 1\,\,\,{\text{ints}}\,\,\left( * \right)} \right)\)

\(? = 4t + 2v\,\,\,\left[ \$ \right]\)

\(\left( 1 \right)\,\,6t + 3v = 42\,\,\,\left[ \$ \right]\,\,\,\,\,\mathop \Rightarrow \limits^{:\,3} \,\,2t + v = 14\,\,\,\left[ \$ \right]\,\,\,\,\,\mathop \Rightarrow \limits^{ \cdot \,2} \,\,\,\,\,? = 28\,\,\,\left[ \$ \right]\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.\)

\(\left( 2 \right)\,\,5t + 7v = 53\,\,\,\left[ \$ \right]\,\,\,\,\, \Rightarrow \,\,53 - 7v = 5t\,\,,\,\,t\mathop \geqslant \limits^{\left( * \right)} 1\,\,\operatorname{int} \,\,\,\, \Rightarrow \,\,1\mathop \leqslant \limits^{\left( * \right)} v\,\,\left( {\operatorname{int} } \right) \leqslant 7\,\,\,\left( {**} \right)\)

\(\frac{{\left( {3 + 50} \right) - \left( {5v + 2v} \right)}}{5} = \operatorname{int} \,\,\,\, \Leftrightarrow \,\,\,\,\frac{{3 - 2v}}{5} = \operatorname{int} \,\,\,\,\mathop \Leftrightarrow \limits^{\left( {**} \right)} \,\,\,\,v = 4\,\,\,\,\mathop \Rightarrow \limits^{\left( 2 \right)} \,\,\,\,t = 5\)

\(\left( {t,v} \right)\,\,{\text{unique}}\,\,\,\, \Rightarrow \,\,\,{\text{?}}\,\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}{\text{.}}\)

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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