A mixture of oil, water, and vinegar contains 10% oil. After all of th

\({\text{Before: x}}\,\,{\text{liters}}\,\,\,\,\left\{ \begin{gathered}\\
{\text{oil}}\,\,:0.1x\,\,{\text{liters}} \hfill \\\\
water\,\,:\,\,k\left( {0.9x} \right)\,\,{\text{liters}} \hfill \\\\
vinegar\,\,:\,\,\,\left( {1 - k} \right)\left( {0.9x} \right)\,\,{\text{liters}}\,\, \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\,\left( {0 < k < 1} \right)\)

\({\text{After}}:\,\,\,{\text{x}}\left( {1 - 0.9k} \right)\,\,\,{\text{liters}}\,\,\,\left\{ \begin{gathered}\\
{\text{oil}}\,\,:0.1x\,\,{\text{liters}} \hfill \\\\
water\,\,:\,\,0\,\,{\text{liters}} \hfill \\\\
vinegar\,\,:\,\,\,\left( {1 - k} \right)\left( {0.9x} \right)\,\,{\text{liters}}\,\, \hfill \\ \\
\end{gathered} \right.\)

\(? = \frac{{0.1x}}{{x\left( {1 - 0.9k} \right)}} = \frac{{0.1}}{{1 - 0.9k}}\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\boxed{? = k}\,\)

\(\left( 1 \right)\,\,\,x = 1\,\,\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,\,k = 0.1 \hfill \\\\
\,{\text{Take}}\,\,\,k = 0.2 \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.\)

\(\left( 2 \right)\,\,\,\,\,\frac{{\left( {1 - k} \right)\left( {0.9x} \right)\,\,}}{x} = \frac{2}{5}\,\,\,\,\, \Rightarrow \,\,\,\,k\,\,{\text{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,{\text{SUFF}}.\,\)


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

Regards,
Fabio.