Bunuel wrote:
If machine J, working alone at its constant rate, takes 2 minutes to wrap 60 pieces of candy, how many minutes does it take machine K, working alone at its constant rate, to wrap 120 pieces of candy?
(1) Machine K, working alone at its constant rate, takes more than 5 minutes to wrap 60 pieces of candy.
(2) Machines J and K, working together at their respective constant rates, take 1 minute and 30 seconds to wrap 60 pieces of candy.
\(J\,\,\, - \,\,\,60\,\,{\text{candies}}\,\,\, - \,\,\,2\,\,{\text{min}}\)
\(K\,\,\, - \,\,60\,\,{\text{candies}}\,\,\left( * \right)\,\,\, - \,\,\,?\,\,\min\)
\(\left( * \right)\,\,{\text{Our}}\,\,{\text{FOCUS}}\,\,{\text{is}}\,\,{\text{half}}\,\,{\text{the}}\,\,{\text{official}}\,\,{\text{one}}!\)
\(\left( 1 \right)\,\,\left\{ \begin{gathered}\\
\,{\text{Take}}\,\,? = 5.5\min \hfill \\\\
\,{\text{Take}}\,\,? = 6\min \hfill \\ \\
\end{gathered} \right.\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{INSUFF}}.\)
\(\left( 2 \right)\,\,\frac{1}{{1\frac{1}{2}}} = \frac{1}{2} + \frac{1}{?}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\frac{1}{?}\,\,\,{\text{unique}}\,\,\,\, \Rightarrow \,\,\,\,?\,\,\,{\text{unique}}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\text{SUFF}}.\)
(This is the "work/rate shortcut".)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.