Abhi077 wrote:

Stark and Roggers are both baking pies. stark can make M pies per hour; Roggers can make J pies per hour. stark begins baking pies alone, but later Roggers joins him. If 100 pies are finished x hours after Roggers joins stark, how many pies has stark already made by the time Roggers begins to help?

(1) Working together stark and Roggers can bake 25 pies per hour.

(2) Once Roggers arrives, Stark and Roggers complete the remainder of the pies in 2 hours.

\(S:\,\,\,{{M\,\,{\rm{pies}}} \over {1\,\,{\rm{h}}}}\,\,\,\,\,;\,\,\,\,\,\,y\,{\rm{h}}\,\,\,\left( {S\,\,{\rm{alone}}} \right)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{UNITS}}\,\,{\rm{CONTROL}}} \,\,\,\,\,\,? = yM\,\,\,\,\left[ {{\rm{pies}}} \right]\)

\(R:\,\,\,{{J\,\,{\rm{pies}}} \over {1\,\,{\rm{h}}}}\,\,\,\,\,;\,\,\,\,\,\,x\,{\rm{h}}\,\,\,\left( {S \cup R} \right)\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{{\rm{UNITS}}\,\,{\rm{CONTROL}}} \,\,\,\,\,\,yM + \left( {M + J} \right)x = 100\,\,\,\,\left( * \right)\,\,\,\,\left[ {{\rm{pies}}} \right]\,\)

\(\left( 1 \right)\,\,\,M + J = 25\,\,\,{\rm{and}}\,\,\,\left( * \right)\,\,\,\,\left\{ \matrix{

\,{\rm{Take}}\,\,\left( {M,y,J,x} \right) = \left( {5,15,20,1} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{75}}\,\, \hfill \cr

\,{\rm{Take}}\,\,\left( {M,y,J,x} \right) = \left( {10,5,15,2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{50}}\,\,\,\, \hfill \cr} \right.\)

\(\left( 2 \right)\,\,\,x = 2\,\,\,{\rm{and}}\,\,\,\left( * \right)\,\,\,\,\,\left\{ \matrix{

\,\left( {{\rm{Re}}} \right){\rm{Take}}\,\,\left( {M,y,J,x} \right) = \left( {10,5,15,2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{50}}\,\, \hfill \cr

\,{\rm{Take}}\,\,\left( {M,y,J,x} \right) = \left( {10,4,20,2} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{?}}\,\,{\rm{ = }}\,\,{\rm{40}}\,\, \hfill \cr} \right.\)

\(\left( {1 + 2} \right)\,\,\,\left\{ \matrix{

\,M + J = 25 \hfill \cr

\,x = 2 \hfill \cr} \right.\,\,\,\,\,\mathop \Rightarrow \limits^{\left( * \right)} \,\,\,\,\,yM\,\,\,{\rm{unique}}\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{SUFF}}{\rm{.}}\)

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

Regards,

Fabio.

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Fabio Skilnik ::

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