subh2273 wrote:For the students in a Physical Fitness Class Section #1, the range of their weights is R kg and the highest amount of weight is X kg. For the students in Physical Fitness Class Section #2, the range of their weights is S kg and the highest amount of weight is Y kg. Is the lowest weight of the students in Section #1 greater than the lowest weight of the students in Section #2 ?
(1) R < S
(2) X > Y
\[{\text{range}} = {V_{\max }} - {V_{\min }}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,{V_{\min }} = {V_{\max }} - {\text{range}}\]
Section 1: weights have minimum value (X-R) kg and maximum value X kg.
Section 2: weights have minimum value (Y-S) kg and maximum Y kg.
\[X - R\,\,\,\mathop > \limits^? \,\,Y - S\]
\[\left( 1 \right)\,\,R < S\,\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {X,R,Y,S} \right) = \,\left( {90,10,90,20} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\left( {80\,\,\mathop > \limits^? \,70} \right) \hfill \\
\,{\text{Take}}\,\,\left( {X,R,Y,S} \right) = \,\left( {80,10,90,20} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\,\left( {70\,\,\mathop > \limits^? \,70} \right) \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,X > Y\,\,\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {X,R,Y,S} \right) = \,\left( {90,10,80,10} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\,\left( {80\,\,\mathop > \limits^? \,70} \right) \hfill \\
\,{\text{Take}}\,\,\left( {X,R,Y,S} \right) = \,\left( {90,20,80,10} \right)\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\,\,\,\left( {70\,\,\mathop > \limits^? \,70} \right) \hfill \\
\end{gathered} \right.\]
\[\left( {1 + 2} \right)\,\,\,\,\left\{ \begin{gathered}
\,\,R < S\,\,\, \Rightarrow \,\,\,\, - R > - S \hfill \\
\,\,X > Y \hfill \\
\end{gathered} \right.\,\,\,\,\,\,\,\,\mathop \Rightarrow \limits^{\left( + \right)} \,\,\,\,\,\,\,X + \left( { - R} \right)\,\, > \,\,Y + \left( { - S} \right)\,\,\,\,\, \Rightarrow \,\,\,\left\langle {{\text{YES}}} \right\rangle \,\]
The correct answer is therefore (C).
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
