Max@Math Revolution wrote:[Math Revolution GMAT math practice question]
Is the sum of 7 different positive integers greater than or equal to 48?
1) Their median is 9
2) The largest number is 12
(At least to my students) If you don´t know how to begin, do not forget:
STRUCTURE rules!
\[1 \leqslant {a_{\,1}} < {a_{\,2}} < \,\, \ldots < {a_{\,7}}\,\,\,{\text{ints}}\]
\[\sum\nolimits_{{\text{them}}\,{\text{7}}} {\,\,\mathop \geqslant \limits^? \,\,\,48} \]
\[\left( 1 \right)\,\,\,{\text{Me}}\,\,{\text{ = }}\,\,{{\text{a}}_{\text{4}}} = 9\,\,\,\,\, \Rightarrow \,\,\,\,\,\,1 + 2 + 3 \leqslant {a_1} + {a_2} + {a_3} \leqslant 6 + 7 + 8\]
\[\mathop \Rightarrow \limits^{{\text{FOCUS}}\,{\text{!}}} \,\,\,\left\{ \begin{gathered}
\,15\,\,\, \leqslant \,\,\,\sum\nolimits_{{\text{small}}\,4} {} \,\,\, \leqslant \,\,30 \hfill \\
33 = \,10 + 11 + 12\,\,\, \leqslant {a_5} + {a_6} + {a_7} = \,\,\,\sum\nolimits_{{\text{big}}\,3} {} \hfill \\
\end{gathered} \right.\,\,\,\mathop \Rightarrow \limits^{{\text{FOCUS}}\,{\text{!}}} \,\,\,\,\,\,48 = 15 + 33\,\,\, \leqslant \,\,\,\,\sum\nolimits_{{\text{them}}\,{\text{7}}} {\,\,\,\,\, \Rightarrow } \,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\,\,\,\,\]
\[\left( 2 \right)\,\,{a_7} = 12\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,\left( {{a_1},{a_2},{a_3},{a_4},{a_5},{a_6}} \right) = \left( {1,2,3,4,5,6} \right)\,\,\,\, \Rightarrow \,\,\,\sum\nolimits_{{\text{them}}\,{\text{7}}} {\,\, = \,\,\,33} \,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,\left( {{a_1},{a_2},{a_3},{a_4},{a_5},{a_6}} \right) = \left( {6,7,8,9,10,11} \right)\,\,\,\, \Rightarrow \,\,\,\sum\nolimits_{{\text{them}}\,{\text{7}}} {\,\, = \,\,\,63} \,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \, \hfill \\
\end{gathered} \right.\]
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
