mshrek wrote:
If s is an integer between 0 and 10, is t less than the average (arithmetic mean) of s and 10?
(1) t is closer to 10 on the number line than it is to s.
(2) t is 5 times as large as s.
\(1 \leqslant s \leqslant 9\,\,\,\,\operatorname{int} \,\,\,\left( * \right)\)
\(t\,\,\mathop < \limits^? \,\,\frac{{s + 10}}{2}\)
\(\left( 1 \right)\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}
\,\,t \geqslant 10\,\,\,\,\mathop \Rightarrow \limits^{{\text{FOCUS}}!} \,\,\,t \geqslant \frac{{10 + 10}}{2}\,\,\,\mathop > \limits^{\left( * \right)} \,\,\,\frac{{s + 10}}{2}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\, \hfill \\
\,\,{\text{OR}} \hfill \\
\,s < \,\,\frac{{s + 10}}{2} < t\,\, < \,\,\,10\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\)
\(\left( 2 \right)\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,s = 1\,\,\,\, \Rightarrow \,\,\,\,t = 5\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,s = 2\,\,\,\, \Rightarrow \,\,\,\,t = 10\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\)
This solution follows the notations and rationale taught in the GMATH method.
Regards,
Fabio.
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