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.

_________________

Fabio Skilnik :: GMATH method creator (Math for the GMAT)

Our high-level "quant" preparation starts here: https://gmath.net