Data Sufficiency

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.

subh2273 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.

The average of s and 10 lies HALFWAY between s and 10 on the number line:
s------------AVERAGE------------10

Statement 1:
Since t closer to 10 than to s, t must be greater than the average of s and 10, as shown here:
s-----------AVERAGE---t----------10
Thus, the answer to the question stem is NO.

Statement 2:
Case 1: s=1 and t = 5*1 = 5, with the result that the average of s and 10 = (1+10)/2 = 5.5.
In this case, t is less than the average of s and 10, so the answer to the question stem is YES.
Case 2: s=2 and t = 5*2 = 10, with the result that the average of s and 10 = (2+10)/2 = 6.
In this case, t is greater than the average of s and 10, so the answer to the question stem is NO.
INSUFFICIENT.

The correct answer is A.

subh2273 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.

\[1 \leqslant s \leqslant 9\,\,\,\,\operatorname{int} \,\,\,\left( * \right)\]
\[t\,\,\mathop < \limits^? \,\,\frac{{s + 10}}{2}\]
\[\left( 1 \right)\,\, \Rightarrow \,\,\,\,\left\{ \begin{gathered}
\,\,t \geqslant 10\,\,\,\,\mathop \Rightarrow \limits^{{\text{FOCUS}}!} \,\,\,t \geqslant \frac{{10 + 10}}{2}\,\,\,\mathop > \limits^{\left( * \right)} \,\,\,\frac{{s + 10}}{2}\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \,\,\,\,\,\, \hfill \\
\,\,{\text{OR}} \hfill \\
\,s < \,\,\boxed{\,\,\frac{{s + 10}}{2} < t\,\,}\,\, < \,\,\,10\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\]
\[\left( 2 \right)\,\,\,\left\{ \begin{gathered}
\,{\text{Take}}\,\,s = 1\,\,\,\, \Rightarrow \,\,\,\,t = 5\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\langle {{\text{YES}}} \right\rangle \,\, \hfill \\
\,{\text{Take}}\,\,s = 2\,\,\,\, \Rightarrow \,\,\,\,t = 10\,\,\,\,\, \Rightarrow \,\,\,\,\left\langle {{\text{NO}}} \right\rangle \hfill \\
\end{gathered} \right.\]


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

Regards,
Fabio.