Which of the following values, when inserted into the set of

AAPL wrote:EMPOWERgmat
$$\{5, 10, 12, 13, 17, 22, 22\}$$
Which of the following values, when inserted into the set of 7 values (above), will cause the median of the new set to become 15?

A. 2
B. 11
C. 15
D. 16
E. 17

$$L = \left\{ {5,10,12,13,17,22,22} \right\}\,\, \cup \left\{ x \right\}\,\,\,\, \to \,\,\,\,\,{\rm{Me}}{{\rm{d}}_{\,{\rm{L}}}} = 15$$
$$?\,\,\,\, = \,\,\,x$$

The list L has 8 numbers, hence its median (15) must be the average of the fourth and fifth (when values are considered in increasing order).

The correct choice must be (E), immediately. Reason:
$$x < 17\,\,\,\,\, \Rightarrow \,\,\,\,17,22,22\,\,\,{\rm{last}}\,\,{\rm{three}}\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,4{\rm{th}}\,{\rm{ }} \le {\rm{13}} \hfill \cr
\,5{\rm{th}}\,{\rm{ }}\,\, < {\rm{17}} \hfill \cr} \right.\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,{\rm{Med}} = {{4{\rm{th + 5th}}} \over 2}\,\,\, < \,\,\,{{13 + 17} \over 2} = 15$$

Obs.: the last equality shows that 17 "works". If you don´t see that, please follow Brent´s last evaluation presented in his post (above).


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

Regards,
Fabio.