For any integers a, b, c, and d, sechigh(a, b, c, d) is the

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For any integers a, b, c, and d, sechigh(a, b, c, d) is the second highest integer when the integers are placed in an ordered list. For example sechigh(2, 5, 4, 6) = 5 and sechigh(6, 5, 3, 6) = 6 . For the integer y, what is the value of sechigh(6, 7, 11, y)?

(1) y = sechigh (7, 13, 12, x) for some integer x.

(2) y = sechigh (7, 13, 8, z) for some integer z.

OA A

Source: Manhattan Prep

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by fskilnik@GMATH » Mon Nov 26, 2018 5:11 am

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BTGmoderatorDC wrote:For any integers a, b, c, and d, sechigh(a, b, c, d) is the second highest integer when the integers are placed in an ordered list. For example sechigh(2, 5, 4, 6) = 5 and sechigh(6, 5, 3, 6) = 6 . For the integer y, what is the value of sechigh(6, 7, 11, y)?

(1) y = sechigh (7, 13, 12, x) for some integer x.

(2) y = sechigh (7, 13, 8, z) for some integer z.
Source: Manhattan Prep
$$? = {\rm{sechigh}}\left( {6,7,11,y} \right)\,\,\,\,\,\,\,\,\left[ {y\,\,{\mathop{\rm int}} } \right]$$

$$\left( 1 \right)\,\,y = {\rm{sechigh}}\left( {7,13,12,x} \right)\,\,\,\,\left[ {x\,\,{\mathop{\rm int}} } \right]\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\left\{ \matrix{
\,y = 13\,\,\,{\rm{if}}\,\,\,x \ge 13\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 11 \hfill \cr
\,y = 12\,\,\,{\rm{if}}\,\,\,x \le 12\,\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 11 \hfill \cr} \right.\,\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{SUFF}}.$$

$$\left( 2 \right)\,\,y = {\rm{sechigh}}\left( {7,13,8,z} \right)\,\,\,\,\left[ {z\,\,{\mathop{\rm int}} } \right]\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\left\{ \matrix{
\,{\rm{Take}}\,\,z = 7\,\,\,\,\, \Rightarrow \,\,\,\,\,y = 8\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 8 \hfill \cr
\,\,{\rm{Take}}\,\,z = 14\,\,\,\,\, \Rightarrow \,\,\,\,\,y = 13\,\,\,\,\, \Rightarrow \,\,\,\,\,? = 11 \hfill \cr} \right.\,\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,{\rm{INSUFF}}{\rm{.}}$$


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

Regards,
Fabio.
Fabio Skilnik :: GMATH method creator ( Math for the GMAT)
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by Jay@ManhattanReview » Mon Nov 26, 2018 9:55 pm

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BTGmoderatorDC wrote:For any integers a, b, c, and d, sechigh(a, b, c, d) is the second highest integer when the integers are placed in an ordered list. For example sechigh(2, 5, 4, 6) = 5 and sechigh(6, 5, 3, 6) = 6 . For the integer y, what is the value of sechigh(6, 7, 11, y)?

(1) y = sechigh (7, 13, 12, x) for some integer x.

(2) y = sechigh (7, 13, 8, z) for some integer z.

OA A

Source: Manhattan Prep
Given: sechigh(a, b, c, d) is the second highest integer when the integers are placed in an ordered list

For example

1. sechigh(2, 5, 4, 6) = sechigh(2, 4, 5, 6) = 5
2. sechigh(6, 5, 3, 6) = sechigh(3, 5, 6, 6) = 6

We have to determine the value of sechigh(6, 7, 11, y).

Question: We have to determine the value of sechigh(6, 7, 11, y).

Let's take each statement one by one.

(1) y = sechigh (7, 13, 12, x) for some integer x.

Case 1: Say y is the greatest term => y = sechigh (7, 12, 13, x) = 13.

Thus, sechigh(6, 7, 11, y) = sechigh(6, 7, 11, 13) = 11.

Case 2: Say y is the second highest term => y = sechigh (7, 12, x, 13) => y = 12 or 13

Thus, sechigh(6, 7, 11, y) = sechigh(6, 7, 11, 12/13) = 11.

Case 3: Say y is the third highest term => y = sechigh (7, x, 12, 13) => y = 12

Thus, sechigh(6, 7, 11, y) = sechigh(6, 7, 11, 12) = 11.

Case 4: Say y is the first term => y = sechigh (x, 7, 12, 13) => y = 12

Thus, sechigh(6, 7, 11, y) = sechigh(6, 7, 11, 12) = 11.

Unique answer. Sufficient.

(2) y = sechigh (7, 13, 8, z) for some integer z.

Case 1: Say y is the highest term => y = sechigh (7, 8, 13, z) => y = 13

Thus, sechigh(6, 7, 11, y) = sechigh(6, 7, 11, 13) = 11.

Case 2: Say y is the second highest term => y = sechigh (7, 8, z, 13) => 8 ≤ y ≤ 13

Thus, if 8 ≤ y ≤ 11, then sechigh(6, 7, 11, y) = sechigh(6, 7, y, 11) = y (= 8/9/10/11). No unique value of y.

Insufficient.

The correct answer: A

Hope this helps!

-Jay
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