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If x, y, and z are integers, and x < y < z, is z y = y

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If x, y, and z are integers, and x < y < z, is z y = y [#permalink] New post 18 Nov 2009, 20:43
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If x, y, and z are integers, and x < y < z, is z – y = y – x?

(1) The mean of the set {x, y, z, 4} is greater than the mean of the set {x, y, z}.
(2) The median of the set {x, y, z, 4} is less than the median of the set {x, y, z}.
[Reveal] Spoiler: OA

Last edited by Bunuel on 04 Jul 2013, 22:56, edited 1 time in total.
Added the OA.
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Re: x < y < z, is z – y = y – x? [#permalink] New post 19 Nov 2009, 05:38
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kairoshan wrote:
If x, y, and z are integers, and x < y < z, is z – y = y – x?

(1) The mean of the set {x, y, z, 4} is greater than the mean of the set {x, y, z}.
(2) The median of the set {x, y, z, 4} is less than the median of the set {x, y, z}.



Question Stem : Is 2y = z + x ; x , y , z , are integers such that x < y < z

St. (1) : \frac{x+y+z+4}{4} > \frac{x+y+z}{3}
This simplifies to : 12 > x + y + z
Consider the following two sets both of which satisfy all the given conditions:
(a) {2, 3, 4} ----> Proves the question stem true.
(b) {2, 3, 5} ----> Proves the question stem false.
Since we get contradicting solutions, statement is Insufficient.

St. (2) : median of the set {x, y, z, 4} < median of the set {x, y, z}
The problem here is that we don't know the relation of 4 with respect to the other numbers. Let us consider all the possible cases and see how St. (2) relates to them :
Case 1 : {4, x, y, z} --> \frac{x+y}{2} < y --> y > x which is already specified.
Case 2 : {x, 4, y, z} --> \frac{4+y}{2} < y --> y > 4
Case 3 : {x, y, 4, z} --> \frac{y+4}{2} < y --> y > 4 which is false since we assumed y < 4 in the set.
Case 4 : {x, y, z, 4} --> \frac{y+z}{2} < y --> y > z which is false since it violates information in question stem.
Thus the only two valid cases are Case 1 and 2. However, they do not give us enough information to answer the question stem.
Hence, Insufficient.

St. (1) and (2) together : 12 > x + y + z ; Case 1 and 2 from above.
Now let us see how Case 1 and 2 relate to St. (1) :
Case 1 : {4, x, y, z} --> y > 4 (assumed for the set)
Case 2 : {x, 4, y, z} --> y > 4 (assumed for the set)
Note: The important thing to note here is that the only valid cases from St. (2) are those in which we have assumed y > 4.
Thus, for the conditions:
(a) y > 4
(b) x + y + z < 12
(c) x < y < z
(d) x, y, z are integers
The only set of values possible are x < ; y >= 5; z >= 6
More importantly, this tells us that x, y and z can never be equidistant from each other (which is a condition necessary for 2y = x + z). Thus the question stem will always be false.
Sufficient.

Answer : C
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Re: x < y < z, is z – y = y – x? [#permalink] New post 19 Nov 2009, 06:40
kairoshan wrote:
sriharimurthy wrote:
kairoshan wrote:
If x, y, and z are integers, and x < y < z, is z – y = y – x?

(1) The mean of the set {x, y, z, 4} is greater than the mean of the set {x, y, z}.
(2) The median of the set {x, y, z, 4} is less than the median of the set {x, y, z}.



Question Stem : Is 2y = z + x ; x , y , z , are integers such that x < y < z

St. (1) : \frac{x+y+z+4}{4} > \frac{x+y+z}{3}
This simplifies to : 12 > x + y + z
Consider the following two sets both of which satisfy all the given conditions:
(a) {2, 3, 4} ----> Proves the question stem true.
(b) {2, 3, 5} ----> Proves the question stem false.
Since we get contradicting solutions, statement is Insufficient.

St. (2) : median of the set {x, y, z, 4} < median of the set {x, y, z}
The problem here is that we don't know the relation of 4 with respect to the other numbers. Let us consider all the possible cases and see how St. (2) relates to them :
Case 1 : {4, x, y, z} --> \frac{x+y}{2} < y --> y > x which is already specified.
Case 2 : {x, 4, y, z} --> \frac{4+y}{2} < y --> y > 4
Case 3 : {x, y, 4, z} --> \frac{y+4}{2} < y --> y > 4 which is false since we assumed y < 4 in the set.
Case 4 : {x, y, z, 4} --> \frac{y+z}{2} < y --> y > z which is false since it violates information in question stem.
Thus the only two valid cases are Case 1 and 2. However, they do not give us enough information to answer the question stem.
Hence, Insufficient.

St. (1) and (2) together : 12 > x + y + z ; Case 1 and 2 from above.
Now let us see how Case 1 and 2 relate to St. (1) :
Case 1 : {4, x, y, z} --> y > 4 (assumed for the set)
Case 2 : {x, 4, y, z} --> y > 4 (assumed for the set)
Note: The important thing to note here is that the only valid cases from St. (2) are those in which we have assumed y > 4.
Thus, for the conditions:
(a) y > 4
(b) x + y + z < 12
(c) x < y < z
(d) x, y, z are integers
The only set of values possible are x < ; y >= 5; z >= 6
More importantly, this tells us that x, y and z can never be equidistant from each other (which is a condition necessary for 2y = x + z). Thus the question stem will always be false.
Sufficient.

Answer : C



Good explanation! OA is C
+1


I don't think it's possible to finish this problem in 2 minutes. However, who knows....:)
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Re: x < y < z, is z – y = y – x? [#permalink] New post 19 Nov 2009, 07:35
lugeruo wrote:
I don't think it's possible to finish this problem in 2 minutes. However, who knows....:)


agree it will take more than 2mins (atleast I took around 5mins to solve) but does GMAC always intend to give questions which can be solved in 2mins?? :?
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Re: x < y < z, is z – y = y – x? [#permalink] New post 19 Nov 2009, 08:13
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kairoshan wrote:
If x, y, and z are integers, and x < y < z, is z – y = y – x?

(1) The mean of the set {x, y, z, 4} is greater than the mean of the set {x, y, z}.
(2) The median of the set {x, y, z, 4} is less than the median of the set {x, y, z}.



Another way of tackling this problem. (Graphical/Diagramatic Method)

Q: ( is z – y = y – x? )
Question is asking whether x,y, z are equidistant, while y being middle number. Means.. Is Y middle number or integer?
Clearly individual statemnets are not sufficient.. they can be equidistance or not .

Combined:
Assume that y is middle number. Now check, whether two statements satisfy this or not.

x---------y---------z
x---------y---------z -----4 (stmt 1 satisfy, 2 not satisfy)
x---------y----4--- z (stmt 1 satisfy, 2 not satisfy)
x-------4--y--------z (stmt 1 not satisfy, 2 satisfy)

Clearly, 4 will fall on right hand side of y (For statment 1 to be true) and on left hand side of y (for statement 2 be true).

Clearly this is not possible.. our assumption (Y is middle integer) is not correct
C is the answer.
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Re: x < y < z, is z – y = y – x? [#permalink] New post 19 Nov 2009, 08:21
kp1811 wrote:
lugeruo wrote:
I don't think it's possible to finish this problem in 2 minutes. However, who knows....:)


agree it will take more than 2mins (atleast I took around 5mins to solve) but does GMAC always intend to give questions which can be solved in 2mins?? :?



Yes, I agree that this takes more than 2 min.
I guess a few problems take more than 2 min on Gmat, so we should save time in easy ones.
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Re: x < y < z, is z – y = y – x? [#permalink] New post 19 Nov 2009, 08:53
kairoshan wrote:
kp1811 wrote:
lugeruo wrote:
I don't think it's possible to finish this problem in 2 minutes. However, who knows....:)


agree it will take more than 2mins (atleast I took around 5mins to solve) but does GMAC always intend to give questions which can be solved in 2mins?? :?



Yes, I agree that this takes more than 2 min.
I guess a few problems take more than 2 min on Gmat, so we should save time in easy ones.


That's the ultimate strategy.
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Means & Medians [#permalink] New post 08 Sep 2010, 23:16
Another fairly tough question :

x,y,z,k are integers such that x<y<z and k>0. Is (y-x) = (z-y) ?

(1) The mean of set \{x,y,z,k\} is greater than the mean of set \{x,y,z\}
(2) The median of set \{x,y,z,k\} is less than the median of set \{x,y,z\}
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Re: Means & Medians [#permalink] New post 09 Sep 2010, 00:04
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Re: x < y < z, is z – y = y – x? [#permalink] New post 09 Sep 2010, 00:12
Yeah, I think so. But there is no reason to take the special case of 4, it can be proven for a general k.

I had an alternate solution :

(1) & (2) are clearly not sufficient as k is arbitrary and can be large enough or small enough to force any condition on mean or median seperately

For (1) + (2) : Statement (2) implies y>k. Statement (1) implies x+y+z<3k. Now if you assume x+z=2y the above two inequalities are inconsistent ==> Contradiction ==> The answer to the question is always "No"

Hence (C)

short & sweet
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Re: x < y < z, is z – y = y – x? [#permalink] New post 09 Sep 2010, 00:36
Expert's post
shrouded1 wrote:
Yeah, I think so. But there is no reason to take the special case of 4, it can be proven for a general k.

I had an alternate solution :

(1) & (2) are clearly not sufficient as k is arbitrary and can be large enough or small enough to force any condition on mean or median seperately

For (1) + (2) : Statement (2) implies y>k. Statement (1) implies x+y+z<3k. Now if you assume x+z=2y the above two inequalities are inconsistent ==> Contradiction ==> The answer to the question is always "No"

Hence (C)

short & sweet


Yes you are right: as "4" in original question is purely arbitrary we can replace it with any other number or with unknown k as you did.

Question: is 2y=x+z?

For (1): x+y+z<3k;
For (2): k can not be the the largest term, thus it must be less than y;

(1)+(2) As x+y+z<3k and k<y then 2y=x+z can not be true. Sufficient.

Answer: C.
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Re: x < y < z, is z – y = y – x? [#permalink] New post 10 Sep 2010, 11:13
(1)+(2) As x+y+z<3k and k<y then 2y=x+z can not be true.

Please elaborate on this...
thanks!
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Re: x < y < z, is z – y = y – x? [#permalink] New post 10 Sep 2010, 11:44
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ahsanmalik12 wrote:
(1)+(2) As x+y+z<3k and k<y then 2y=x+z can not be true.

Please elaborate on this...
thanks!


x+y+z<3k and k<y, or 3k<3y --> sum these two x+y+z+3k<3k+3y --> x+z<2y, so x+z=2y is not true.
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Re: If x, y, and z are integers, and x < y < z, is z y = y [#permalink] New post 22 Feb 2014, 10:30
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Re: If x, y, and z are integers, and x < y < z, is z y = y   [#permalink] 22 Feb 2014, 10:30
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