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Is z equal to the median of the three positive integers, x, y, and z?

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LM
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Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   Updated on: 28 Nov 2017, 01:19
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Is z equal to the median of the three positive integers, x, y, and z?

(1) x < y + z
(2) y = z

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B

Originally posted by LM on 10 May 2010, 09:33.
Last edited by Bunuel on 28 Nov 2017, 01:19, edited 2 times in total.
Renamed the topic, edited the question and added the OA.
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Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   10 May 2010, 14:18
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Is z equal to the median of the three positive integers, x, y, and z?

Median of the three numbers is the middle term, hence z would be the median in two cases: \(x\leq{z}\leq{y}\) or \(x\geq{z}\geq{y}\).

(1) x < y + z --> clearly insufficient. If x=2, y=10, z=1, then answer would be NO but x=1, y=10, z=2, then answer would be YES.

(2) y = z --> either the three numbers are z, z, x (in ascending order) --> media=z or the three numbers are x, z, z (in ascending order) --> median=z. Sufficient.

Answer: B.
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   10 May 2010, 21:03
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Bunuel wrote:
Is z equal to the median of the three positive integers, x,y, and z?

Median of the three numbers is the middle term, hence z would be the median in two cases: \(x\leq{z}\leq{y}\) or \(x\geq{z}\geq{y}\).

(1) x<y+z --> clearly insufficient. If x=1, y=10, z=0, then answer would be NO but x=1, y=10, z=2, then answer would be YES.

(2) y=z --> either the three numbers are z, z, x (in ascending order) --> media=z or the three numbers are x, z, z (in ascending order) --> median=z. Sufficient.

Answer: B.


I just could not think that Z=0 is also possible.
IN this question will it be fair enough to assume that X=Y=Z is also one possibility because it does not say that they each is unique and different!
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   11 May 2010, 03:35
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LM wrote:
Bunuel wrote:
Is z equal to the median of the three positive integers, x,y, and z?

Median of the three numbers is the middle term, hence z would be the median in two cases: \(x\leq{z}\leq{y}\) or \(x\geq{z}\geq{y}\).

(1) x<y+z --> clearly insufficient. If x=1, y=10, z=0, then answer would be NO but x=1, y=10, z=2, then answer would be YES.

(2) y=z --> either the three numbers are z, z, x (in ascending order) --> media=z or the three numbers are x, z, z (in ascending order) --> median=z. Sufficient.

Answer: B.


I just could not think that Z=0 is also possible.
IN this question will it be fair enough to assume that X=Y=Z is also one possibility because it does not say that they each is unique and different!


z may or may not be zero. For (1) you can pick infinite examples x<y+z to hold true. Another example: x=5, y=10, z=3, then answer would be NO but x=6, y=10, z=8, then answer would be YES.

About x=y=z. For statement (2) x=y=z is possible --> three numbers would be z, z, z --> median still z.
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   22 Aug 2010, 21:37
kwhitejr wrote:
Is z equal to the median of the three positive integers x, y, and z?

(1) x < y + z
(2) y = z


Statement 1:

If we pick numbers we find that z may or may not be the median.

Hence insufficient.

Statement 2:

y = z then irrespective of x, z would be the median since there are only three integers.

Hence sufficient.

Answer: B
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   19 Oct 2010, 10:44
Median of the three numbers is the middle term, hence z would be the median in two cases: \(x\leq{z}\leq{y}\) or \(x\geq{z}\geq{y}\).
[/quote]

Great. Remembering that line would help alot in solving such questions.
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   20 Oct 2010, 06:45
If I go with the values for y and Z in (2) .. I would get the ans as (B) only...
lets say y=z=5

then Place the values in one order either desc or asc
X,5,5 ..so median is : 5

or 5,5,X again median is : 5

or 5,X,5 again ,median would be 5 and even X = 5 since X is a positive interger and X is btwn 5 and 5 ...so it should be equal to 5 only ...

From (2) only i can get the ans .... So B wins
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   24 Apr 2011, 00:57
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If we place these numbers in the increasing order, then the median will be the second number.

(a) x<y+z
Assume that x=y=z. Then x<y+z. However since all three numbers are equal, then z is equal to the median
Now assume, that y<x, but x<z. Then obviously, x<y+z, but the median is x.

So (i) is not sufficient.
If y=z, then numbers in increasing order are either x y z or y z x. However, since y=z, the median in both cases is equal to z. So (ii) is sufficient.

The answer is B
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   24 Apr 2011, 19:41
(1)

x = 1, y = 2, z = 3

but median is y

z = 2, x = 1, y = 3
then z is median

(1) is insufficient


(2)

x = 2, y = 1, z = 1

z is the median when we arrange the numbers as 1,1,2

x = 1, y = 1 z = 1

z is median

x = 1, y = 2, z = 2

z is median when we arrange numbers as 1,2,2

(2) is sufficient



Answer - B
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   19 May 2011, 23:26
a 1,2,3 for x,y,z plays around with the median. not sufficient.

b x,z,z means z is definitely the median.
B
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   11 Nov 2014, 15:05
LM wrote:
Bunuel wrote:
Is z equal to the median of the three positive integers, x,y, and z?

Median of the three numbers is the middle term, hence z would be the median in two cases: \(x\leq{z}\leq{y}\) or \(x\geq{z}\geq{y}\).

(1) x<y+z --> clearly insufficient. If x=1, y=10, z=0, then answer would be NO but x=1, y=10, z=2, then answer would be YES.

(2) y=z --> either the three numbers are z, z, x (in ascending order) --> media=z or the three numbers are x, z, z (in ascending order) --> median=z. Sufficient.

Answer: B.


I just could not think that Z=0 is also possible.
IN this question will it be fair enough to assume that X=Y=Z is also one possibility because it does not say that they each is unique and different!


Is the median the middle term true even if the integers are not consecutive? Meaning, if it's 1 4 4 -- is the median still 4?
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   12 Nov 2014, 04:57
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russ9 wrote:
LM wrote:
Bunuel wrote:
Is z equal to the median of the three positive integers, x,y, and z?

Median of the three numbers is the middle term, hence z would be the median in two cases: \(x\leq{z}\leq{y}\) or \(x\geq{z}\geq{y}\).

(1) x<y+z --> clearly insufficient. If x=1, y=10, z=0, then answer would be NO but x=1, y=10, z=2, then answer would be YES.

(2) y=z --> either the three numbers are z, z, x (in ascending order) --> media=z or the three numbers are x, z, z (in ascending order) --> median=z. Sufficient.

Answer: B.


I just could not think that Z=0 is also possible.
IN this question will it be fair enough to assume that X=Y=Z is also one possibility because it does not say that they each is unique and different!


Is the median the middle term true even if the integers are not consecutive? Meaning, if it's 1 4 4 -- is the median still 4?


If a set has odd number of terms the median of the set is the middle number when arranged in ascending or descending order. So, the median of {1, 4, 4} is 4.

If a set has even number of terms the median of the set is the average of the two middle terms when arranged in ascending or descending order. For example, the median of {1, 1, 4, 4} is (1 + 4)/2 = 2.5.

Hope it's clear.
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   12 Nov 2014, 10:28
Bunuel wrote:
russ9 wrote:
LM wrote:

I just could not think that Z=0 is also possible.
IN this question will it be fair enough to assume that X=Y=Z is also one possibility because it does not say that they each is unique and different!


Is the median the middle term true even if the integers are not consecutive? Meaning, if it's 1 4 4 -- is the median still 4?


If a set has odd number of terms the median of the set is the middle number when arranged in ascending or descending order. So, the median of {1, 4, 4} is 4.

If a set has even number of terms the median of the set is the average of the two middle terms when arranged in ascending or descending order. For example, the median of {1, 1, 4, 4} is (1 + 4)/2 = 2.5.

Hope it's clear.


Thanks. Makes sense. I thought that rule only applied to consecutive terms, but thanks for clarifying!
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   28 Nov 2017, 01:18
Bunuel
Are we assuming that x, y and z are not jumbled so as any integer can take any place? Is that why x can not be in the middle and only two possible order can be x,y,z or z,y,x ?

Thanks
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   28 Nov 2017, 01:22
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TaN1213 wrote:
Bunuel
Are we assuming that x, y and z are not jumbled so as any integer can take any place? Is that why x can not be in the middle and only two possible order can be x,y,z or z,y,x ?

Thanks


We don't know the order of the variables. Therefore, x, y, and z could have 6 possible orderings.
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   23 Jun 2018, 02:33
Bunuel wrote:
LM wrote:
Bunuel wrote:
Is z equal to the median of the three positive integers, x,y, and z?

Median of the three numbers is the middle term, hence z would be the median in two cases: \(x\leq{z}\leq{y}\) or \(x\geq{z}\geq{y}\).

(1) x<y+z --> clearly insufficient. If x=1, y=10, z=0, then answer would be NO but x=1, y=10, z=2, then answer would be YES.

(2) y=z --> either the three numbers are z, z, x (in ascending order) --> media=z or the three numbers are x, z, z (in ascending order) --> median=z. Sufficient.

Answer: B.


I just could not think that Z=0 is also possible.
IN this question will it be fair enough to assume that X=Y=Z is also one possibility because it does not say that they each is unique and different!


z may or may not be zero. For (1) you can pick infinite examples x<y+z to hold true. Another example: x=5, y=10, z=3, then answer would be NO but x=6, y=10, z=8, then answer would be YES.

About x=y=z. For statement (2) x=y=z is possible --> three numbers would be z, z, z --> median still z.



if x=y=z, couldn't three numbers be (x,x,x), (y, y, y), or (z,z,z)? Hence x could be viewed as median as well.
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   23 Jun 2018, 06:54
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Andy24 wrote:
Bunuel wrote:
LM wrote:

I just could not think that Z=0 is also possible.
IN this question will it be fair enough to assume that X=Y=Z is also one possibility because it does not say that they each is unique and different!


z may or may not be zero. For (1) you can pick infinite examples x<y+z to hold true. Another example: x=5, y=10, z=3, then answer would be NO but x=6, y=10, z=8, then answer would be YES.

About x=y=z. For statement (2) x=y=z is possible --> three numbers would be z, z, z --> median still z.



if x=y=z, couldn't three numbers be (x,x,x), (y, y, y), or (z,z,z)? Hence x could be viewed as median as well.


Yes, but if x = z, then x is the median is the same as z is the median.
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   14 Nov 2020, 10:56
Thanks for the explanation. One question though; why can't x be the middle number/median? It doesn't specify anywhere that it has to be the order xyz or zyx (in the case of statement 2: xzz or zzx)
Couldn't the order be zxz? Bunuel

Thank you in advance!
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   14 Nov 2020, 11:11
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dortinator1234923 wrote:
Thanks for the explanation. One question though; why can't x be the middle number/median? It doesn't specify anywhere that it has to be the order xyz or zyx (in the case of statement 2: xzz or zzx)
Couldn't the order be zxz? Bunuel

Thank you in advance!


The median is a middle number when arranged in ascending or descending order. When you arrange x, z, and z, in ascending order you get z, z, x or x, z, z. How can you have z, x, z?
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink] New post   14 Nov 2020, 12:18
Bunuel wrote:
dortinator1234923 wrote:
Thanks for the explanation. One question though; why can't x be the middle number/median? It doesn't specify anywhere that it has to be the order xyz or zyx (in the case of statement 2: xzz or zzx)
Couldn't the order be zxz? Bunuel

Thank you in advance!


The median is a middle number when arranged in ascending or descending order. When you arrange x, z, and z, in ascending order you get z, z, x or x, z, z. How can you have z, x, z?


Thank you for your quick response. Was not aware of the fact that ascending/descending order was necessary, get it now.
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Re: Is z equal to the median of the three positive integers, x, y, and z? [#permalink]
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