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Is z equal to the median of the three positive integers, x, y, and z?
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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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Re: Is z equal to the median of the three positive integers, x, y, and z?
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10 May 2010, 14:18
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.
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Re: Is z equal to the median of the three positive integers, x, y, and z?
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10 May 2010, 21:03
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?
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11 May 2010, 03:35
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?
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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?
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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?
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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?
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24 Apr 2011, 00:57
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?
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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?
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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?
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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?
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12 Nov 2014, 04:57
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?
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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?
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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?
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28 Nov 2017, 01:22
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?
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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?
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23 Jun 2018, 06:54
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?
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24 Jun 2018, 12:18
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. Bunuel In the highlighted part z can't be 0. The question stem says positive number. It should be another positive integer. Thanks




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