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GMAT Diagnostic Test Question 21 Updated on: 28 Sep 2013, 21:33
Originally posted by bb on 06 Jun 2009, 22:59.
Last edited by bb on 28 Sep 2013, 21:33, edited 1 time in total.
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GMAT Diagnostic Test Question 21
Field: word problems (min/max)
Difficulty: 700

A set of 11 different integers has a median of 25 and a range of 50. What is the greatest possible integer that could be in this set?

A. 65
B. 70
C. 75
D. 80
E. 85

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GMAT Diagnostic Test Question 21 08 Jul 2009, 06:56
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Consider 11 numbers in ascending order to be \(x_1\), \(x_2\), \(x_3\), ..., \(x_{11}\).

The median of a set with odd number of elements is the middle number (when arranged in ascending or descending order), so the median of given set is \(x_{6}=25\);

The range of a set is the difference between the largest and the smallest numbers of a set, so the range of given set is \(50=x_{11}-x_{1}\) --> \(x_{11}=50+x_{1}\);

We want to maximize \(x_{11}\), hence we need to maximize \(x_{1}\). Since all integers must be distinct then the maximum value of \(x_{1}\) will be \(median-5=25-5=20\) and thus the maximum value of \(x_{11}\) is \(x_{11}=50+20=70\).

The set could be {20, 21, 22, 23, 24, 25, 26, 27, 26, 29, 70}

Answer: B.
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GMAT Diagnostic Test Question 21 20 Nov 2013, 12:24
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Why must the numbers in the set be different in order to calculate the median?

Couldn't I arrange a set like the following? S={25, 25, 25, 25, 25, 25, 75, 75, 75, 75, 75}?
(Note that after x6, the follwoing numbers could be anything between 26 and 74)

In this way, the median is still 25, but the máximum value of x would now be 75, making "C" the correct answer.

Thanks for your comments!
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GMAT Diagnostic Test Question 21 21 Nov 2013, 01:55
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peti
Why must the numbers in the set be different in order to calculate the median?

Couldn't I arrange a set like the following? S={25, 25, 25, 25, 25, 25, 75, 75, 75, 75, 75}?
(Note that after x6, the follwoing numbers could be anything between 26 and 74)

In this way, the median is still 25, but the máximum value of x would now be 75, making "C" the correct answer.

Thanks for your comments!
D.
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The number is the set must be different because we are told that they are different, check the stem: "A set of 11 different integers has ..."
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GMAT Diagnostic Test Question 21 05 Jan 2014, 17:45
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[quote="dzyubam"]Consider 11 numbers in ascending order to be \(x_1\), \(x_2\), \(x_3\), ..., \(x_{11}\).

The median of a set with odd number of elements is the middle number (when arranged in ascending or descending order), so the median of given set is \(x_{6}=25\);

The range of a set is the difference between the largest and the smallest numbers of a set, so the range of given set is \(50=x_{11}-x_{1}\) --> \(x_{11}=50+x_{1}\);

We want to maximize \(x_{11}\), hence we need to maximize \(x_{1}\). Since all integers must be distinct then the maximum value of \(x_{1}\) will be \(median-5=25-5=20\) and thus the maximum value of \(x_{11}\) is \(x_{11}=50+20=70\).

The set could be {20, 21, 22, 23, 24, 25, 26, 27, 26, 29, 70}

its clear till since....distict
i was not able to follow after that..??? can u please explain just that area
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GMAT Diagnostic Test Question 21 05 Jan 2014, 19:03
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lindt123
dzyubam
Consider 11 numbers in ascending order to be \(x_1\), \(x_2\), \(x_3\), ..., \(x_{11}\).

The median of a set with odd number of elements is the middle number (when arranged in ascending or descending order), so the median of given set is \(x_{6}=25\);

The range of a set is the difference between the largest and the smallest numbers of a set, so the range of given set is \(50=x_{11}-x_{1}\) --> \(x_{11}=50+x_{1}\);

We want to maximize \(x_{11}\), hence we need to maximize \(x_{1}\). Since all integers must be distinct then the maximum value of \(x_{1}\) will be \(median-5=25-5=20\) and thus the maximum value of \(x_{11}\) is \(x_{11}=50+20=70\).

The set could be {20, 21, 22, 23, 24, 25, 26, 27, 26, 29, 70}

its clear till since....distict
i was not able to follow after that..??? can u please explain just that area
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We know that the median, \(x_6\), is 25. What is the maximum value of \(x_5\). Since \(x_5<x_6\), then the maximum value of \(x_5\) is 24. Similarly the maximum value of \(x_4\) is 23, the maximum value of \(x_3\) is 22, the maximum value of \(x_2\) is 21 and the maximum value of \(x_1\) is 20.

Hope it's clear.
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GMAT Diagnostic Test Question 21 05 Jan 2014, 21:51
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dzyubam
Consider 11 numbers in ascending order to be \(x_1\), \(x_2\), \(x_3\), ..., \(x_{11}\).

The median of a set with odd number of elements is the middle number (when arranged in ascending or descending order), so the median of given set is \(x_{6}=25\);

The range of a set is the difference between the largest and the smallest numbers of a set, so the range of given set is \(50=x_{11}-x_{1}\) --> \(x_{11}=50+x_{1}\);

We want to maximize \(x_{11}\), hence we need to maximize \(x_{1}\). Since all integers must be distinct then the maximum value of \(x_{1}\) will be \(median-5=25-5=20\) and thus the maximum value of \(x_{11}\) is \(x_{11}=50+20=70\).

The set could be {20, 21, 22, 23, 24, 25, 26, 27, 26, 29, 70}

its clear till since....distict



i was not able to follow after that..??? can u please explain just that area

We know that the median, \(x_6\), is 25. What is the maximum value of \(x_5\). Since \(x_5<x_6\), then the maximum value of \(x_5\) is 24. Similarly the maximum value of \(x_4\) is 23, the maximum value of \(x_3\) is 22, the maximum value of \(x_2\) is 21 and the maximum value of \(x_1\) is 20.

Hope it's clear.
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THANX A LOT....IT DID HELP A LOT!!!

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