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# A set of 15 different integers has median of 25 and a range

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Re: A set of 15 different integers has median of 25 and a range  [#permalink]

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19 Jan 2018, 20:44
Rocket7 wrote:
So does that mean that this set does not contain any -ve integers? I am assuming it can not but why?

The set cannot have negative integers because of the given data points.
The median is 25 so the middle element is 25.
The range is 25 so greatest - smallest = 25
We know that 25 is there in the set so the smallest element can certainly not be less than 0 since the range of the entire set is 25.
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A set of 15 different integers has median of 25 and a range  [#permalink]

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24 Nov 2018, 14:56
A video explanation can be found here:

Median is 25. In a set of 15 integers, the median will be the 8th number (take 15 – 1, then divide that by 2. You have seven numbers to left of the median and seven to the right.)

Maximum value of the “greatest” possible integer would equal maximum value of the “least” possible integer, plus 25.

Since all integers are different, least possible integer is 15 - 7 = 18.

Maximum possible integer is therefore 18 + 25 = 43
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Re: A set of 15 different integers has median of 25 and a range  [#permalink]

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25 Dec 2018, 10:28
bhushangiri wrote:
A set of 15 different integers has median of 25 and a range of 25. What is greatest possible integer that could be in this set?

A. 32
B. 37
C. 40
D. 43
E. 50

WLOG (without loss of generality) we may assume that:

$$a = {x_1} < {x_2} < \ldots < {x_7} < {x_8} = 25 < {x_9} < \ldots < {x_{14}} < {x_{15}} = a + 25\,\,\,\,\,{\text{ints}}$$

Considering this powerful structure, the problem is trivialized:

$$?\,\, = \,\,\left( {a + 25} \right)\,\,\max \,\,\,\,\,\,\, \Leftrightarrow \,\,\,\,\,\,a\,\,\max$$

$$a\,\,\max \,\,\,\,\, \Leftrightarrow \,\,\,\,\left( {{x_7},{x_6},{x_5},{x_4},{x_3},{x_2},{x_1} = a} \right) = \left( {24,23,22,21,20,19,18} \right)\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,a\,\,\max \,\, = \,\,18\,\,\,\,\,\,\,\, \Rightarrow \,\,\,\,\,\,\,\,? = 43$$

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

Regards,
Fabio.
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Re: A set of 15 different integers has median of 25 and a range  [#permalink]

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21 Apr 2020, 00:33
1
may i know is this a vice-versa rule i.e , when asked to find the greatest possible value of the smallest integer in the set, do we need to maximize the greater value of the set ?

EMPOWERgmatRichC wrote:
Hi Pratyaksh2791,

This question asks us to find the greatest possible number that could be in this set. Since 25 is the MEDIAN of the group of 15 INTEGERS, we know that 7 integers are greater than 25 and 7 integers are less than 25. We're told that the largest integer is exactly 25 more than the smallest integer, so to maximize the biggest value, we also have to maximize the smallest value. Since we're restricted to INTEGERS, the only way to get that maximum result is if the 7 integers that are less than 25 are CONSECUTIVE integers:

24, 23, 22, 21, 20, 19 and 18

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Re: A set of 15 different integers has median of 25 and a range  [#permalink]

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21 Apr 2020, 11:27
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SuryaNouliGMAT wrote:
may i know is this a vice-versa rule i.e , when asked to find the greatest possible value of the smallest integer in the set, do we need to maximize the greater value of the set ?

EMPOWERgmatRichC wrote:
Hi Pratyaksh2791,

This question asks us to find the greatest possible number that could be in this set. Since 25 is the MEDIAN of the group of 15 INTEGERS, we know that 7 integers are greater than 25 and 7 integers are less than 25. We're told that the largest integer is exactly 25 more than the smallest integer, so to maximize the biggest value, we also have to maximize the smallest value. Since we're restricted to INTEGERS, the only way to get that maximum result is if the 7 integers that are less than 25 are CONSECUTIVE integers:

24, 23, 22, 21, 20, 19 and 18

GMAT assassins aren't born, they're made,
Rich

Hi SuryaNouliGMAT,

When a GMAT question asks you to consider a 'set' of numbers, you have to pay careful attention to the information that you're given about the set. In this prompt, we're told that the RANGE = 25 which means something really specific (re: the largest number is "25 more" than the smallest number - so if you change one of those two numbers, then you have to change the other as well). With this question, to maximize one number, you also have to maximize the other.

In other situations, to make the smallest number as big as possible, you have to make the other numbers as SMALL as possible. For example:

"The sum of 5 distinct integers is 100. What is the maximum possible value of the smallest number in this group?"

Here, we know that the 5 integers are DIFFERENT and that the sum of the integers is 100. To maximize the smallest value, we have to make the other 4 integers as small as possible (while still making sure that they are each bigger than the smallest integer). That group would be 18, 19, 20, 21 and 22.

GMAT assassins aren't born, they're made,
Rich
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Re: A set of 15 different integers has median of 25 and a range   [#permalink] 21 Apr 2020, 11:27

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