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Median Maniac

by gmat6087 » Sat Oct 06, 2012 7:40 am
I am totally lost in this problem. Even though questions similar to this have been posted earlier but none of the explanations have been satisfactory.


As per median definition if arrange all the given numbers in ascending order if n is the number of elements and n is odd then middle term's value is median if n is even then median is average of the 2 middle terms.


Q)A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?

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

Here 32 can be the greatest possible integer eg: 7,7,7,7...7,25(median),25,25,..25,25,32

Here 37 can be the greatest possible integer eg: 12,13,14,15...19,25(median),25,25,..25,37

Here 40 can be the greatest possible integer eg: 15,15,15,15....25(median),25,25,..26,40

Here 43 can be the greatest possible integer eg: 18,19,20,21....25(median),25,25,..26,43

So how do I approach the problem.
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by Brent@GMATPrepNow » Sat Oct 06, 2012 8:16 am
gmat6087 wrote: Q)A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?
A) 32
B) 37
C) 40
D) 43
E) 50
Let's tackle this one step at a time.

First, we have 15 different integers.
We can let these 15 spaces represent the 15 numbers written in ascending order: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

If the median is 25, we can add this as the middle value: _ _ _ _ _ _ _ 25 _ _ _ _ _ _ _
Notice that 7 of the remaining numbers must be greater than 25 and the other 7 remaining number must be less than 25.

Since, we are told that the range is 25, we know that the greatest number minus the smallest number = 25

Now notice two things:
1) Once we know the value of the smallest number, the value of the greatest number is fixed.
For example, if the smallest number were 10, then the greatest number would have to be 35 in order to have a range of 25
Similarly, if the smallest number were 12, then the greatest number would have to be 37 in order to have a range of 25

2) If we want to maximize the value of the greatest number, we need to maximize the value of the smallest number.

So, how do we maximize the value of the smallest number in the set?
To do this, we must maximize each of the 7 numbers that are less than the median of 25.

Since the 15 numbers are all different, the largest values we can assign to the numbers less than the median of 25 are as follows:
18 19 20 21 22 23 24 25 _ _ _ _ _ _ _ (this maximizes the value of the smallest number)

If 18 is the maximum value we can assign to the smallest number, and if the range of the 15 numbers is 25, then greatest number must equal 43 (since 43 - 18 = 25)

So, the numbers are as follows: 18 19 20 21 22 23 24 25 _ _ _ _ _ _ 43 (the missing numbers don't really matter here)

This means the answer is [spoiler]43 = D[/spoiler]

Cheers,
Brent
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by anuprajan5 » Sat Oct 06, 2012 8:27 am
gmat6087 wrote:I am totally lost in this problem. Even though questions similar to this have been posted earlier but none of the explanations have been satisfactory.


As per median definition if arrange all the given numbers in ascending order if n is the number of elements and n is odd then middle term's value is median if n is even then median is average of the 2 middle terms.


Q)A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?

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

Here 32 can be the greatest possible integer eg: 7,7,7,7...7,25(median),25,25,..25,25,32

Here 37 can be the greatest possible integer eg: 12,13,14,15...19,25(median),25,25,..25,37

Here 40 can be the greatest possible integer eg: 15,15,15,15....25(median),25,25,..26,40

Here 43 can be the greatest possible integer eg: 18,19,20,21....25(median),25,25,..26,43

So how do I approach the problem.
Hey - the key takeaways in this question are different integers and greatest integer

Taking your cases -

Here 32 can be the greatest possible integer eg: 7,7,7,7...7,25(median),25,25,..25,25,32 - Not possible as they are different integers in the set

Here 37 can be the greatest possible integer eg: 12,13,14,15...19,25(median),25,25,..25,37 - Possible but will not gives you the largest integer in the set. here you only get 37.

Here 40 can be the greatest possible integer eg: 15,15,15,15....25(median),25,25,..26,40 - Not possible as they are different integers in the set.

Here 43 can be the greatest possible integer eg: 18,19,20,21....25(median),25,25,..26,43 - Perfect


As an addition, vision this:

Since the range is constant ie: 25 and the range is the difference between the largest and smallest numbers, in this case, since the integers are different, the range is essentially the distance between the largest and smallest integers.

If you make the smallest integer even smaller, since the distance is 25, the largest integer will get smaller.

If you bring the smallest integer closer to the median and stay within the rule that the integers have to be different, the greatest will be 43.

I hope that makes some sense :)

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by gmat6087 » Sat Oct 06, 2012 8:36 am
Brent@GMATPrepNow wrote:
gmat6087 wrote: Q)A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?
A) 32
B) 37
C) 40
D) 43
E) 50
Let's tackle this one step at a time.

First, we have 15 different integers.
We can let these 15 spaces represent the 15 numbers written in ascending order: _ _ _ _ _ _ _ _ _ _ _ _ _ _ _

If the median is 25, we can add this as the middle value: _ _ _ _ _ _ _ 25 _ _ _ _ _ _ _
Notice that 7 of the remaining numbers must be greater than 25 and the other 7 remaining number must be less than 25.

Since, we are told that the range is 25, we know that the greatest number minus the smallest number = 25

Now notice two things:
1) Once we know the value of the smallest number, the value of the greatest number is fixed.
For example, if the smallest number were 10, then the greatest number would have to be 35 in order to have a range of 25
Similarly, if the smallest number were 12, then the greatest number would have to be 37 in order to have a range of 25

2) If we want to maximize the value of the greatest number, we need to maximize the value of the smallest number.

So, how do we maximize the value of the smallest number in the set?
To do this, we must maximize each of the 7 numbers that are less than the median of 25.

Since the 15 numbers are all different, the largest values we can assign to the numbers less than the median of 25 are as follows:
18 19 20 21 22 23 24 25 _ _ _ _ _ _ _ (this maximizes the value of the smallest number)

If 18 is the maximum value we can assign to the smallest number, and if the range of the 15 numbers is 25, then greatest number must equal 43 (since 43 - 18 = 25)

So, the numbers are as follows: 18 19 20 21 22 23 24 25 _ _ _ _ _ _ 43 (the missing numbers don't really matter here)

This means the answer is [spoiler]43 = D[/spoiler]

Cheers,
Brent
Brent,
Thanks a lot for the quick response. I just have one doubt if a median of a number is 25 and range is 25 then the numbers left to the median except to the smallest number can be 25 right? Because it is not mandatory that numbers should be ascending order(or lesser than the value of median)

so,
(lowest number)25,25,25,25,25.....25(median),25,25,25,25...50(Highest number) is this sequence valid?.
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by anuprajan5 » Sat Oct 06, 2012 9:02 am
gmat6087 wrote: Brent,
Thanks a lot for the quick response. I just have one doubt if a median of a number is 25 and range is 25 then the numbers left to the median except to the smallest number can be 25 right? Because it is not mandatory that numbers should be ascending order(or lesser than the value of median)

so,
(lowest number)25,25,25,25,25.....25(median),25,25,25,25...50(Highest number) is this sequence valid?.
Hi,

They are different integers. SO the numbers to the left and right of the median should each be different.

Q)A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?


Regards
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by gmat6087 » Sat Oct 06, 2012 9:05 am
anuprajan5 wrote:
gmat6087 wrote: Brent,
Thanks a lot for the quick response. I just have one doubt if a median of a number is 25 and range is 25 then the numbers left to the median except to the smallest number can be 25 right? Because it is not mandatory that numbers should be ascending order(or lesser than the value of median)

so,
(lowest number)25,25,25,25,25.....25(median),25,25,25,25...50(Highest number) is this sequence valid?.
Hi,

They are different integers. SO the numbers to the left and right of the median should each be different.

Q)A set of 15 different integers has a median of 25 and a range of 25. What is the greatest possible integer that could be in this set?


Regards
Anup

oops my mistake, i overlooked "the 15 diff integers"
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