Mean Median problem

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Mon Jul 30, 2012 12:19 am

Hello,

Can you please assist here:

S = {1,2,5,7,x}

If x is a positive integer, is the mean of set S greater than 4?

1) The median of set S is greater than 2
2) The median of set S is equal to the mean of set S

Thanks a lot for your help

Best Regards,
Sri

Ans: B

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Source: Beat The GMAT — Data Sufficiency | Mon Jul 30, 2012 12:19 am

xesh: Newbie | Next Rank: 10 Posts; Posts: 2; Joined: Mon Jul 30, 2012 12:53 am; Thanked: 1 times

by xesh » Mon Jul 30, 2012 1:11 am

gmattesttaker2 wrote:Hello,

Can you please assist here:

S = {1,2,5,7,x}

If x is a positive integer, is the mean of set S greater than 4?

1) The median of set S is greater than 2
2) The median of set S is equal to the mean of set S

Thanks a lot for your help

Best Regards,
Sri

Ans: B

for the mean to be greater than 4,
(1+2+5+7+x)/5 > 4
15+x > 20
x>5
Statement 1 :

statement 1 says median is greater than 2 it implies that the least possible value of x can be 3, consider case S=(1,2,3,5,7), here x = 3, and mean is less than 4
S=(1,2,5,7,10), here x = 10, and mean is greater than 4
both these sets satisfy statement 1
hence insufficient

Statement 2 :
statement 2 says mean = median
(15 + x)/5 = median (note since the no of terms in S is odd the median has to be an integer)
so least possible value of x can be 5, in this case mean will be 4 and median will be 5
next value of x can be 10, in this case mean = median = 5
hence B is enough to solve the above question.

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alex.gellatly: Master | Next Rank: 500 Posts; Posts: 435; Joined: Wed Nov 16, 2011 7:27 amFollowed by: 16 members; Thanked: 48 times

by alex.gellatly » Mon Jul 30, 2012 8:38 pm

gmattesttaker2 wrote:Hello,

Can you please assist here:

S = {1,2,5,7,x}

If x is a positive integer, is the mean of set S greater than 4?

1) The median of set S is greater than 2
2) The median of set S is equal to the mean of set S

Statement 1:
If the median of the set is greater than 2, then x could take the following values:
1,2,3,5,7
1,2,5,5,7
In both of these examples the median is greater than 2, but we get different values of X (3,5). Thus, this statement is insufficient.

Statement 2:
This one is a bit more tricky. The key to note that the question stem stated the X is an integer. You should always read GMAT questions very careful as they always state some important thing like that...
Anyway, if the median=mean then...
(15+x)/5=mean=median [Do you understand where this came from]
So, 15+x=5*median.. Now, normally you would think this can't be solved. To unknown variables (X and median). But remember X must be an integer. Thus, if 5*median is a multiple of 5, then 15+x must be a multiple of five, and so x must be 5, 10, 15, ect. Note this least value for X is 5, which satisfies the question. Hence, sufficient. Remember, we don't have to solve for x, just say if it is greater than 4.
I hope this helps

A useful website I found that has every quant OG video explanation:

https://www.beatthegmat.com/useful-websi ... tml#475231

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GMAT/MBA Expert

Brent@GMATPrepNow: GMAT Instructor; Posts: 16207; Joined: Mon Dec 08, 2008 6:26 pmFollowed by: 1268 members; Location: Vancouver, BC; Thanked: 5254 times; GMAT Score:770

by Brent@GMATPrepNow » Tue Jul 31, 2012 6:55 am

gmattesttaker2 wrote: S = {1,2,5,7,x}
If x is a positive integer, is the mean of set S greater than 4?

1) The median of set S is greater than 2
2) The median of set S is equal to the mean of set S

Ans: B

Target question: Is the mean of set S greater than 4?

In other words, is (1+2+5+7+x)/5 > 4?
Simplify: Is 1+2+5+7+x > 20?
Simplify: Is 15+x > 20?
Simplify: Is x > 5?

Now that we've rephrased the target question as "Is x > 5?", the question is much easier to handle.

Aside: If a set consists of an odd number of elements, the median will be the middle number.

Statement 1: The median of set S is greater than 2
So, the middle number (the median) is not 2. So, it must be either 5 or x (if x between 2 and 5).
What does this tell us about x? Here are two possibilities.
case a: x=3, which gives us {1,2,3,5,7}. In this case, the median is 3 and x is not greater than 5.
case b: x=6, which gives us {1,2,5,6,7}. In this case, the median is 5 and x is greater than 5.
Since we have conflicting answers to the rephrased target questions, statement 1 is NOT SUFFICIENT

Statement 2: The median of set S is equal to the mean of set S
We know that the mean = (x+15)/5,.
We can rewrite this as: the mean = (x/5) + (15/5) or the mean = (x/5) + 3

Important: Since x must be a positive integer, and since the other four numbers are positive integers, we know that the median must be a positive integer. If the median = mean, then the mean is also a positive integer.

For (x/5) + 3 to be a positive integer, x must be divisible by 5.
Let's see what happens when x=5.
When x=5, the median=5 and the mean=4. Nope. x cannot equal 5.

Let's see what happens when x=10.
When x=10, the median=5 and the mean=5.
Great, that works.
Now, should we keep checking other values of x to see what happens? No, we don't need to.
Remember that our rephrased target question: Is x > 5?
We already saw that x can't equal 5, and we've shown that x must be a positive integer that's divisible by 5. Since x could equal 10, we've already shown that x must be greater than 5.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT.

Answer = B

Cheers,
Brent

Brent Hanneson - Creator of GMATPrepNow.com

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gmattesttaker2: Legendary Member; Posts: 641; Joined: Tue Feb 14, 2012 3:52 pmFollowed by: 8 members; Thanked: 11 times

by gmattesttaker2 » Fri Aug 03, 2012 7:46 pm

Brent@GMATPrepNow wrote:
gmattesttaker2 wrote: S = {1,2,5,7,x}
If x is a positive integer, is the mean of set S greater than 4?

1) The median of set S is greater than 2
2) The median of set S is equal to the mean of set S

Ans: B
Target question: Is the mean of set S greater than 4?

In other words, is (1+2+5+7+x)/5 > 4?
Simplify: Is 1+2+5+7+x > 20?
Simplify: Is 15+x > 20?
Simplify: Is x > 5?

Now that we've rephrased the target question as "Is x > 5?", the question is much easier to handle.

Aside: If a set consists of an odd number of elements, the median will be the middle number.

Statement 1: The median of set S is greater than 2
So, the middle number (the median) is not 2. So, it must be either 5 or x (if x between 2 and 5).
What does this tell us about x? Here are two possibilities.
case a: x=3, which gives us {1,2,3,5,7}. In this case, the median is 3 and x is not greater than 5.
case b: x=6, which gives us {1,2,5,6,7}. In this case, the median is 5 and x is greater than 5.
Since we have conflicting answers to the rephrased target questions, statement 1 is NOT SUFFICIENT

Statement 2: The median of set S is equal to the mean of set S
We know that the mean = (x+15)/5,.
We can rewrite this as: the mean = (x/5) + (15/5) or the mean = (x/5) + 3

Important: Since x must be a positive integer, and since the other four numbers are positive integers, we know that the median must be a positive integer. If the median = mean, then the mean is also a positive integer.

For (x/5) + 3 to be a positive integer, x must be divisible by 5.
Let's see what happens when x=5.
When x=5, the median=5 and the mean=4. Nope. x cannot equal 5.

Let's see what happens when x=10.
When x=10, the median=5 and the mean=5.
Great, that works.
Now, should we keep checking other values of x to see what happens? No, we don't need to.
Remember that our rephrased target question: Is x > 5?
We already saw that x can't equal 5, and we've shown that x must be a positive integer that's divisible by 5. Since x could equal 10, we've already shown that x must be greater than 5.
Since we can answer the target question with certainty, statement 2 is SUFFICIENT.

Answer = B

Cheers,
Brent

Hello Brent,

Hope all is well. Thanks a lot for the excellent and detailed explanation. Your explanation is awesome (as always!). Thank you very much again.

Best Regards,
Sri

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