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TOUGH : Manhattan Challenge Problem

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by GmatKiss » Sat Oct 15, 2011 11:44 am
Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, x}. For how many values of x does the mean of set S equal the median of set S?

(A) Zero
(B) One
(C) Two
(D) Three
(E) More than three
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by gmatclubmember » Sat Oct 15, 2011 12:43 pm
The mean will be 8+x/5--------first equation
median = 8 if x < 4-------------second equation
=12 if 4 <= x <8---------third eq.
=x if 8 <= x < 12--------fourth eqn.
=12 if x >= 12-----------fifth eqn
If the mean and median are equal then

--"either" 8 = 8+x/5 =>x=0 (from first and second eqn)

--"or" 12 = 8+x/5 =>x=20 (from first and third eqn), but as per third eqn. x cannot be equal to 20, so this is an invalid value.

--"or" x=8+x/5 =>x=10 (from first and fourth eqn)

--"or" 12=8+x/5 => x=20 (from first and fifth eqn).

So X can take 3 values (0,10 and 20).

Answer is D.
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by GMATGuruNY » Sat Oct 15, 2011 2:10 pm
GmatKiss wrote:Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, x}. For how many values of x does the mean of set S equal the median of set S?

(A) Zero
(B) One
(C) Two
(D) Three
(E) More than three
When a set is SYMMETRICAL about the median, the mean = the median.
Since the given numbers are evenly spaced, there are only 3 options:

x = the smallest value:
0,4,8,12,16

x = the median, halfway between 8 and 12:
4,8,10,12,16

x = the greatest value:
4,8,12,16,20

The correct answer is D.
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by GmatKiss » Sat Oct 15, 2011 2:20 pm
GMATGuruNY wrote:
GmatKiss wrote:Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, x}. For how many values of x does the mean of set S equal the median of set S?

(A) Zero
(B) One
(C) Two
(D) Three
(E) More than three
When a set is SYMMETRICAL about the median, the mean = the median.
Since the given numbers are evenly spaced, there are only 3 options:

x = the smallest value:
0,4,8,12,16

x = the median, halfway between 8 and 12:
4,8,10,12,16

x = the greatest value:
4,8,12,16,20

The correct answer is D.
Hi Mitch,

why not between 12 and 16 (14) ? Am i missing anything!

TIA,
GK
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by GMATGuruNY » Sat Oct 15, 2011 2:50 pm
GmatKiss wrote:
GMATGuruNY wrote:
GmatKiss wrote:Set S consists of 5 values, not necessarily in ascending order: {4, 8, 12, 16, x}. For how many values of x does the mean of set S equal the median of set S?

(A) Zero
(B) One
(C) Two
(D) Three
(E) More than three
When a set is SYMMETRICAL about the median, the mean = the median.
Since the given numbers are evenly spaced, there are only 3 options:

x = the smallest value:
0,4,8,12,16

x = the median, halfway between 8 and 12:
4,8,10,12,16

x = the greatest value:
4,8,12,16,20

The correct answer is D.
Hi Mitch,

why not between 12 and 16 (14) ? Am i missing anything!

TIA,
GK
If x is halfway between 12 and 16, the numbers will NOT be symmetrical about the median:
4,8,12,14,16

The distances are 4-4-median-2-2.
Not symmetrical about the median.

Each of the sets in my post above is symmetrical about the median:

{0,4,8,12,16}
The distances are 4-4-median-4-4.

{4,8,10,12,16}
The distances are 4-2-median-2-4.

{4,8,12,16,20}
The distances are 4-4-median-4-4.
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My students have been admitted to HBS, CBS, Tuck, Yale, Stern, Fuqua -- a long list of top programs.

As a tutor, I don't simply teach you how I would approach problems.
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by saketk » Mon Oct 17, 2011 9:40 pm
the set is {4,8,12,16,x}

Mean is (40+x)/5 = Median [given]

To find Median we can place X before 4 or between 4 and 8 -- median in this case will be 8

This will give value of x = 0 [means we cannot put X between 4 and 8]

Next, put X between 8 and 12. This will give median = X

Subsitute this value in equation 1 we will get x =10

Next, after 16 we will get Median = 12 and corresponding value of X = 20

Hence total values = 3

Answer option D
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