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# A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang

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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang [#permalink]
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vtran wrote:
A set S = {x, -8, -5, -4, 4, 6, 9, y} with elements arranged in increasing order. If the median and the mean of the set are the same, what is the value of |x|-|y|?

(A) -1
(B) 0
(C) 1
(D) 2
(E) Cannot be determined.

First off, the question tells us that the numbers are arranged in ascending order.
So, we know that x ≤ -8, and y ≥ 9

There are 8 elements in the set. So, the median = the average of the two middlemost values.
Here, the two middlemost values are -4 and 4
So, the median = (-4 + 4)/2 = 0/2 = 0

Since the median and the mean of the set are EQUAL, we know that the mean is also 0

That is, [x + (-8) + (-5) + (-4) + 4 + 6 + 9 + y]/8 = 0
Multiply both sides by 8 to get: x + (-8) + (-5) + (-4) + 4 + 6 + 9 + y = 0
Simplify: x + y + 2 = 0
This means x + y = -2

So, here's what we know:
x + y = -2
x ≤ -8
y ≥ 9

Let's find some values of x and y and see where this leads us....

x = -12 and y = 10
In this case, |x|-|y|= |-12|-|10| = 12 - 10 = 2

x = -13 and y = 11
In this case, |x|-|y|= |-13|-|11| = 13 - 11 = 2

x = -12.5 and y = 10.5
In this case, |x|-|y|= |-12.5|-|10.5| = 12.5 - 10.5 = 2

x = -100 and y = 98
In this case, |x|-|y|= |-100|-|98| = 100 - 98 = 2

As we can see, the answer will always be 2

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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang [#permalink]
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A set S = {x, -8, -5, -4, 4, 6, 9, y} with elements arranged in increasing order. If the median and the mean of the set are the same, what is the value of |x|-|y|?
(A) -1
(B) 0
(C) 1
(D) 2
(E) Cannot be determined.

Median of the set = (-4+4)/2 = 0
As per statement, Mean of the set = 0

Mean of the set
|y|- |x| +19-17 = 0 (where x is negative n y is positive)
|y|- |x| = -2

So the absolute difference between two numbers is 2
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A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang [#permalink]
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Kudos
Hi,

This question is built around a number of statistics concepts and you have to pay careful attention to how you organize your work.

The prompt gives us a number of Facts to work with:
1) We're given the following set of values: {X, -8, -5, -4, 4, 6, 9, Y}
2) We're told that they are in INCREASING order
3) We're told the Median and the Mean are the SAME

We're asked for the value of |X| - |Y|

Since there are 8 terms, the Median will equal the average of the 'middle two' terms. Those 'middle two' terms are -4 and 4, so the Median is 0 (and since the Median = the Mean, the overall average is 0). Since the overall average is 0, the sum of the 8 terms MUST be 0...

Adding up the terms, we have...
X + Y + 2

So, since X+Y+2 = 0....

X+Y = -2

At this point, since X and Y have an established relationship, we can use any pair of values that fits all of the facts...We can TEST VALUES to prove the answer....

IF....
X = -12
Y = 10

|-12| - |10| = +2

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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arranged in incr [#permalink]
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This is what I did, we know Median which is 0 ( -4+ 4/2),. since Mean is same, it has to be 0. ( x, -8, -5, -4, 4, 6, 9, y ) so how will it be 0. if x, -8, -5, -4 = 4, 6, 9, y .

hence the number is x -12 and and y 10. D is the answer.
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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang [#permalink]
A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arranged in increasing order. If the median and the mean of the set are the same, what is the value of |x|-|y|?

A. -1
B. 0
C. 1
D. 2
E. Cannot be determined.

S = { x, -8, -5, -4, 4, 6, 9, y } with elements arranged in increasing order
This means x < 0 and y > 0.

The median of a series with even numbers is the average of the middle two numbers
Hence the median here is (-4 + 4)/2 = 0

Mean = sum of all the terms/ no. of terms
(x - 8 -5 -5 +4 +6 +9 + y)/8 = 0 (Given that median = mean)

Hence, x + y = 2
We need to find |x| - |y|
and we know that x < 0 and y > 0

Opening the modulus with appropriate signs:
Modulus of any number is the absolute value of the number, or simply the positive value
Always remember that the modulus of a negative number opens with a negative sign and of a positive number opens with a positive sign

We have ,
|x| - |y| = -x - y = -(x + y) = -(-2) = 2
Hence Option D
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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arranged in incr [#permalink]
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A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arranged in increasing order. If the median and the mean of the set are the same, what is the value of |x|-|y|?

A. -1
B. 0
C. 1
D. 2
E. Cannot be determined.

Hi,
the set is
{ x, -8, -5, -4, 4, 6, 9, y }
median is the center of middle 2 numbers as number of elements in set is EVEN..
median= (-4+4)/2=0..
it is given MEAN = MEDIAN..
so MEDIAN= MEAN = { x +(-8)+( -5)+( -4)+ 4+ 6+ 9+ y }/8 = 0
$$\frac{(x+y+2)}{8}=0$$..
or x+y+2=0..
x= -(y+2)
|x|=(y+2)..
so
|x|-|y|= |y+2|-|y|= y+2-y=2
D
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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arranged [#permalink]
We know that the median in this set is 0
If the mean of this set must be equal to 0,
the sum of elements in the set must always be zero

Adding the terms, we get x-17+19+y = 0
This is only possible when -x = 2+y eg, if y = 10, x=-12
Since x will always bme greater than y & we will have a difference of 2.

mod(x) - mod(y) = 2 always(Option B)
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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang [#permalink]
Quote:
Now, since the elements in the set are arranged in increasing order, then x<0x<0 and y>0y>0, so |x|−|y|=−x−y=−(x+y)=−(−2)=2|x|−|y|=−x−y=−(x+y)=−(−2)=2.

Can you please elaborate more on how you came up with -x-y from |x|−|y|?
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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang [#permalink]
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Zoser wrote:
Quote:
Now, since the elements in the set are arranged in increasing order, then x<0x<0 and y>0y>0, so |x|−|y|=−x−y=−(x+y)=−(−2)=2|x|−|y|=−x−y=−(x+y)=−(−2)=2.

Can you please elaborate more on how you came up with -x-y from |x|−|y|?

For x<0, |x|= -x because anything thatches out of || has to be a non negative number (and we know that x is negative).
Since, y is positive,, |y| = y

However, a more intuitive way to look at this problem is that it is asking you for the difference of magnitudes of x and y.

We know that, the remaining elements (except x and y) add upto 2 and to nullify the same you need the magnitude of X greater than that of Y by 2 units.
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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang [#permalink]
vtran wrote:
A set S = {x, -8, -5, -4, 4, 6, 9, y} with elements arranged in increasing order. If the median and the mean of the set are the same, what is the value of |x|-|y|?

(A) -1
(B) 0
(C) 1
(D) 2
(E) Cannot be determined.

Thus, Mean is also = 0
Solving we get.....--> x+y+2=0,

==> x+y=-2

x= -2-y

as x<0, mod x = 2+y
as y>0, mod y = y

Therefeore, mod x - mod y= 2+y-y ==> 2
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Re: A set S = { x, -8, -5, -4, 4, 6, 9, y } with elements arrang [#permalink]
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