Set S consists of 5 values, not necessarily in ascending

Question

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

Vips0000 · Accepted Answer

There could be 3 medians of this set. 
8 , when x <8
x , when 8<=x<=12
and
12, when x>12

There could be 1 mean = (40+x)/5

Now, we can solve (40+x)/5 = 8 , x, 12 to find out values.. and if valid values are found that solution counts. 
We get x= 0 ,10, 20. Hence 3 solutions.

Ans D it is.

Vips0000 · Answer

Actually that (such numbers) could be a trap in this question.

You might remember - 
If set is evenly spaced then for sure mean=median
But, if mean=median, it doesnt mean that set is evenly spaced. It may or may not be.

So, if we make x=0 and x=20, and find mean and median =10. We are pratically limiting our scope only to evenly spaced sets. And thus we get a few solutions-but not all.

We still need to look for solution for the other case where set is not evenly spaced. (Also x=10 doesnt give u an evenly spaced set.. so concept cant be applied directly, it would be more evident if the number of elements in set is even).

One easy concept that u can use though is, if median =mean then numbers in the set will be symmetrical around median. Why it helps is that finding median is usually very simple,without calculation.

Bunuel · Answer

Similar questions to practice: what-is-the-median-of-the-numbers-4-5-6-7-9-and-x-141730.html if-set-s-consists-of-the-numbers-1-5-2-8-and-n-is-132570.html what-is-the-value-of-n-in-the-list-above-137225.html if-the-range-of-the-set-containing-the-numbers-x-y-and-z-127089.html the-sum-of-the-integers-in-list-s-is-the-same-as-the-sum-of-127755.html given-the-ascending-set-of-positive-integers-a-b-c-d-e-115675.html Hope it helps.

Marcab · Answer

@vips and @bunuel:
Since mean and median are the same for evenly spaced sets, so to make the entire set an evenly spaced set x can be made either 0 or 20.
Can this concept be used to find x=10 as the mean and median?

abrakadabra21 · Answer

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

Mean = 8+x/5

x<=4
median =8

x >8 & x<12
median = X

8+x/5 =X
x=10

x>12
8+x/5 =12
x=20

So 3

Vorovski · Answer

The mean is always equal to median when numbers in a set have equal deviation. For instance, in our case if deviation is  4 , the  x  should  0  or  20 . 
Another case when a set is ascending from the mean out both sides.  x  should be  10 . Please, look the atachments. Hope, it helps.

The answer is  D

gracie · Answer

if mean=(x+40)/5,
then x must be an even multiple of 5
let x=0: mean=8; median=8 yes
let x=10: mean=10; median=10 yes
let x=20; mean=12; median=12 yes
let x=30; mean=14; median=12 no
3 values of x work
D

hellosanthosh2k2 · Answer

First rearrange the given set in ascending order = {4,8, 12,16} {x}
 average = (40 + x) / 5

now depending on value of x, the median will change, also
median can be one of 8, 12, x

(1) suppose median = x, then average = median
 (40 + x)/5 = x => 4x = 40 => x = 10

(2) suppose median = 12, then average = 12
 (40 + x)/5 = 12 => x = 20

(3) suppose median = 8, then average = 8
 (40 + x)/5 = 8 => x = 0

so there are three possible values of x {0, 10, 20} for which median = mean => (D)

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Set S consists of 5 values, not necessarily in ascending

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