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Bunuel
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The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to 14 Mar 2019, 03:07
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75% (hard)

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54% (02:17) correct 46% (02:36) wrong based on 187 sessions

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The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of n?

(A) 7
(B) 9
(C) 19
(D) 24
(E) 26
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E
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The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to 14 Mar 2019, 03:21
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A is the answer. The sum of thev1st 4 numbers is 28. The mean of the resulting equals its median which is an integer and thereby the mean is an integer. Sum of 28 and 7 is divided by 5 and thereby it is tye only valid answer

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The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to 15 Mar 2019, 22:44
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Bunuel
The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of n?

(A) 7
(B) 9
(C) 19
(D) 24
(E) 26
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sum of no ; 3+6+9+10 ; 28
now
avg 28+n/5 = a digit which should be not same as given set values
so
at n =2 ; avg= 6
n=7; avg = 7
n=17; avg = 9
n=12; avg = 8
sum ; possible values of n =2+7+17+12 ; 38
Bunuel ; please provide solution on why answer is given as 26 not 38 ? seems n= 12 has not been considered?
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The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to 15 Mar 2019, 23:56
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Archit3110
Bunuel
The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to any of the other four. The median of the resulting set is equal to its mean. What is the sum of all possible values of n?

(A) 7
(B) 9
(C) 19
(D) 24
(E) 26

sum of no ; 3+6+9+10 ; 28
now
avg 28+n/5 = a digit which should be not same as given set values
so
at n =2 ; avg= 6
n=7; avg = 7
n=17; avg = 9
n=12; avg = 8
sum ; possible values of n =2+7+17+12 ; 38
Bunuel ; please provide solution on why answer is given as 26 not 38 ? seems n= 12 has not been considered?
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there are 3 possibilities ...

1) n<3 and 6

(N+3+6+9+10)/5 = 6
N= 30-28 =2

2) 9>n>6
(3+6+n+9+10)/5 = n
28+n = 5n
N= 7

3) n>10
(3+6+9+10+n)/5 = 9
N= 17.

If 12 is included as one of the value of n, then the median has to be 8 .... Because median is equal to mean... But as per the question, there are only 4 specific values...

Here it is
(3+6+9+10+12)/5 is equal to 8.. and the median of this set is 9... This won't be true...
So the possible values of n are 2+7+17 = 26

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The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to 17 Jan 2026, 11:53
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Archit3110


sum of no ; 3+6+9+10 ; 28
now
avg 28+n/5 = a digit which should be not same as given set values
so
at n =2 ; avg= 6
n=7; avg = 7
n=17; avg = 9
n=12; avg = 8
sum ; possible values of n =2+7+17+12 ; 38
Bunuel ; please provide solution on why answer is given as 26 not 38 ? seems n= 12 has not been considered?
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12 is not considered as question states "The median of the resulting set is equal to its mean".
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The set {3, 6, 9, 10} is augmented by a fifth element n, not equal to 17 Jan 2026, 23:14
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In such cases, one of the simplest approaches to solve would be to form cases to figure out the answer.

Let's first find out the value of the mean of the set: (3+6+9+10+n)/5 = (28+n)/5

Case 1: n > 10
If n > 10, then the median would be 9, as n would be placed after 10 and the middle-most element would be 9 in this case.

Equating mean and median => (28+n)/5 = 9, thus n = 17 (which is greater than 10), hence one possible value of n is 17.

Case 2: 6 < n < 9
When n is between 6 and 9, the median would be n as n would be the middle-most element of the resultant set.

Equating mean and median => (28+n)/5 = n, thus n = 7 (which is between 6 and 9), hence another possible value of n is 7.

Case 3: n < 6
When n is lesser than 6, the median would be 6 as it would be the middle-most element of the new set.

Equating mean and median => (28+n)/5 = 6, thus n = 2 (which is lesser than 6), hence another possible value of n is 2.

Thus, we get 3 such values of n, and the sum of them all = 17+7+2 = 26

Hence, the correct answer is Option E, i.e. 26 (Answer)
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