List Q consists of {12, 17, 21, 24, 30 and x}. For how many integer va

Question

List Q consists of {12, 17, 21, 24, 30 and x}. For how many integer values of x is the mean of S equal to the median of S?

(A) None 
(B) One 
(C) Two 
(D) Three 
(E) More than three

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CareerGeek · Accepted Answer

Mean = (12 + 17 + 21 + 24 + 30 + x)/6 = (104 + x)/6

Median is average of 2 middle numbers.
Possible Middle numbers are: 
Case 1: When x ≥ 24; (21, 24)
Case 2: When 21 ≤ x ≤ 24; (21, x)
Case 3: When x ≤ 21; (17, 21)

Case 1: Median = (21 + 24)/2 = 45/2
Mean = Median
--> (104 + x)/6 = 45/2
--> 104 + x = 135
--> x = 31; An integer

Case 2: Median = (21 + x)/2
Mean = Median
--> (104 + x)/6 = (21 + x)/2
--> 104 + x = 63 + 3x
--> 2x = 41
--> x = 20.5; Not an integer

Case 3: Median = (17 + 21)/2 = 38/2 = 19
Mean = Median
--> (104 + x)/6 = 19
--> 104 + x = 114
--> x = 10; An integer

Possible integral values of x = 10, 31

Option C

GMATinsight · Answer

Please check the video for the step-by-step solution.

GMATinsight's Solution

https://www.youtube.com/watch?v=f0moI4RSn14

ScottTargetTestPrep · Answer

Solution:

The mean is (12 + 17 + 21 + 24 + 30 + x)/6 = (104 + x)/6. Since there are 6 values, the median is the average of the two values in the middle. It could be one of the following:

(17 + 21)/2 = 19 (if x is the smallest value or the second smallest value)

(x + 21)/2 (if x is the third smallest value or the third largest value)

(21 + 24)/2 = 22.5 (if x is the second largest value or the largest value)

We are given that the mean is equal to the median. If the median is 19, we have:

(104 + x)/6 = 19

104 + x = 114

x = 10

We see that if x = 10, it will be indeed the smallest value in the set. So 10 could be a value of x.

If the median is (x + 21)/2, we have:

(104 + x)/6 = (x + 21)/2

6(x + 21) = 2(104 + x)

6x + 126 = 208 + 2x

4x = 82

x = 20.5

Since 20.5 is not an integer, 20.5 could not be a value of x.

If the median is 22.5, we have:

(104 + x)/6 = 22.5

104 + x = 135

x = 31

We see that if x = 31, it will be indeed the largest value in the set. So 31 could be a value of x.

Answer: C

sayan640 · Answer

GMATinsight  , Kindly review what has been described for case no 2. 
If 21<x < 24 , then the series should be like this :- 12 ,17, 21, x, 24, 30 
And hence the median should (21 +x) / 2 ; Not (24+x ) / 2 as mentioned in the video.
Since we will get x=20.5 which does not lie in the range assumed , we will discard the value.

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List Q consists of {12, 17, 21, 24, 30 and x}. For how many integer va

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List Q consists of {12, 17, 21, 24, 30 and x}. For how many integer va

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