If the mean of the set of 5 distinct positive integers (p,q,2,4,6) is

Mean(m)= 12+p+q /5 and Median M is the middle integer in the set arranged in ascending or descending order since the set has odd number of elements.
We need information on p and q to be able to answer the question.

1) If the highest term in the set is 6, then there can be two different sets of numbers possible for p and q like (1 and 3) and (3 and 5)
If p and q are 1 and 3, then m would be 16/5 which is more than the median of 3.
If p and q are 3 and 5, then m would be 20/5 which is equal to the median of 4.
Not sufficient.

2) If the lowest term in the set is 2, then then there can be two different sets of numbers possible for p and q like (7 and 8) and (10 and 11)
If p and q are 7 and 8, then m would be 27/5 which is less than the median of 6.
If p and q are 10 and 11, then m would be 33/5 which is more than the median of 6.
Not sufficient.

1&2) p and q can only have one combination possible which is 3 and 5. Thus mean= median and we can successfully answer the question.

Answer: C

#1
The highest term in the set is 6
we get following
(1,2,3,4,6) ; 3.2 mean and median 3
(1,2,4,5,6) ; 3.6 mean and median 4
(0,2,3,4,6) ; 3 mean and median 3
(2,3,4,5,6) ; 4 mean and median 4
insufficient
#2
The lowest term in the set is 2
(2,3,4,5,6) ; 4 mean and median 4
(2,4,6,8,10) ; 6 mean and median 6
(2,3,4,6,10) ; 5 mean and median 4
insufficient
from1 &2
(2,3,4,5,6) ; 4 mean and median 4
sufficient
OPTION C


If the mean of the set of 5 distinct positive integers (p,q,2,4,6) is m and the median is M, is m > M?
1) The highest term in the set is 6
2) The lowest term in the set is 2

If the mean of the set of 5 distinct positive integers (p,q,2,4,6) is m and the median is M, is m > M?
1) The highest term in the set is 6
2) The lowest term in the set is 2

Statement 1 will produce different results if we take different combinations from 1, 3, 5 as the value of p and q. Not sufficient
Statement 2 makes the set (2, 3, 4, 5, 6). we can find the mean and median from here. sufficient
B is the answer

Given: The mean of the set of 5 distinct positive integers (p,q,2,4,6) is m and the median is M
Asked: Is m > M?
mean(p,q,2,4,6) = (p+q+12)/5=m

1) The highest term in the set is 6
{p,q} < 6; p & q can take values {1,3,5}
If (p,q) = (1,5); m=3.6; M=4; m<M
If (p,q) = (1,3); m=3.2; M=3; m>M
If (p,q) = (3,5); m=4; M=4; m=M
NOT SUFFICIENT

2) The lowest term in the set is 2
{p,q}>2; p & q can take values {3,5,7,8……}
If (p,q) = (3,5); m=4; M=4; m=M
If (p,q) = (3,7); m=4.4; M=4; m>M
NOT SUFFICIENT

(1) + (2)
1) The highest term in the set is 6
{p,q} < 6;
2) The lowest term in the set is 2
{p,q} > 2
(p,q) = (3,5)
If (p,q) = (3,5); m=4; M=4; m=M
SUFFICIENT

IMO C

If the mean of the set of 5 distinct positive integers (p,q,2,4,6) is m and the median is M, is m > M?
1) The highest term in the set is 6
2) The lowest term in the set is 2

Statement 1
Since the set must be 5 distinct positive integers. There are only 3 possible numbers to replace p and q, which are 1,3, and 5.
If p and q are 5 and 3, mean = \(\frac{6+4+2+5+3}{5}=4 \) and the median is 4. In this case, M = m
If p and q are 5 and 1, mean = \(\frac{6+4+2+5+1}{5}=3.6 \) and the median is still 4. In this case, M>m

Since the relation between M and m are different, statement 1 is not sufficient. A and D are incorrect

Statement 2
Since the set must be 5 distinct positive integers. There are many possible numbers to replace p and q, such are 3 and 5 or 3 and 8
If p and q are 5 and 3, mean = \(\frac{6+4+2+5+3}{5}=4 \) and the median is 4. In this case, M = m
If p and q are 8 and 3, mean = \(\frac{6+4+2+8+3}{5}=4.6 \) and the median is still 4. In this case, m>M

Hence, both statements alone are not sufficient.

If both statements are used,
the highest term is 6 and the lowest term is 2,
2, p, 4, q, 6 --> p and q must be 3 and 5. Hence, with both statements, the relation between M and m can be found (C)

5 distinct positive integers (p,q,2,4,6)

Q. Mean (m) > Median (M)?

(1) The highest term in the set is 6

If the set is (1,2,3,4,6) then mean (m) = 3.2 > median (M) = 3
If the set is (2,3,4,5,6) then mean (m) = 4 is equal to median (M) = 4
--> We need more information to determine whether mean (m) > median (M) or not
NOT SUFFICIENT

(2) The lowest term in the set is 2
If the set is (2,3,4,6,8) then mean (m) = 4.6 > median (M) = 4
If the set is (2,3,4,5,6) then mean (m) = 4 is equal to median (M) = 4
--> We need more information to determine whether mean (m) > median (M) or not
NOT SUFFICIENT

Combined
We now know that the lowest and the highest terms in the distinct positive integer set are 2 and 6, respectively. As a result, the only possible set is (2,3,4,5, 6). Thus, the mean (m) = 4 is equal to the median (M) = 4.
--> We can deduce that mean of the set (m) is NOT greater than median of the same set (M).
SUFFICIENT

FINAL ANSWER IS (C)

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If the mean of the set of 5 distinct positive integers (p,q,2,4,6) is m and the median is M, is m > M?

(1) The highest term in the set is 6
(2) The lowest term in the set is 2

i)
Case I) :-
S = {1,2,3,4,6} => mean = 16/5 = 3.2; Median = 3 => m > M - Yes

Case II)
S = {2,3,4,5,6} => mean = Median = 4 => is m > M - No

Insufficient

ii)
Case I) :-
S = {2,3,4,5,6} => mean = Median = 4 => is m > M - No

Case II)
S = {2,4,6,10,12} => mean = 34/5 = 6.8; Median = 6 - is m>M - Yes

Insufficient

Combining,
S = {2,3,4,5,6} - The only set possible - mean = Median - No

Sufficient

Answer - C