Q is greater than the largest number in list P

[jain2016]

List P contains m numbers; list Q contains n numbers. If the two lists are combined to produce list R, containing m + n numbers, is the median of list R greater than the median of list P ?

(1) The smallest number in list Q is greater than the largest number in list P.

(2) m = n

OAC

Hi Experts ,

Please explain

[GMATGuruNY]

List P contains m numbers; list Q contains n numbers. If the two lists are combined to produce list R, containing m + n numbers, is the median of list R greater than the median of list P ?

(1) The smallest number in list Q is greater than the largest number in list P.

(2) m = n

Statement 1: The smallest number in list Q is greater than the largest number in list P.
Test an EASY CASE that satisfies BOTH statements.
Case 1: P = {0} and Q = {1}, implying that R = {0, 1}.
Median of P = 0.
Median of R = 1/2.
In this case, median of R > median of P, so the answer to the question stem is YES.

Test an easy case that satisfies ONLY STATEMENT 1.
Case 2: P = {0, 0} and Q = {1}, implying that R = {0, 0, 1}.
Median of P = 0.
Median of R = 0.
In this case, median of R = median of P, so the answer to the question stem is NO.

Since the answer is YES in Case 1 but NO in Case 2, INSUFFICIENT.

Statement 2: m=n
Case 1 satisfies statement 2.
In Case 1, the answer to the question stem is YES.

Test a case that satisfies ONLY STATEMENT 2.
Case 3: P = {0} and Q = {0}, implying that R = {0, 0}.
Median of P = 0.
Median of R = 0.
In this case, median of R = median of P, so the answer to the question stem is NO.

Since the answer is YES in Case 1 but NO in Case 3, INSUFFICIENT.

Statements combined:
Let P = {a, b, c}, where a≤b≤c.
Let Q = {d, e, f}, where d≤e≤f.

Statement 1 indicates that d > c.
Thus:
R = {a, b, c, d, e, f}, where a ≤ b ≤ c < d ≤ e ≤ f}.

Adding together c≥b and d>b, we get:
c+d > 2b
(c+d)/2 > b.

Since the median of R = (c+d)/2, and the median of P = b, the resulting inequality implies that the median of R > median of P.
SUFFICIENT.

The correct answer is C.

[MartyMurray]

List P contains m numbers; list Q contains n numbers. If the two lists are combined to produce list R, containing m + n numbers, is the median of list R greater than the median of list P ?

(1) The smallest number in list Q is greater than the largest number in list P.

(2) m = n

The median of a list is the middle value in the list or, in a list with an even number of values, the mean of the two middle values in the list.

Statement 1:

This tells us that all of numbers in list Q are greater than those in list P.

Consider the following.

If list P contains 100 numbers, all of which are the same, and list Q contains just 1 number, the median of R will still be somewhere in the middle of list P, and it still be the same number that it was.

Answer to the question is No.

If list P contains 1 number and list Q contains 100 numbers, the median of R will be somewhere in the middle of list Q, all the values in which are greater than those in P.

Answer to the question is Yes.

Two different answers.

Insufficient.

Statement 2:

Use extreme examples.

Extreme example 1: P and Q are identical. In that case m = n and the median of R is the same as the median of P.

Answer is No.

Extreme Example 2: Duplicate Statement 1, saying that all of the values of Q are greater than those in P.

If m = n, the median of R will be the mean of the highest value in P and the lowest value in Q, and will obviously be greater than the median of P.

Answer is Yes.

Two different answers.

Insufficient.

Statements Combined:

The second extreme example used for analyzing Statement 2 showed that when m = n and all values of Q are greater than all values of P, the median of R is greater than any value, including the median, of list P.

Sufficient.

The correct answer is C.