Set S consists of five consecutive integers and set T consists of seven consecutive integers. Is the median of the numbers in set S equal to the median of the numbers in set T?
(1) The median of the numbers in Set S is 0
(2) The sum of the numbers in set S is equal to the sum of the numbers in set T
Good Question +1.
Let's start by revisiting the concepts of mean and median in the set of consecutive integers.
For any set of consecutive integers, or for any set of evenly spaced numbers, the mean and the median are equal. If there's an odd number of terms in the set, then the median or mean is the middle number.
As Set S and T consist of the odd no of consecutive integers, we can conclude that the median and the mean of each set are equal and will be the middle term in each set.
For example let's take any set of 5 consecutive integers: 3,4,5,6,7
Median = middle term in the set i.e 5
Since the terms in the set as evenly spaced, Average = (first term + last term)/2 = 3+7 /2 = 10/2 =5
Now, it's time to analyze the Question stem: Is the median of the numbers in set S equal to the median of the numbers in set T?
Can we re-frame Q.stem as 'Is the mean of the numbers in set S equal to the mean of the numbers in set T?' Yes, you can.
Statement 1: The median of the numbers in Set S is 0
Since the median of set S is the same as its middle term, we can conclude that Set S = -2,-1,0,1,2
But, we don't have any info regarding Set T, which clearly indicates that Statement 1 alone is not sufficient.
Statement 2: The sum of the numbers in set S is equal to the sum of the numbers in set T
Since it's given that the sum of each set is equal, it is better to stick to the re-framed Q.stem form: Is the mean of Set S = the mean of Set T?
Let's check.
Is Sum of Set S/5 =Sum of Set T/7?
Case 1 :
If Sum of Set S = Sum of Set T = 0,then the mean of Set S =0 and the mean of Set T = 0,
Set S:-2,-1,0,1,2
Set T:-3,-2,-1,0,1,2,3
Since the mean of Set S and T are equal in this case and it gives a YES to the Q.stem
Is the mean of Set S and T are always equal, when both the sets have the same sum?
Most of the students will miss thinking beyond case 1. That makes this question a 700 level one.
Let's say the sum of Set S = the Sum of Set T = LCM ( 5,7 ) =35
Why did I pick 35 as the sum?
Mean of Set S = Sum of T /5 and the Mean of Set T = Sum of T /7
The Mean of odd no of consecutive integers is always an integer and the sum has to be a multiple of 5 and 7 . so, 35 would be a good number to start with.
Mean of Set S = 35/5 = 7
Set S: 5,6,7,8,9
Mean of Set T = 35/7 =5
Set T: 2,3,4,5,6,7,8
Here both the sets have different mean or median, even though they had the same sum. Here you are getting a No as the answer to the Q.stem
Another list that gives the same result: Sum of Set S = Sum of Set T = 70
Mean of Set S = 70/5 = 14
Set S: 12,13,14,15,16
Mean of Set T = 70/7 =10
Set T: 7,8,9,10,11,12,13
Since we are getting both Yes as well as a No to the Q.stem, Statement 2 alone is not sufficient.
Next is combining both statements...
St 1. clearly says that Set S : -2,-1,0,1,2
St 2. says that Sum of set S = Sum of Set T.
From St 1, the sum of set S = 0 and the Sum of Set T has to be 0. Hence the mean of Set T has to be 0 or the middle term in the set T is 0.
Set S:-2,-1,0,1,2
Set T:-3,-2,-1,0,1,2,3
Now we are limited to only one case and the median of set S and set T are equal, therefore Option C is the correct answer.
Thanks,
Clifin J Francis,
GMAT Quant Mentor