Set S consists of five consecutive integers, and set T consi

For the above question, [C] seems logical since:

Statement 1: gives median of Set S = 0, Hence if S = {x,x+1,....x+4} median = x+2 => x+2 = 0, Hence the starting number is -2. [Not sufficient Alone]
Statement 2: gives Sum of numbers in S = Sum of numbers in y i.e. Avg/Median of Set S * 5 = Avg/Median of Set T*7
=> (2x+4)*5 = (2y+6)*7, where y is the starting element of Set T.
=> y+3 = 5*(x+2)/7, where y+3 is the median of Set T.
From above, the only way the medians can be equal, is when both of the values are 0. For all other values of x and y, the medians are not equal. The choice presents a level of uncertainty, and therefore can't be sufficient.

Since x+2 = 0, from Statement 1, putting values, y+3 = 0. Hence, the median of both the sets are equal. Please verify!

Regards,
Arpan

I'm happy to help with this. This is a really good question.

Statement #1 --- no info about T, obviously not sufficient by itself.

Statement #2 --- this is tricky ---- we could have two sets, say S = {5, 6, 7, 8, 9} and T = {2, 3, 4, 5, 6, 7, 8}, both of which have a sum of 35, and different medians, or we could have positives & negative centered around zero, with sums of zero and medians of zero. By itself, this statement is not sufficient.

Combined statements ---- very interesting now. If the median of S is 0, then S must be {-2, -1, 0, 1, 2}, which has a sum of zero. If the two sets have the same sum, then T must have a sum of zero, and the only way a set of an odd number of consecutive integers can have a sum of zero is if the list is centered on zero with the positives and negatives cancelling out. T must be {-3, -2, -1, 0, 1, 2, 3}, with a median of zero. The two lists have the same median. We are able to answer the prompt question, so the information is now sufficient.

Does all this make sense?

Mike

GIVEN: elements are consecutve integers.

ST1: Median of S = 0
No information about set T hence Insufficient.

ST2: Sum of elements in Set S = Sum of Numbers in Set T
If Set s = {x-2, x-1, x, x+1, x+2} Sum = 5x
If Set T = {y-3, y-2, y-1, y, y+1, y+2, y+3} Sum = 7y
We are given that 5x = 7y hence CASE 1: x = y = 0 or CASE 2: X = 7, Y = 5

Case1: If S = {-2,-1,0,1,2} median = 0 and Set T = {-3,-2,-1,0,1,2,3} median = 0. the sum of elements in each set = 0 and median of S = median of T
Case2: S = {5,6,7,8,9} median = 7 and Set T = {2,3,4,5,6,7,8} median = 5. the sum of elements in each set = 35 but median of S <> median of T
Hence Insufficient

Together, Only Case 1 possible
hence Sufficient
Ans C

Even though the answer to this question has been provided in detail, I would like to point out a takeaway here:

Sometimes, one statement helps you while 'analyzing the other statement alone'. It's good to forget the data of one statement while analyzing the other but don't forget the learning from the statement.

Let me explain:

S = {.........}
T = {.............}
Both S and T have consecutive integers. We want to know if they have the same median. If they have the same median, it means their middle number must be the same. Let's look at the statements:

(1) The median of the numbers in Set S is 0
Doesn't tell us anything about T so ignore it.

(2) The sum of the numbers in set S is equal to the sum of the numbers in set T

It's improbable that the medians will be same now, right? Can we say that certainly the median will not be the same and hence this statement alone is sufficient? To have the same median, their middle number must be the same. Say S = {2, 3, 4, 5, 6} and T = {1, 2, 3, 4, 5, 6, 7}. In this case, they will not have the same sum, right? Are we missing something? What about a case when the numbers are negative integers? Even if it doesn't occur to you by just looking at this statement, it must occur to you after reading the first statement. If 0 is the middle number, their sum could be the same while median is same. Hence you know that this statement by itself is not sufficient.

Using both statements, you can say that the sum of all members of S and sum of all members of T will be 0 and hence they DO have the same median.

Answer (C)

Forget conventional ways of solving math questions. In DS, Variable approach is the easiest and quickest way to find the answer without actually solving the problem.
Remember equal number of variables and independent equations ensures a solution.


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

Transforming the original condition we have 1 variable (the first term of set S) and since we need to know the first term of set T there is also 1 variable here, making it overall 2. We need 2 equations to match the number of variables and equations, and since there is 1 each in 1) and 2), there is high probability that C is the answer. Actual calculation gives us, using both 1) & 2), S={-2,-1,0,1,2}, T={-3,-2,-1,0,1,2,3} where both median is 0=0. The answer is yes, and thus the conditions are sufficient. Therefore the answer is C.

Why is it so easy to fall for a tricky second statement? Clearly it is not a very tough question at all. Is it because there is some residue memory from st1? Is it a better strategy to always start with st2 on DS questions? thanks

Yes, after you are done with Statement 1, it is normal for people to use its data while analysing the second statement "alone". The reason is simply that you have that piece of information and it is hard to ignore it now. I suggest my students to start from the beginning with stmnt 2 - that is, after getting done with stmnt 1 and reaching the required conclusion (sufficient/not sufficient), ignore it and go back to the question stem. Read it again like you would read a new question and read only statement 2. It just takes a few seconds extra. Even if you don't read the question stem completely, at least look at it so that you can see the data points it has and the data points present in stmnt 2 alone.
Reading stmnt 2 first may not be the solution since its data might linger in your mind while working on stmnt 1.
Usually, it is a good idea to start with the statement that "looks" easier. Once that is out of the way, it's easier to think clearly and you might get some hints for the other statement.

Hi Karishma,

I just wanted to point out a flaw in your logic on statement 2. We want to know whether the sum of S and T could be the same under 2 scenarios: with the same median and with different medians. Only this case proves that statement 2 is insufficient. Your logic is backwards: you're figuring out if the median could be the same with the same and different sums. Note that it's not quite the same question.

However, if by noticing that {5 6 7 8 9} and {2 3 4 5 6 7 8} have the same sums but different medians, and that any two set of consecutive integers centered around zero will have the same sum but DIFFERENT medians, then that is enough to say that statement 2 is insufficient.

To reiterate: you have to prove that you can find two pairs of sets with the same sum, one pair with the same median and one pair with different medians. Your logic does not rule out the case that if two sets have the same sum, they have the same median - which would make statement 2 sufficient.

There is no flaw in the logic. Please note how my solution begins:

"Even though the answer to this question has been provided in detail, I would like to point out a takeaway here:"

I am not looking at providing a detailed solution. I am only focusing on the takeaway from this question. The takeaway is "don't forget the learning from stmnt 1 though you need to forget the data"
In stmnt 2, I focus on the case where the learning from stmnt 1 should be utilised. The intent of the post is different and hence the solution focuses on that.

Further, my opening sentences on stmnt 2 should give you enough hint that it is easy to find cases where sum is same but median is not (also shown in solutions by others).

"It's improbable that the medians will be same now, right? Can we say that certainly the median will not be the same and hence this statement alone is sufficient?"

So all I worry about is the case where sum is same and so is the median. Median same but sum different is only a step towards arriving at median same and sum same case.

Below solutions is also very helpful to understand the problem.

For any set of consecutive integers:
Sum = (number of integers)(median).

Let s = the median of Set S and t = the median of set T.
Then:
Sum of the 5 integers in Set S = 5s.
Sum of the 7 integers in Set T = 7t.

Statement 1:
No information about t.
INSUFFICIENT.

Statement 2:
5s = 7t.
It's possible that s=t=0, in which case the medians are equal.
It's possible that s=7 and t=5, in which case the medians are NOT equal.
INSUFFICIENT.

Statements combined:
Since 5s=7t and s=0, we know that s=t=0.
SUFFICIENT.

The correct answer is C.

Is the contrary also true?

No. The mean = median does not necessarily means that the set is evenly spaced. For example, {0, 0, 1, 1, 2, 2} --> mean = median =1, but the set is not evenly spaced.

Hi,

Statement 1 : We have no info about T; therefore, it is insufficient.

Statement 2 :

Case 1:
sequence S looks like this : -2 -1 0 1 2
and sequence T looks like this : -3 -2 -1 0 1 2 3
Does the sum of S equals the sum of T? YES
Is the median the same ? YES

Case 2:
s+s+1+s+2+s+3+s+4 = t+t+1+t+2+t+3+t+4+t+5+t+6
5s+10=7t+21
5s=7t+11 => YES the two sequences are equal when s=5 and t=2. Is the median the same? NO

Getting Yes and NO means it's insufficient

Now if we combine the 2 statements, it becomes sufficient.

Answer C

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

ANSWER IS C THOUGH...