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Set S consists of five consecutive integers, and set T consi [#permalink]

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19 Feb 2011, 09:43

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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

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?

Sets S and T are evenly spaced. In any evenly spaced set (aka arithmetic progression): (mean) = (median) = (the average of the first and the last terms) and (the sum of the elements) = (the mean) * (# of elements).

So the question asks whether (mean of S) = (mean of T)?

(1) The median of the numbers in Set S is 0 --> (mean of S) = 0, insufficient as we know nothing about the mean of T, which may or may not be zero.

(2) The sum of the numbers in set S is equal to the sum of the numbers in set T --> 5*(mean of S) = 7* (mean of T) --> answer to the question will be YES in case (mean of S) = (mean of T) = 0 and will be NO in all other cases (for example (mean of S) =7 and (mean of T) = 5). Not sufficient.

(1)+(2) As from (1) (mean of S) = 0 then from (2) (5*(mean of S) = 7* (mean of T)) --> (mean of T) = 0. Sufficient.

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.

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!

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.

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
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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

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.

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

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.

Re: Set S consists of five consecutive integers, and set T consi [#permalink]

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08 Sep 2013, 09:32

For those who wants to verify the insufficiency of statement 2: Infinite data sets are possible, following are few data sets for which sum of 5 consecutive numbers and 7 consecutive numbers is same. Previously I assumed that no such set is possible except a set uniform around zero. I selected B

Match found Sum -70=-70 5 consecutive no = -16,-15,-14,-13,-12 7 consecutive no = -13,-12,-11,-10,-9,-8,-7

Match found Sum -35=-35 5 consecutive no = -9,-8,-7,-6,-5 7 consecutive no = -8,-7,-6,-5,-4,-3,-2

Match found Sum 0=0 5 consecutive no = -2,-1,0,1,2 7 consecutive no = -3,-2,-1,0,1,2,3

Match found Sum 35=35 5 consecutive no = 5,6,7,8,9 7 consecutive no = 2,3,4,5,6,7,8

Match found Sum 70=70 5 consecutive no = 12,13,14,15,16 7 consecutive no = 7,8,9,10,11,12,13
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Re: Set S consists of five consecutive integers, and set T consi [#permalink]

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24 Sep 2014, 06:22

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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.
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Re: Set S consists of five consecutive integers, and set T consi [#permalink]

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10 Jan 2016, 06:52

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
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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.
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Re: Set S consists of five consecutive integers, and set T consi [#permalink]

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11 Jan 2016, 11:56

1

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VeritasPrepKarishma wrote:

mydreambschool wrote:

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

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)

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.

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.
_________________

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

When there are consecutive integers in an original condition, you only need to know the first number of variable. So, there is 1 variable in this case. That is, you need to know the first starting number of set S and set T, which is S={n-2,n-1,n,n+1,n+2}, T={m-3,m-2,m-1,m,m+1,m+2,m+3}. And there are 2 variables(n, m), which should match with the number of equations. So you need 2 equations. For 1) 1 equation, for 2) 1 equation, which is likely to make C the answer. When 1) & 2), it becomes S={-2,-1,0,1,2,3}, T={-3,-2,-1,0,1,2,3} -> yes. In S={n-2,n-1,n,n+1,n+2}, T={m-3,m-2,m-1,m,m+1,m+2,m+3}, 5n=7m. That is, n=7, m=5. So, S={5,6,7,8,9}, T={2,3,4,5,6,7,8} is no and not sufficient. Therefore, the answer is C.

For cases where we need 2 more equations, such as original conditions with “2 variables”, or “3 variables and 1 equation”, or “4 variables and 2 equations”, we have 1 equation each in both 1) and 2). Therefore, there is 70% chance that C is the answer, while E has 25% chance. These two are the majority. In case of common mistake type 3,4, the answer may be from A, B or D but there is only 5% chance. Since C is most likely to be the answer using 1) and 2) separately according to DS definition (It saves us time). Obviously there may be cases where the answer is A, B, D or E.
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Re: Set S consists of five consecutive integers, and set T consi [#permalink]

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13 Jan 2016, 09:13

I would go for B. Set S (s, s+1, s+2, s+3, s+4) has sum= 5s+10. Set S has median s+2 Set T (t, t+1, t+2, t+3, t+4, t+5, t+6) has sum= 7t+21. Set T has median t+3 Since 5s+10=7t+21, thus s=1.4t+2.2. Obvisous s+2 = (1.4t +4.2) > (t+3) It means median of set S always > median of set T.