CAT2 #10: If S is a finite set of consecutive even numbers..

[ssr174]

DS QUESTION

If S is a finite set of consecutive even numbers, is the median of S an odd number?

(1) The mean of set S is an even number.
(2) The range of set S is divisible by 4.

Can you pls explain why st. 2 is sufficient?

[jnelson0612]

DS QUESTION

If S is a finite set of consecutive even numbers, is the median of S an odd number?

(1) The mean of set S is an even number.
(2) The range of set S is divisible by 4.

Can you pls explain why st. 2 is sufficient?

Sure! Okay, let's start by rephrasing the question. I'm going to make up some example sets so we can see what is going on.
Set: 2, 4, 6
Median is 4
NOT odd
Set: 2, 4, 6, 8
Median is 5 (average of two middle numbers)
IS odd

What's the difference? If I have an odd number of members of the set, the median will be the middle value, which I know is even.

If I have an even number of members of the set, the median will be the average of the two middle values. This average will be odd.

So our rephrase is: Does the set have an even number of integers?

Statement 2 tells me that the range of the set is divisible by 4. Thus, the largest member minus the smallest member is a multiple of 4. Given that these are consecutive even integers, the set members must be:
x, x+2, x+4
OR x, x+2, x+4, x+6, x+8
OR x, x+2, x+4, x+6, x+8, x+10, x+12
etc.
Note that the set always has an odd number of integers.

Since the median will always be a member of the set, and thus an even integers, we can answer NO to the question "is the median odd?". Remember, NO is a sufficient answer; maybe is not a sufficient answer.

[thapliyalabhi]

DS QUESTION

If S is a finite set of consecutive even numbers, is the median of S an odd number?

(1) The mean of set S is an even number.
(2) The range of set S is divisible by 4.

Can you pls explain why st. 2 is sufficient?

Sure! Okay, let's start by rephrasing the question. I'm going to make up some example sets so we can see what is going on.
Set: 2, 4, 6
Median is 4
NOT odd
Set: 2, 4, 6, 8
Median is 5 (average of two middle numbers)
IS odd

What's the difference? If I have an odd number of members of the set, the median will be the middle value, which I know is even.

If I have an even number of members of the set, the median will be the average of the two middle values. This average will be odd.

So our rephrase is: Does the set have an even number of integers?

Statement 2 tells me that the range of the set is 4. Thus, the largest member minus the smallest member is 4. Given that these are consecutive even integers, the set members must be:
x, x+2, x+4

Since we have to have a set of three integers we know that the median, x+2, will be even. Thus, we can answer NO to the question "is the median odd?". Remember, NO is a sufficient answer; maybe is not a sufficient answer.

I have a query. In the official explanation as well as the explanation given above, we have considered even INTEGERS, while the question talks about even NUMBERS.
Why are we not considering numbers such as 2.2, 2.4, 2.42 etc. I guess these are also even numbers. Please correct me if I am wrong.
Although, I believe that considering the numbers mentioned by me won't affect the answer, but still want to get my doubt clarified.

[jnelson0612]

DS QUESTION

If S is a finite set of consecutive even numbers, is the median of S an odd number?

(1) The mean of set S is an even number.
(2) The range of set S is divisible by 4.

Can you pls explain why st. 2 is sufficient?

Sure! Okay, let's start by rephrasing the question. I'm going to make up some example sets so we can see what is going on.
Set: 2, 4, 6
Median is 4
NOT odd
Set: 2, 4, 6, 8
Median is 5 (average of two middle numbers)
IS odd

What's the difference? If I have an odd number of members of the set, the median will be the middle value, which I know is even.

If I have an even number of members of the set, the median will be the average of the two middle values. This average will be odd.

So our rephrase is: Does the set have an even number of integers?

Statement 2 tells me that the range of the set is 4. Thus, the largest member minus the smallest member is 4. Given that these are consecutive even integers, the set members must be:
x, x+2, x+4

Since we have to have a set of three integers we know that the median, x+2, will be even. Thus, we can answer NO to the question "is the median odd?". Remember, NO is a sufficient answer; maybe is not a sufficient answer.

I have a query. In the official explanation as well as the explanation given above, we have considered even INTEGERS, while the question talks about even NUMBERS.
Why are we not considering numbers such as 2.2, 2.4, 2.42 etc. I guess these are also even numbers. Please correct me if I am wrong.
Although, I believe that considering the numbers mentioned by me won't affect the answer, but still want to get my doubt clarified.

An "even" number MUST be an integer, because the definition of an even number is a number evenly divisible by 2 with no remainder. As you can see, 2.2 and the other numbers you listed would not fit this definition. :-)

[Macaroni]

"(2) The range of set S is divisible by 4."

"Statement 2 tells me that the range of the set is 4. Thus, the largest member minus the smallest member is 4. Given that these are consecutive even integers, the set members must be:
x, x+2, x+4

Since we have to have a set of three integers we know that the median, x+2, will be even. Thus, we can answer NO to the question "is the median odd?". Remember, NO is a sufficient answer; maybe is not a sufficient answer."

My Question:

Was this question explained wrong? I feel like Jamie explained this as if statement 2 said that the range was 4, not that the range of set is DIVISIBLE by 4?

Or I could be mistaken...?

[tim]

Yes, it appears that Jamie overlooked the "divisible by 4" part. Please let us know if you have any further questions on this one.

[Macaroni]

Yes, how do you explain the answer to the actual question, the statement two part?

Thank-you in advance!

[RonPurewal]

Yes, how do you explain the answer to the actual question, the statement two part?

Thank-you in advance!

Have you tried investigating specific cases?

In problems about number properties, even a small bit of investigation tends to reveal the key notion(s)/pattern(s) quite readily.
If you just sit there and scratch your head, the mysteries will remain mysterious. If you toss in a handful of numbers, though, you'll see what's happening, often with surprising ease.

If the range is actually 4, then the set is something like 2, 4, 6, with median = 4.
If you write a few more such sets——4, 6, 8; then 6, 8, 10; etc.——you'll notice that their behavior is all the same, since you're just taking 2, 4, 6 and adding an even number to all of them.

If the range is 8, the set will be something like 2, 4, 6, 8, 10, with median = 6. All of the other possibilities behave identically, since they're the same as this one + some even-number increment.

Pretty soon you'll see that it's always the same. Your set will always contain an odd # of evens, the middle one of which will be the median.

[FarahM108]

In consecutive sets of numbers, whether odd or even or neither (1,4,7,10)...The rule is that mean=median.
so given statement (1) we know that the mean is even, therefore we can conclude that the median is also even and NOT odd.
So statement (1) is sufficient alone.

Now considering statement (2) : if the range is divisible by 4, then the set must contain an odd number of even integers, like (2,4,6,8,10)...where the range is 10-2=8 which is divisible by 4. and the median =6 which is even and NOT odd.
Therefore, statement (2) is also sufficient alone.

So the answer is D. (Either of the statements is sufficient to find that the median is NOT odd)

Farah

Yes, how do you explain the answer to the actual question, the statement two part?

Thank-you in advance!

Have you tried investigating specific cases?

In problems about number properties, even a small bit of investigation tends to reveal the key notion(s)/pattern(s) quite readily.
If you just sit there and scratch your head, the mysteries will remain mysterious. If you toss in a handful of numbers, though, you'll see what's happening, often with surprising ease.

If the range is actually 4, then the set is something like 2, 4, 6, with median = 4.
If you write a few more such sets——4, 6, 8; then 6, 8, 10; etc.——you'll notice that their behavior is all the same, since you're just taking 2, 4, 6 and adding an even number to all of them.

If the range is 8, the set will be something like 2, 4, 6, 8, 10, with median = 6. All of the other possibilities behave identically, since they're the same as this one + some even-number increment.

Pretty soon you'll see that it's always the same. Your set will always contain an odd # of evens, the middle one of which will be the median.

[jnelson0612]

Nice work, Farah. :-)

[FarahM108]

Thanks :)

Nice work, Farah. :-)

[tim]

:)