If the median of 127 numbers is 35, which of the following

Question

If the median of 127 numbers is 35, which of the following must be true?

I. At least 64 of the numbers are greater than or equal to 35
II. At least 64 of the numbers are smaller than 35
III. At most 64 numbers are greater than or equal to 35

A. I only
B. II only
C. III only
D. I and II only
E. I and III only

Bunuel · Answer

Not a good question because answer choices are rather odd for the GMAT.

Anyway,  if a set has odd number of terms the median of a set is the middle number when arranged in ascending (or descending) order , so the median of 127 numbers is 35 means that 35 is 64th number when arranged in ascending order. Therefore 63 numbers are less than or equal to 35 and 63 numbers are more than or equal to 35. Therefore I must be true: at least 64 of the numbers (the median itself plus 63 other numbers) are greater than or equal to 35.

Answer: A

vikram4689 · Answer

But at least 64 of the numbers cannot be greater than 35...
.....63 numbers...35 ...63 numbers ... later 63 no's can be greater or equal but median has to be 35

Bunuel · Answer

Not so. 63 numbers are more than or equal to 35 plus the median itself (35), so 64 of the numbers are greater than  or equal to 35 .

vikram4689 · Answer

63 numbers are more than or equal to 35 plus the median itself (35).... this is exactly what i mean
BUT isnt 
64 of the numbers are greater than or equal to 35 ==  64 of the numbers are greater than 35  OR  64 of the numbers are equal to 35

Bunuel · Answer

No it does not mean that.

At least 64 of the numbers are greater than or equal to 35 means that we can have for example the following cases:

{..., median=35, 35, 35, ..., 35}
{..., median=35, 36, 37, ..., 98}
{..., median=35, 35, 35, 100, 100, ..., 1000}

vikram4689 · Answer

So you are saying that  "63 numbers are more than or equal to 35 plus the median itself (35)."  ==  "64 of the numbers are greater than or equal to 35"

there is contradiction, when you are saying  63 numbers are more than or equal to 35 , you are intending that numbers can be greater than 35 (as shown in above examples). but when you are saying  "64 of the numbers are greater than or equal to 35"  you are NOT taking that all of 64 numbers can be greater than 35.

I hope there is no difference between more than and greater than in above statements

Bunuel · Answer

I'm not quite sure I understand your question. But anyway consider the following set: {1, 2, 3}. If I say that all numbers in the set are more than or equal to -10 would it be correct? Obviously yes. The same with the question above.

vikram4689 · Answer

I think i am getting hold of the statement, so original statement means
at least 64 numbers >= 35 ... which is correct.

I was considering that one facet of statement is "at least 64 numbers > 35"

Thanks

Sukant2010 · Answer

Hi,
When you're saying atleast 64 nos are greater than or equal to 35, dont you mean that min 64 nos are greater than 35 i.e. a min of 64 nos are greater than 35. According to this statement, we can have even 100 nos greater than or equal to 35, which is not the case here, right??
Isnt the third option that atmost 64 nos can be greater than or equal to 35 right?, as there can be a max of 64 nos such as 64, 63, 62 whatever..., greater than or equal to 35.
Please correct me if I am wrong.
Thanks in advance.

Bunuel · Answer

III says: at most 64 numbers are greater than or equal to 35. The question asks which of the options MUST be true, not COULD be true.

The third option is not always true. For example, the set could consist of all identical elements, 35, and in this case all 127 would be greater than or equal to 35.

Does this make sense?

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If the median of 127 numbers is 35, which of the following

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