If the median annual salary of the 127 employees in a certain

Question

If the median annual salary of the 127 employees in a certain department of Company X is $35,000, which of the following must be true?

I. At least 64 of the employees have an annual salary that is $35,000 or more

II. At least 63 of the employees have an annual salary that is less than $35,000

III. At most 64 of the employees have an annual salary that is $35,000 or more.

A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III

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Official Answer

A

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Bunuel · Accepted Answer

The median of an odd count of terms is the middle number when the numbers are in order. Thus, the 64th highest salary is $35,000:

x_1 \leq x_2 \leq ... \leq x_{63} \leq (x_{64} = $35,000)\leq x_{65} \leq ... \leq x_{126} \leq x_{127} .

Let's evaluate the options:

I. At least 64 of the employees have an annual salary that is $35,000 or more

This must be true, since terms from  x_{64}  to  x_{127}  must be at least $35,000.

II. At least 63 of the employees have an annual salary that is less than $35,000

This is not necessarily true. For example, all terms from  x_1  till  x_{64}  can be $35,000, making this statement false.

III. At most 64 of the employees have an annual salary that is $35,000 or more.

This is also not necessarily true. For example, all terms can be $35,000, making this statement false.

Answer: A.

ostrick5465 · Answer

The median is the 64th in the set. the 64th number in the set is 35,000

I. Each of number from the 64th to the 127th is $35,000 or more. There are 64 values from 64 to 127. => True
II. The 63rd, 62nd, 61st values can be 35,000 => False
III. Same to II, The 63rd, 62nd, 61st values can be 35,000, so the number of employees who have an annual salary that is $35,000 or more can be more than 64. => False

=> Choice A

GMAT1034 · Answer

Hi  Bunuel , can you please explain the difference bw "Atleast" and "At most" here in this question. In option 1 too, cant the values be all equal to 35,000. Thanks

Bunuel · Answer

"At least"  means  greater than or equal to  — so “at least 64 employees earn $35,000 or more” allows for exactly 64, or more than 64.

"At most"  means  less than or equal to  — so “at most 64 employees earn $35,000 or more” means 64 or fewer.

Now your example:
 
Yes,  all  127 salaries could be equal to $35,000. That would still satisfy  Statement I  (at least 64 earning $35,000 or more), but would  violate Statement III , because in that case  more than 64  employees earn $35,000 or more — not “at most” 64.

So the confusion clears up when you realize:

"At least"  is fine with equality or more. 
  "At most"  becomes false if you go over that number.

GMAT1034 · Answer

A follow up to the discussion  Bunuel , in the 3rd statement we are said that its At most 35000, meaning that it has to be 35000 or more only 64 times max. So, if we take the more than 35000 case, that can only be 64 times max right? So doesnt this fulfill the condition statement. Am I inferring something incorrectly here?

Bunuel · Answer

Yes - you're making a mistake in how you're interpreting “at most.”

Statement III says: 
 
"At most 64 employees have a salary of $35,000 or more."

That means the number of employees earning  $35,000 or more must be ≤ 64 .

But we know the  64th employee  earns exactly $35,000 (since it's the  median  in a list of 127). That alone already gives  64 people earning $35,000 or more  - and if even  one more  person earns $35,000 or more, it becomes  65 , violating the “at most 64” condition.

But it's entirely possible for  more than 64  people to earn $35,000 or more - for example, all 127 salaries could be $35,000 or more. That means Statement III  does not have to be true  in every case, so it's  not a must .

Hope it's clear.

MartyMurray · Answer

If the median annual salary of the 127 employees in a certain department of Company X is $35,000, which of the following must be true?

127 is an odd number. So, the median value is the middle value, specifically the 64th value, under the assumption that the values are ordered from least to greatest.

So, the 64th value is 35,000.

Accordingly, we know the following:

Values 1 through 63 must be below or equal to 35,000.

Values 65 through 127 must be equal to or above 35,000.

I. At least 64 of the employees have an annual salary that is $35,000 or more

We know that value 64 must be 35,000 and that values 65 through 127 must be equal to or above 35,000.

That's 64 values that must be equal to or above 35,000.

This must be true.

II. At least 63 of the employees have an annual salary that is less than $35,000

Values 1 through 63 could be less than 35,000.

At the same time, they don't have to be. Some or all of them could equal the median of 35,000.

This does not have to be true.

III. At most, 64 of the employees have an annual salary that is $35,000 or more.

All of the employees could have the same salary, $35,000.

So, more than 64 could have an annual salary that is $35,000 or more.

This does not have to be true.

A) I only
B) II only
C) III only
D) I and II only
E) I, II, and III

Correct answer:   A

If the median annual salary of the 127 employees in a certain

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If the median annual salary of the 127 employees in a certain

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