What is the median number of employees assigned per project

What is the median number of employees assigned per project for the projects at Company Z?

(1) 25 percent of the projects at Company Z have 4 or more employees assigned to each project. Not sufficient on its own.

(2) 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project. Not sufficient on its own.

(1)+(2) Since 35% of of the projects have 2 or fewer (\(\leq{2}\))employees and 25% of the projects have 4 or more (\(\geq{4}\)) employees, then 100%-(25%+35%)=40% of the projects have exactly 3 employees assigned to each of them. So, the median number of employees assigned per project is 3. Sufficient.

Answer: C.

To elaborate more: consider there are 100 projects: \(\{p_1, \ p_2, \ ... , \ p_{100}\}\). The values of \(p_1\) to \(p_{35}\) will be 0, 1, or 2; the values of \(p_{36}\) to \(p_{75}\) will be exactly 3; the values of \(p_{76}\) to \(p_{100}\) will be 4 or more. \(Median=\frac{p_{50}+p_{51}}{2}=\frac{3+3}{2}=3\).

For example list can be: \(\{2, \ 2, \ 2, \ ..., \ (p_{35}=2), \ (p_{36}=3), \ 3, \ ..., \ (p_{75}=3), \ (p_{76}=4), \ 4, \ ..., \ (p_{100}=4)\}\);
OR:
\(\{0, \ 0, \ 1, \ 1, \ 1, \ 2, \ 2, \ ..., \ (p_{35}=2), \ (p_{36}=3), \ 3, \ ..., \ (p_{75}=3), \ (p_{76}=4), \ 5, \ 7, \ 27, \ ..., \ (p_{100}=10000)\}\) (of course there are a lot of other breakdowns).

In any case median=3.

Hope it's clear.

P.S. Please post PS questions in the PS subforum: gmat-problem-solving-ps-140/ and DS questions in the DS subforum: gmat-data-sufficiency-ds-141/ No posting of PS/DS questions is allowed in the main Math forum.
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Video solution from Quant Reasoning:

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I've solved it this way (see attachment) -> the mid section of 40% =3 (it must be an integer, you cannot have fractions when dealing with persons So Statement 1+2 are sufficient (C)

Will vote for C

let total be "x" projects i.e 100% projec

(A) 25% projects has number of employees >= 4

- we dont know anything about other 75% projects

(B) 35% project has number of employees <= 2


- we dont know anything about other 65% projects

(C)

for 35% --> No. of emp <= 2
for 25% --> No. of emp >= 4

based on above data we can calculate that
Remaining 40% of projects --> No. of emp = 3 (which is between 2 < no. of emp < 4)

What is the median number of employees assigned per project for the projects at Company Z?

(1) 25 percent of the projects at Company Z have 4 or more employees assigned to each project.
(2) 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project


Ruled out quickly A,B,D.
remained with C, E.
chose E.
then looked here:
what-is-the-median-number-of-employees-assigned-per-project-131892.html

and i still dont understand bunuel's way.
why the employees who are not in the 25% or 35% must have exactly 3 employees?
thanks

25% of company Z projects have 4 or more that (4,5,6,....)

35% of company z projects have 2 or less (0,1,2 )...

there must b 3 employees per project of company z which consists of (25+35=60) and 100-60=40%..

0,1,2 must be first 35 projects
3 employees for next 40 project
and 4 or more than 4 employees for remaning 25 projects..
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Ans: : Median is the middle value. Statements 1 and 2 alone give us insufficient data but when we combine both of them we see that 25 % have 4 or more and 35% have 2 or less therefore 40 %have 3 employees. Therefore the median would be 3 and the answer is (C).

I Did not understand this. How can you deduce about the 40% of the projects based on the information given. The problem just says 25% of the projects have 4 or more. 35% of the projects have 2 or less. So it does not talk anything about 40%. 40 % can be 5,6,7 or anything right?? Could you explain it in detail.

35 of the projects have 2 or fewer employees.
25 of the projects have 4 or more employees.

How many employees can be assigned to the remaining 40 projects? The ranges \(\leq{2}\) and \(\geq{4}\) are covered, thus the remaining 40 projects have 3 employees assigned to them.

Hope it's clear.
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Aren't we assuming here that, statement 1 and 2 refer to highest 25 and bottom 35 percent of the employees, couldn't it be first 35 % then the next 25% and the remaining 40% with unknown number of employees per project, then the median will lie in 4 or more employees, we do not have any particular value and hence E as the correct answer.

Statement 2 and 1 must refer to the bottom and to the highest part.
"couldn't it be first 35 % then the next 25% and the remaining 40%"? NO.
The first 35% have 2 or fewer (till here correct), but then your reasoning goes against the info in statement 1.
What you are saying is that 25%+40%=65% has 4 or more employees => wrong, look at statement 1:
(1) 25 percent of the projects at Company Z have 4 or more employees assigned to each project.

Hope it's clear

P.S: Welcome to GMAT Club!

Detailed Solution

Step-I: Given Info

The question asks us to find the median number of employees assigned per project for projects at Company Z, using the 2 statements given.

Step-II: Interpreting the Question Statement

To find the median number of employees, we need to somehow extract the central value for employees, when arranged in ascending order. Central value or median value will be the exact determination of value at the 50%th observation.

Step-III: Statement-I

Statement- I gives us the information about the 25% of projects have 4 or more employees assigned to each project.
Now let us say we have 100 projects. There could be a possibility that remaining 75 projects (>50 projects) have 3 employees. Then the central observation (50%th observation- in this case 50th project value in total 100 projects) will have a median value of 3 employees. But the remaining 75 projects can have 2 employees as well. Then the median will be 2. So we cannot determine with exact surety, the exact value of median at 50%th observation.

Hence, statement 1 is not sufficient to answer the question.

Step-IV: Statement-II

Statement- II gives us information that 35% of the projects have 2 or fewer employees assigned to each project. Now, the remaining 65 projects (>50 projects) can have 3 employees. Then the central observation (50%th observation- in this case 50th project value in total 100 projects) will have a median value of 3 employees. But the remaining 65 projects(>50 projects) can have a median value of 4 employees as well. Then the median value will be 4 employees. So we cannot determine with exact surety, the exact value of median at 50%th observation.

Hence, statement 2 is not sufficient to answer this question.

Step-V: Combining Statements I & II

When we combine both the statements we know the value of 25% of projects (4 or more employees) and 35% projects(2 or fewer employees). So the remaining 40 projects (<50 projects) will have 3 employees. When the observations are arranged in ascending order. The first 35 observations will have 2 or fewer employees and next 40 observations will have 3 employees and then next 25 observations will have 4 or more employees. Thus, the median number of employees will be 3.

Hence, the exact determination of central value or value at 50%th observation is 3.

So the correct answer is C. Both the statements together are sufficient.

Key Takeaways

Use of concept of median and evaluating all possible value of central value (median value) for observations to be able to find exact value at 50%th observation (when values arranged in ascending order).

Dear Friends,

I need your help, coz I'm having problem
I have two questions:

1) What if there is no project with THREE employees?
Why can't 40% of projects have 5 or more employees or only 1 employee for that matter

2) I couldn't understand why 40% must be between 2 (35%) and 4 (25%): please see the attachment below.
I would to know the reasoning behind why 3 employees (40%) must be between 2 and 4.
The question does not say that 25% is at the higher band or that the 35% is at the lower band. (Although we can deduce 25% band is higher than the 35% band).
Why can't the rest 100-(25+35)=40% be at an even higher band (eg 5 employees or more) ?

It hasn't been stated nowhere that 35% is the 1st and 25% is the last :


I would be so grateful, if u could explain me thoroughly.
Thank you very much!

Because "5 or more" is already a part of "4 or more" set.


Look, the number of employees assigned to a project could be any of the following:
0 (debatable but ok), 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 ... and so on

When you say 35% project have 2 or fewer employees assigned, it means that 25% projects have either 0 or 1 or 2 employees assigned. All projects that have 1 employee assigned to them are a part of this 25%. Say, this is the first group.

When you say 25% projects have 4 or MORE employees assigned, it includes all projects with 4 or 5 or 6 or 7 or 8 etc employees assigned to them. Any project that has 5 or more employees is already a part of this 25%. This is the second group.

So what about the rest of the 40% projects? Can they have 1 employee assigned? No. All those projects are accounted for in first group. Can they have 6 employees assigned? No. All those projects are accounted for in second group. The only number left that is not accounted for in wither group is 3.
So 40% projects must have exactly 3 employees assigned.

Makes sense, now?
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Target question: What is the median number of employees assigned per project?

Statement 1: 25 percent of the projects at Company Z have 4 or more employees assigned to each project.
Let's pretend that there are 4 projects altogether.
There are several sets of values that meet this condition. Here are two:
Case a: the set of numbers representing employees per project are {1, 1, 1, 4} in which case the median is 1
Case b: the set of numbers representing employees per project are {2, 2, 2, 4} in which case the median is 2
Since we cannot answer the target question with certainty, statement 1 is NOT SUFFICIENT

Statement 2: 35 percent of the projects at Company Z have 2 or fewer employees assigned to each project.
Using logic similar to the above, we can show that statement 2 is NOT SUFFICIENT

Statements 1 and 2 combined:
35% projects have 2 employees OR FEWER, and 25% of the projects have 4 employees OR MORE. So, 40% projects have EXACTLY 3 employees.

To find the median, we must find the middlemost value when all of the values are listed in ASCENDING order.
So, the first 35% of the numbers will be 1's and 2's.
Then next 40% of the numbers will be 3's
And the last 25% of the numbers will be 4, 5's, etc

As you can see, the MIDDLEMOST value will be 3. In other words, the median must be 3.
Since we can answer the target question with certainty, the combined statements are SUFFICIENT

Answer:

RELATED VIDEO

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I understand that we're looking for the median and that each statement alone only tell us PART of what we want to know...further, I understand adding up both %s [25 & 35] and deducing that 40% have 3 employees.
- HOWEVER, how do we know that 40% contains the median? What if we changed the numbers in Statement 1 & 2 both to 35? In this case, 30% would be left for 3 employees...how would this affect the answer?

It won't change the answer. Say there are 20 projects.
If 35% have 4 or more employees, it means 7 projects have 4 or more employees.
If 35% have 2 or fewer employees, it means 7 projects have 2 or fewer employees.
The leftover 30% i.e. 6 projects have 3 employees.

To find the median, you will need to arrange the number of employees in increasing order

1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 4, 4, 4, 5, 6, 6, 7

The one in the middle will be 3.
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If a set has odd number of terms the median of a set is the middle number when arranged in ascending or descending order;
If a set has even number of terms the median of a set is the average of the two middle terms when arranged in ascending or descending order.

So if there are 20 projects,
35% i.e. 7 projects have 2 or fewer employees assigned to each project, for example: 0, 0, 1, 1, 1, 2, 2;
40% i.e. 8 projects have exactly 3 employees assigned to each project: 3, 3, 3, 3, 3, 3, 3, 3;
25% i.e. 5 projects have 4 or more employees assigned to each project, for example: 4, 5, 6, 7, 8;

0, 0, 1, 1, 1, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 5, 6, 7, 8.

So, as 20 is even, then the median number of employees per project would be the average of two middle terms: 10th and 11th --> (3+3)/2=3.

Hope it helps.
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Please check the video for the step-by-step solution.

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