let us suppose total number of men working in the department is m
and total number of women working in the department is w

Of the total women working in the department let us assume those that have a phd is A
hence those that do not have a phd = A+10

we are given 1/3 M have phd
hence 2/3 M do not have a phd

we have 1/3 M + A = 14 (those with phd).....i
and 2/3 M + A+10 = 30 >> 2/3 M + A = 20 (those without phd)...ii

subtracting eqn i from ii we get,
1/3 M = 6 hence, M = 18

Total number of people that work in the department = 44
Total number of Men that work in the department = 18
hence, total number of women working in the department = 44-18 = 26
(c)
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Let's use the double matrix method. We will use W for women, M for men, and T for total.

Total number who have PhDs is 14 so we put that in the bottom left. Total number with no PhD is 30 so bottom middle. Total of course should add to 44.

"In the department, the number of women that do not have a PhD degree is 10 greater than the number of women who do have a PhD," thus, we will denote number of women with PhD as X and # of women with no PhD as 10+x.

"If a third of the men working in the department have a PhD degree," so we will denote total number of men with y, then we should put y/3 accordingly for men with PhDs, which leaves 2y/3 for men without phDs.



| PhD | No PhD | T
W | x | 10+x | 2x+ 10
M | y/3 | 2/3y | y
T | 14 | 30 | 44

Now we have a system of linear equations that we need to solve.

x+ 2/3 y = 20
x + y/3 = 14

Upon solving, we get y/3 = 6. We don't need to simplify that, go ahead and plug that in the table.

| PhD | No PhD | T
W | x | 10+x | 2x+ 10
M | 6 | 2/3y | y
T | 14 | 30 | 44

Thus, x = 8.

Then the total number of women is 8 + 18 = 26.

C

How about this
Women with PHD =Wp
Women without PHD is Wp+10
That means total women is Wp+Wp+10=2Wp+10
Men with PHD=M/3
Total Men is M

Men+Women=44
M+2Wp+10=44

M=34-2Wp, so we have a substitution for Women to men
Then lets look at the PHD's
M/3 + Wp = 14
(34-2Wp)/3+ Wp = 14

(34-2wp +3Wp)/3= 14
34+Wp=42
Wp=8
Wp+10=18
Total women = 2Wp+10= 26

The choice is C

Let's use the Double Matrix Method.
This technique can be used for most questions featuring a population in which each member has two characteristics associated with it (aka overlapping sets questions)..

Here, we have a population of workers, and the two characteristics are:
- has a PhD or doesn't have a PhD
- woman or man

In a research and development department, 14 workers have a PhD, and 30 workers don't have a PhD degree.
So there are 44 workers in TOTAL.
Our diagram looks like this:




In the department, the number of women that do not have a PhD degree is 10 greater than the number of women who do have a PhD.
Let x = the number of women who DO have a PhD degree is 10 greater
So, x+10 = the number of women who do NOT have a PhD degree is 10 greater
We can add this information to the diagram to get:




A third of the men working in the department have a PhD degree
Let y = the total number of men working in the department
So, y/3 = the number of men WITH a PhD degree
And 2y/3 = the number of men WITHOUT a PhD degree
We get:



How many women work in the department?
On the left-side column, we see that: x + y/3 = 14
Multiply both sides by 3 to get: 3x + y = 42

On the right-side column, we see that: (x + 10) + 2y/3 = 30
Simplify to get: x + 2y/3 = 20
Multiply both sides by 3 to get: 3x + 2y = 60

We have:
3x + 2y = 60
3x + y = 42

Subtract the bottom equation from the top equation to get: y = 18
This means there are 18 MEN in the department.
Since there are 44 workers in total, the remaining 26 workers are WOMEN

Answer: C

This question type is VERY COMMON on the GRE, so be sure to master the technique.

To learn more about the Double Matrix Method, watch this video:

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