In a research and development department, 14 workers have a PhD, and 30 workers don't have a PhD degree. 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. If a third of the men working in the department have a PhD degree, then how many women work in the department?

A) 16
B) 18
C) 26
D) 28
E) 32

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