How to solve it using set theory.

Instead of using Set theory / Venn diagram using tables to solve this kind of question is much simpler.

Construct a table as per the data

Table ........ Windows || No Windows || Total
-----------------------------------------------------------------
HardFloor.....| 50%............ || a %........... || X%
No HardFloor| b% ............ || c %........... || 40%
---------------------------------------------------------------
Total .............. | Y %........... || Z%........... || 100%


Here 50% + a% = X%, 50% +b% = Y% and so on

Clearly X% = 100%-40% = 60%
In that case a% = 10% i.e. (60%- 50%)

There is a relation between a% and c%, it is given that "25% of the apartments without windows have hardwood floors". So 75% of apartments without windows will have no hard floors.
So if a% = 10% then c% will be 30%.
Then we can get b% = 40% - c% = 10%

Finally, percent of the apartments with windows have hardwood floors

= 50% / (Y%) => 50% / 60% => 83 1/3%

For this one, I think you nee to approach it as a quadrant-type system where one dimension is "Windows" and the other dimension is "HW floors" so you have 4 possible combinations:

W(yes), HW(Yes) - A
W(Yes), HW (No) - B
W(No), HW (Yes) - C
W(No, HW (No ) -D

Then this is what we know:

A+B+C+D = 100


50% have windows and hardwood floors, so A = 50

50 + B + C + D = 100
B+C+D = 50

25% of the apartments without Win have HW Floors
C + D = the % without Windows - 25% of these (C + D) have HW floors, which is what we have labeled C so
C = (C + D)/4
4C = C+D
3C = D

40% of the apartments do not have HW floors
B + D are the quadrants that don't have HW floors, so
B + D = 40

If B + C + D = 50
and B + D = 40
Then (B + C + D) - (B + D) = 50-40
C = 10

D = 3C so D = 3 (10) = 30

B + C + D = 50
B + 10 + 30 = 50
B = 10

So now, back to our quadrants:

W(yes), HW(Yes) - A = 50
W(Yes), HW (No) - B = 10
W(No), HW (Yes) - C = 10
W(No, HW (No ) -D = 30

The quetsion is asking what % of all apartments with windows are the ones what have Windows and hardwood floors.

The aparts with Win are A + B = 60
The aprts with Win and HW are A = 50

% of Win+HW floors of the aparts that have Win is 50/60 = 5/6 = 83 1/3%

Answer is E

-Carrie