50% of the apartments in a certain building have windows and hardwood floors. 25% of the apartments without windows have hardwood floors. If 40% of the apartments do not have hardwood floors, what percent of the apartments with windows have hardwood floors?
A. 10%
B. 16 2/3 %
C. 40%
D. 50%
E. 83 1/3 %
How to solve it using set theory.
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- Master | Next Rank: 500 Posts
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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%
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%
- gaggleofgirls
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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
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