GMAT Official Guide
Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?
(A) 10
(B) 45
(C) 50
(D) 55
(E) 65
D
GMAT OG Of the 150 houses in a certain development
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E
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We can break out our trusty formula for dealing with three-set overlaps:AbeNeedsAnswers wrote:Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?
(A) 10
(B) 45
(C) 50
(D) 55
(E) 65
D
Total = Group 1 + Group 2 + Group 3 - [# in exactly 2 groups] - 2[# in all 3 groups] + # in none of the 3 groups
Total = 150
Group 1 = 60% of 150 = 90
Group 2 = 50% of 150 = 75
Group 3 = 30% of 150 = 45
# with all three = 5
# with one of the three = 5
Plug and chug: 150 = 90 + 75 + 45 - x - 2[5] + 5
150 = -x + 205
-55 = -x
55 = x; The answer is D
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We can create the following equation:AbeNeedsAnswers wrote:Of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, and 30 percent have a swimming pool. If 5 of the houses have all three of these amenities and 5 have none of them, how many of the houses have exactly two of these amenities?
(A) 10
(B) 45
(C) 50
(D) 55
(E) 65
Total houses = number with air conditioning + number with sunporch + number with pool - number with only two of the three things - 2(number with all three things) + number with none of the three things
150 = 0.6(150) + 0.5(150) + 0.3(150) - D - 2(5) + 5
150 = 90 + 75 + 45 - D - 10 + 5
150 = 205 - D
D = 55
Answer: D
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Hi All,
We're told that of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, 30 percent have a swimming pool, 5 of the houses have ALL three of these amenities and 5 have NONE of them. We're asked for the number of houses that have EXACTLY TWO of these amenities. This question is a 3-group Overlapping Sets question and can be solved with either a 3-circle Venn Diagram or the 3-group Overlapping Sets Formula:
Total = (None) + (Group 1) + (Group 2) +(Group 3) - (Gp 1 & Gp 2) - (Gp 1 & Gp 3) - (Gp 2 & Gp 3) - 2(All 3)
Based on the percentages in the prompt, we can fill in most of the formula:
150 = (5) + (90) + (75) + (45) - (Gp 1 & Gp 2) - (Gp 1 & Gp 3) - (Gp 2 & Gp 3) - 2(5)
150 = 205 - (Gp 1 & Gp 2) - (Gp 1 & Gp 3) - (Gp 2 & Gp 3)
(Gp 1 & Gp 2) + (Gp 1 & Gp 3) + (Gp 2 & Gp 3) = 55
Thus, the sum of the three "groups of 2" is 55.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich
We're told that of the 150 houses in a certain development, 60 percent have air-conditioning, 50 percent have a sunporch, 30 percent have a swimming pool, 5 of the houses have ALL three of these amenities and 5 have NONE of them. We're asked for the number of houses that have EXACTLY TWO of these amenities. This question is a 3-group Overlapping Sets question and can be solved with either a 3-circle Venn Diagram or the 3-group Overlapping Sets Formula:
Total = (None) + (Group 1) + (Group 2) +(Group 3) - (Gp 1 & Gp 2) - (Gp 1 & Gp 3) - (Gp 2 & Gp 3) - 2(All 3)
Based on the percentages in the prompt, we can fill in most of the formula:
150 = (5) + (90) + (75) + (45) - (Gp 1 & Gp 2) - (Gp 1 & Gp 3) - (Gp 2 & Gp 3) - 2(5)
150 = 205 - (Gp 1 & Gp 2) - (Gp 1 & Gp 3) - (Gp 2 & Gp 3)
(Gp 1 & Gp 2) + (Gp 1 & Gp 3) + (Gp 2 & Gp 3) = 55
Thus, the sum of the three "groups of 2" is 55.
Final Answer: D
GMAT assassins aren't born, they're made,
Rich