Problem Solving

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

We can create the following equation:

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

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

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