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?

Accepted Solution

Answer: There are 70 houses that have exactly two of these amenities.Step-by-step explanation:Number of house in a certain development = 150Number of houses have air conditioning n(A)= [tex]0.6\times 150=90[/tex]Number of houses have sunporch n(B) = [tex]0.5\times 150=75[/tex]Number of houses have swimming pool n(C) = [tex]0.3\times 150=45[/tex]Number of houses have all three amenities = 5Number of houses have none of them = 5So, remaining houses = 150-5=145As we know the rule of sets:[tex]n(A\cup B\cup C)=n(A)+n(B)+n(C)-n(A\cap B)-n(B\cap C)-n(C\cap A)+n(A\cap B\cap C)\\\\145=90+75+45-n(A\cap B)-n(B\cap C)-n(C\cap A)+5\\\\145=215-n(A\cap B)-n(B\cap C)-n(C\cap A)\\\\145-215=-n(A\cap B)-n(B\cap C)-n(C\cap A)\\\\-70=-n(A\cap B)-n(B\cap C)-n(C\cap A)\\\\n(A\cap B)+n(B\cap C)+n(C\cap A)=70[/tex]Hence, there are 70 houses that have exactly two of these amenities.