Of the 150 Houses in a Certain Development GMAT Problem Solving

Question: 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

“Of the 150 houses in a certain development”- is a topic of the GMAT Quantitative reasoning section of GMAT. This question has been taken from the book "GMAT Official Guide Quantitative Review". To solve GMAT Problem Solving questions a student must have knowledge about a good number of qualitative skills. GMAT Quant section consists of 31 questions in total. The GMAT quant topics in the problem-solving part require calculative mathematical problems that should be solved with proper mathematical knowledge.

Solution and Explanation:

Approach Solution 1:

This particular problem can be solved using the Venn Diagram.
Let us consider,
AC = 60% of 150 = 90;
Sunporch = 50% of 150 = 75
SP = 30% of 150 = 45
So, using the Venn diagram we can form the following equations:
a+b+c+d+e+f+g+h = 150
a+b+c+d+f+g = 140 ----- (1) (as e = h = 5)
Taking one circle at a time
AC = a+b+d = 85 ---- (2) .. (as e = 5)
SunPorch = c+b+f = 70 ---- (3) .. (as e = 5)
SP = g+d+f = 40 ---- (4) .. (as e = 5)
equating- (2) + (3) + (4)
This implies a+c+g+ 2(d+b+f) = 195
subtracting this with equation (1)
Hence, d+b+f = 195 - 140 = 55
Hence, the required number of houses that have exactly two of these amenities is 55.

Correct Answer: D

Approach Solution 2:

This problem can be solved using mathematical equations. 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
Hence, the required number of houses that have exactly two of these amenities is 55.

Correct Answer: D

Approach Solutio 3:

Equations in mathematics can be used to address this issue. We may construct the equation shown below:
Total homes are those having air conditioning, a sundeck, a pool, and the others have only two of those features. - 2(number including all three conditions) + number omitting any of the three conditions
150 = 0.6(150) + 0.5(150) + 0.3(150) - D - 2(5) + 5\s150 = 90 + 75 + 45 - D - 10 + 5\s150 = 205 - D\sD = 55
Therefore, 55 homes are needed to have precisely two of these features.

Correct Answer: D

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