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? [Official GMAT-2018]

Hint:

When working with set problems, use the principle of inclusion and exclusion to find the number of elements in overlapping sets.

Answer 1

Step 1: Calculate the number of houses with each amenity.
- Air-Conditioning: 60% of 150 = 90
- Sun Porch: 50% of 150 = 75
- Swimming Pool: 30% of 150 = 45
Step 2: Use the formula for the total number of houses.
Let \( x \) be the number of houses with exactly two amenities. The formula for the total number of houses is: \[ \text{Total} = (\text{All Single}) - 2 \times (\text{All Three}) + N \] Where:
- All Single: The total number of houses with at least one amenity, which is 150.
- All Three: The number of houses with all three amenities (given as 5).
- \( N \) is the number of houses with none of the amenities (also given as 5).
Step 3: Solve the equation.
Using the information, we set up the following equation: \[ 150 = 90 + 75 + 45 - (\text{Exactly Two}) - 2 \times 5 + 5 \] Simplifying this gives: \[ \text{Exactly Two} = 205 - 150 = 55 \] Step 4: Conclusion.
Therefore, 55 houses have exactly two of the amenities.