Of the 150 houses in a certain development, 60% have air-conditioning, 50% have a sunporch, and 30% have a swimming...

1. Translate the problem requirements: We have 150 houses total. We know the percentages with each amenity (AC: 60%, sunporch: 50%, pool: 30%), plus houses with all three amenities (5) and none (5). We need to find houses with exactly two amenities.
2. Calculate total amenities distributed: Find the total count of individual amenities across all houses to establish our baseline.
3. Apply the counting principle: Use the fact that total amenities = (houses with 1 amenity) + 2×(houses with exactly 2 amenities) + 3×(houses with all 3 amenities) to set up our equation.
4. Solve for houses with exactly two amenities: Substitute known values and solve for the unknown quantity.

Execution of Strategic Approach

1. Translate the problem requirements

Let's start by organizing what we know in simple terms:

• Total houses: 150
• Houses with air-conditioning: \(60\% \times 150 = 90\) houses
• Houses with sunporch: \(50\% \times 150 = 75\) houses
• Houses with swimming pool: \(30\% \times 150 = 45\) houses
• Houses with all three amenities: 5 houses
• Houses with none of the amenities: 5 houses

We need to find: Houses with exactly two amenities (any combination of AC + sunporch, AC + pool, or sunporch + pool, but not all three)

Process Skill: TRANSLATE - Converting percentages to actual counts and clearly defining what "exactly two" means

2. Calculate total amenities distributed

Now let's think about this step by step. When we count amenities, we're counting each individual feature across all houses.

Total individual amenities = \(90 + 75 + 45 = 210\) amenities

This means if we went house by house and counted every air conditioner, every sunporch, and every pool, we'd count 210 total amenities across the development.

But here's the key insight: some houses have multiple amenities, so when we count this way, we're counting some houses more than once!

3. Apply the counting principle

Let's think about how amenities get counted:

• Houses with exactly 1 amenity: Each contributes 1 to our total count of 210
• Houses with exactly 2 amenities: Each contributes 2 to our total count (since each house has 2 features)
• Houses with exactly 3 amenities: Each contributes 3 to our total count
• Houses with 0 amenities: These contribute 0

So we can write: Total amenities = \(1 \times (\text{houses with exactly 1}) + 2 \times (\text{houses with exactly 2}) + 3 \times (\text{houses with exactly 3})\)

We know:

• Houses with 0 amenities = 5
• Houses with exactly 3 amenities = 5
• Total houses = 150

Therefore: Houses with 1 or 2 amenities = \(150 - 5 - 5 = 140\) houses

Let's call houses with exactly 1 amenity = x, and houses with exactly 2 amenities = y

So: \(\mathrm{x} + \mathrm{y} = 140\) (total houses with 1 or 2 amenities)
And: \(1 \times \mathrm{x} + 2 \times \mathrm{y} + 3 \times 5 = 210\) (total amenities counted)

Process Skill: INFER - Recognizing that the total amenity count includes multiple counting of houses with multiple amenities

4. Solve for houses with exactly two amenities

Now we have our system of equations:

• \(\mathrm{x} + \mathrm{y} = 140\)
• \(\mathrm{x} + 2\mathrm{y} + 15 = 210\)

From the second equation: \(\mathrm{x} + 2\mathrm{y} = 195\)

Subtracting the first equation from this:
\((\mathrm{x} + 2\mathrm{y}) - (\mathrm{x} + \mathrm{y}) = 195 - 140\)
\(\mathrm{y} = 55\)

Let's verify: If y = 55, then \(\mathrm{x} = 140 - 55 = 85\)
Check: \(85 + 2(55) + 3(5) = 85 + 110 + 15 = 210\) ✓

Therefore, 55 houses have exactly two amenities.

4. Final Answer

The answer is 55 houses have exactly two amenities.

This matches choice (D) 55.

Common Faltering Points

Errors while devising the approach

1. Misinterpreting "exactly two amenities"
Students often confuse "exactly two" with "at least two" amenities. They might include houses with all three amenities in their count, not recognizing that houses with all three amenities have MORE than exactly two.

2. Attempting to use Venn diagram without proper setup
Many students jump straight into drawing a three-circle Venn diagram but struggle because they don't have enough information about the overlapping regions. They get stuck trying to fill in individual intersection values when the problem requires a different strategic approach.

3. Not recognizing the "counting principle" insight
Students miss the key insight that when we add up all individual amenities \((90 + 75 + 45 = 210)\), we're actually counting each house multiple times based on how many amenities it has. This leads them to try more complex approaches instead of the elegant counting method.

Errors while executing the approach

1. Arithmetic errors in percentage calculations
Students make basic calculation mistakes when converting percentages to actual house counts: \(60\% \times 150 = 90\), \(50\% \times 150 = 75\), \(30\% \times 150 = 45\). Simple errors here cascade through the entire solution.

2. Setting up incorrect equations
When applying the counting principle, students might incorrectly set up their system of equations. For example, writing \(\mathrm{x} + \mathrm{y} + 5 + 5 = 150\) but then writing \(\mathrm{x} + 2\mathrm{y} + 15 = 210\), forgetting that the second equation shouldn't include the houses with 0 amenities.

3. Solving the system of equations incorrectly
Students make algebraic errors when solving \(\mathrm{x} + \mathrm{y} = 140\) and \(\mathrm{x} + 2\mathrm{y} = 195\). Common mistakes include incorrect substitution or subtraction, leading to wrong values for x and y.

4. Errors while selecting the answer
No likely faltering points