Of the 75 houses in a certain community, 48 have a patio. How many of the houses in the community...

Understanding the Question

We have 75 houses in a community, and 48 of them have patios. We need to find the exact number of houses with swimming pools.

What Makes This Sufficient?

For sufficiency, we need to determine one specific value for the number of houses with swimming pools. If we can narrow it down to exactly one number, the statement(s) will be sufficient.

Key Insight from Our Analysis

This is a classic overlapping sets problem. Think of the houses divided into four groups:

1. Patio only
2. Pool only
3. Both patio and pool
4. Neither patio nor pool

Since we know 48 houses have patios, groups 1 and 3 together must equal 48.

Analyzing Statement 1

Statement 1 tells us that 38 houses have a patio but no swimming pool.

What This Reveals

Since 48 houses have patios total, and 38 of these don't have pools, we can determine that exactly 10 houses must have both features (\(48 - 38 = 10\)).

Testing Different Scenarios

But here's what we still don't know: how many houses have pools but no patio? Let's test some possibilities to see if we can pin down a unique answer:

• Scenario 1: If 0 houses have pool only → Total with pools = 10
• Scenario 2: If 20 houses have pool only → Total with pools = 30
• Scenario 3: If 27 houses have pool only → Total with pools = 37

All these scenarios work with our given information! We get different answers each time, which means we cannot determine a unique value.

Conclusion

Statement 1 alone is NOT sufficient because multiple values for the total number of houses with pools are possible.

This eliminates choices A and D.

Analyzing Statement 2

Now let's forget Statement 1 completely and analyze Statement 2 independently.

Statement 2 tells us that the number of houses with both patio and pool equals the number of houses with neither feature.

The Balance Principle

This creates a perfect balance: if we call the size of each group x, then:

• Houses with both patio and pool = x
• Houses with neither feature = x

Think about what this means for our 75 houses:

• Some have patio only (let's call this P)
• Some have pool only (let's call this S)
• Some have both (which is x)
• Some have neither (also x)

So our total is: \(\mathrm{P} + \mathrm{S} + \mathrm{x} + \mathrm{x} = 75\)

Why This Constrains Everything

Here's the breakthrough insight: We also know that houses with patios = 48, which means:

• Patio only + Both = 48
• \(\mathrm{P} + \mathrm{x} = 48\)
• Therefore, \(\mathrm{P} = 48 - \mathrm{x}\)

Substituting back into our total:

• \((48 - \mathrm{x}) + \mathrm{S} + \mathrm{x} + \mathrm{x} = 75\)
• \(48 + \mathrm{S} + \mathrm{x} = 75\)
• \(\mathrm{S} + \mathrm{x} = 27\)

Since S represents "pool only" and x represents "both patio and pool," the total number of houses with pools is \(\mathrm{S} + \mathrm{x} = 27\).

[STOP - Sufficient!] We can determine the exact number: 27 houses have pools.

Conclusion

Statement 2 alone is sufficient to determine that exactly 27 houses have swimming pools.

This eliminates choices C and E.

The Answer: B

Statement 2 alone provides the unique constraint needed to determine that exactly 27 houses have swimming pools, while Statement 1 alone allows for multiple possible values.

Answer Choice B: "Statement 2 alone is sufficient, but Statement 1 alone is not sufficient."