There are 210 households in a certain residential complex. All households in the complex that have more than one dog...

Understanding the Question

We need to find the exact number of households with pet rodents in a complex of 210 households.

Given Information

• Total households: 210
• All households with more than one dog → also have at least one cat
• All households with at least one cat → have a pet rodent

Key Relationships

From these facts, we can see a chain:
\(\text{Households with } >1 \text{ dog} \subseteq \text{Households with cats} \subseteq \text{Households with rodents}\)

This means:

• Every household with more than one dog MUST have a cat
• Every household with a cat MUST have a rodent

To answer the question, we need to determine exactly how many households have pet rodents. Having just a minimum number or a range won't be sufficient.

Analyzing Statement 1

Statement 1: 18 households have two or more dogs.

Since households with more than one dog must have cats, and households with cats must have rodents, we know at least 18 households have rodents.

However, this doesn't tell us:

• How many households have cats but only one dog (or no dogs)
• How many households have rodents but no cats

Let's test different scenarios to see if we get different answers:

• Scenario A: If exactly 18 households have cats (all being the multi-dog households), then exactly 18 households have rodents
• Scenario B: If 50 households have cats (including the 18 multi-dog households), then 50 households have rodents
• Scenario C: If some households have rodents without cats, the number could be even higher

Since we can have different valid numbers of households with rodents (18, 50, or more), Statement 1 is NOT sufficient.

This eliminates choices A and D.

Analyzing Statement 2

Let's forget Statement 1 and analyze Statement 2 independently.

Statement 2: 90% of households with pet rodents have at least one cat.

This means 10% of households with rodents don't have cats.

Combined with our knowledge that all households with cats must have rodents, we get:

• All cat households are rodent households (\(\text{cats} \subseteq \text{rodents}\))
• Cat households make up 90% of rodent households

But without knowing either the actual number of households with cats or rodents, we can't determine the exact count.

For example:

• If 100 households have rodents → 90 have cats
• If 200 households have rodents → 180 have cats

Both scenarios are valid. Since multiple answers are possible, Statement 2 is NOT sufficient.

This eliminates choice B.

Combining Both Statements

Let's use both statements together:

• From Statement 1: 18 households have ≥2 dogs (and therefore have cats)
• From Statement 2: Number of cat households = 90% of rodent households
• From the question: Multi-dog households ⊆ Cat households ⊆ Rodent households

Since the 18 multi-dog households must have cats, and cat households equal 90% of rodent households:

• Number of cat households ≥ 18
• Number of cat households = 0.9 × (number of rodent households)

This gives us: \(18 \leq 0.9\mathrm{R}\), so \(\mathrm{R} \geq 20\).

But this only provides a minimum. Let's check if different values work:

Scenario 1: Exactly 18 households have cats (all being the multi-dog ones)

• Then: \(18 = 0.9\mathrm{R}\)
• Solving: \(\mathrm{R} = 20\) households with rodents ✓

Scenario 2: 27 households have cats (18 multi-dog + 9 others)

• Then: \(27 = 0.9\mathrm{R}\)
• Solving: \(\mathrm{R} = 30\) households with rodents ✓

Both 20 and 30 are valid answers that satisfy all constraints. Since we cannot determine a unique value for the number of households with rodents, the statements together are NOT sufficient.

This eliminates choice C.

The Answer: E

Even with both statements combined, we cannot determine the exact number of households with pet rodents. Multiple values (20, 30, and any multiple of 10/9 above 20) satisfy all the given constraints.

Answer Choice E: The statements together are not sufficient.