To determine the relationship between Quantity A and Quantity B, we need to look at what information is provided and, more importantly, what is missing.

Analysis of the Ratios
A ratio tells us the relative proportion of one group to another, but it does not tell us the actual number of individuals in those groups.
- Townville: The ratio 4 : 11 means that for every 15 animals (cats + dogs), 4 are cats and 11 are dogs. The actual number of cats could be $4,40,400$, or any multiple of 4 .
- Villageton: The ratio $3: 8$ means that for every 11 animals, 3 are cats and 8 are dogs. The actual number of cats could be $3,30,3,000$, or any multiple of 3 .

Comparison
Because we do not know the total number of pets (or the number of dogs) in either Townville or Villageton, we cannot calculate the actual number of cats.

Consider these two conflicting scenarios:
1. Scenario 1: Townville is a massive city with $1,500,000$ pets. In this case, there would be $\mathbf{4 0 0 , 0 0 0}$ cats. Villageton is a tiny village with only 11 pets, meaning it has only $\(\mathbf{3}\)$ cats.
- Result: Quantity A is much larger.
2. Scenario 2: Townville is a small neighborhood with only 15 pets, meaning it has $\mathbf{4}$ cats. Villageton is a large city with 110,000 pets, meaning it has 30,000 cats.
- Result: Quantity B is much larger.

Conclusion
Since we can create scenarios where Quantity A is larger, scenarios where Quantity B is larger, and even scenarios where they are equal, there is no way to determine the relationship.

The relationship cannot be determined from the information given.