The relationship cannot be determined.

Here's why:

The area of a rectangle is determined by its length and width ( $\(A=l \times w\)$ ). While we are given the perimeter, there are many possible combinations of length and width that can result in the same perimeter, and each combination will yield a different area. The closer a rectangle's shape is to a square, the larger its area for a given perimeter.

Quantity A
Perimeter $=20$
Possible dimensions:
- Length $=9$, Width $=1$. Area $\(=9 \times 1=9\)$
- Length $=8$, Width $=2$. Area $\(=8 \times 2=16\)$
- Length $=6$, Width $=4$. Area $\(=6 \times 4=24\)$
- Length $=5$, Width $=5$ (a square). Area $\(=5 \times 5=25\)$

The area of a rectangle with a perimeter of 20 can range from a value close to zero up to 25 .

Quantity B

Perimeter $=24$
Possible dimensions:
- Length $=11$, Width $=1$. Area $\(=11 \times 1=11\)$
- Length $=10$, Width $=2$. Area $\(=10 \times 2=20\)$
- Length $=9$, Width $=3$. Area $\(=9 \times 3=27\)$
- Length $=8$, Width $=4$. Area $\(=8 \times 4=32\)$
- Length $=6$, Width $=6$ (a square). Area $\(=6 \times 6=36\)$

The area of a rectangle with a perimeter of 24 can range from a value close to zero up to 36 .

Conclusion

Since it's possible for the area in Quantity A (e.g., 25) to be larger than an area in Quantity B (e.g., 20), and for an area in Quantity A (e.g., 9) to be smaller than an area in Quantity B (e.g., 36), the relationship cannot be determined.