If the length of a rectangle is represented by L and its width by W, its area is LW and its perimeter 2(L+W).
If LW=20, then if we look at the number 20 we have the factors (whole numbers):
1×20=20, 2×10=20, 4×5=20. These are the dimensions of three rectangles with the same area but different perimeters, which are 42, 24 and 18.
We need 4 rectangles so we want two numbers whose product is 20. There are many of these because from LW=20, we get W=20/L or L=20/W. For example, if L=8, W=20/8=5/2, same as 2.5 or 2½. So we could make a rectangle with length 8 and width 2½. This rectangle would have a perimeter of 21.
Areas of 12, 18 and 20 can be used to make 3 rectangles with whole number side lengths. The 4th rectangle would need to have at least one side length which wasn't a whole number.
If we take any whole number between 1 and 20, there are none for which more than 3 rectangles with L and W have whole number side lengths.
If the area was 24, we would have 1×24, 2×12, 3×8, 4×6 as 4 rectangles with whole number side lengths.
