In MS SQL Server, the fill factor determines the percentage of space on each page that is initially allocated to store data. A lower fill factor (e.g., 20%) leaves more space empty on each page, while a higher fill factor (e.g., 40%) allocates less empty space. The purpose of the fill factor is to allow for page splits and data growth without requiring frequent page adjustments, which can be costly in terms of auto performance.

The relationship between A and B can be calculated as follows:

Let's assume that the total number of rows per page is C.

For the first design (20% fill factor): A = 0.20 * C (20% of the page is empty)

For the second design (40% fill factor): B = 0.40 * C (40% of the page is empty)

Now, we want to find the relationship between A and B:

A = 0.20 * C B = 0.40 * C

To find the ratio of A to B, we can divide the equation for A by the equation for B:

A / B = (0.20 * C) / (0.40 * C)

C cancels out, and you get:

A / B = 0.20 / 0.40 = 1/2

So, A is half of B:

A = 0.5 * B

If you want to express the relationship between B and A, you can take the reciprocal:

B = 2 * A

Therefore, the correct relationship between A and B is:

B = 2A