As long as we're not doing maths, my basic intuition about the matter is that
should give such a line: there is definitely one bisecting plane with that line on it (cut along one edge, which is perpendicular to the other edge). Then, rotating that cutting plane about the axis of the special line seems to move symmetrical amounts of volume from one side to ...
@PaulPanzer made a summary of existing answers here (including some they gave themselves).
I want to provide a clearer visualization for each of the existing solutions:
Source code for the above program
No. What you are asking is not possible. Because if an integer and a non-integer pair up on two opposite sides of the new rectangle, then two of the sides will have non-integer values, but the other two will be forced to have integer values.
Given: Rectangle with area of 5N (n is a non-integer number)
& Rectangle with area of 4X (x is a non-...
Just a summary of the answers by @hexomino, @Zoir and @mpasko256, a few notes and perhaps (if nobody else finds it within the next few days) one more solution.
And there is one more solution which I'll add in a few days.