You can run an analysis on each file to determine the average level in dB. Then adjust those numbers based on how you've mixed to figure out the relative difference in dB.
e.g. board has avg. level -15dB and you mix it straight
aud has avg. level -12dB and you mix it +2 dB
relative difference in sources is the difference between -15 and (-12+2) = 5dB
Then you either whip out your log formulas or you can cheat by going into something like CoolEditPro, and take any section of music and pull up the amplitude adjustment, set the 5dB and then uncheck the little box that switches between dB and %. That will show you the equivalent percentage for a dB difference - then do the math.
In the above example, the board signal that is 5dB lower is at 56% of the board so you're mixing at 56/(100+56) = 36%
Then the aud is 100/(100+56) = 64%
So it's 64/36 (aud/board)
To save you the math - here's the scale in 1 dB increments (look up the net difference between the two sources, using average level before mixdown and adjusting for whatever you did to it when mixing).
difference
Same (0dB) = 50/50
1dB = 53/47
2dB = 56/44
3dB = 59/41
4dB = 61/39
5dB = 64/36
6dB = 67/33
7dB = 69/31
8dB = 72/28
9dB = 74/26
or put the other way
50/50 = 0 dB
60/40 = 3.5 dB difference
70/30 = 7.4 dB difference