Well I do not know Gwyen, but I'll try to explain the way as I understand it.
It basically shows how to identify the shot as a quart, half or 3 quarts.
Think of them as ratios. I am using the page titled IDENTIFYING ANGLES to explain.
Quart or 1/4 cuts, from the top RHS table example. Look at where the cue ball is sitting on.
it is 2 diamonds distance to the long rail on the right
and look at spot of the 9ball, it is 2 diamonds to the same spot on the long rail.
This forms a triangle with equal lengths 2 diamonds to 2 diamonds. hence 2:2 or 1:1 ratio.
Half cut
From the center pocket example. the 9 ball is sitting inbetween 2 circles. Distance from the 9 ball to the rail is 1 diamond and that distance to the pocket is 1/2 diamond. Hence you form an imagined triangle with 1 diamond distance to the rail and 1/2 diamond to the pocket. so 1/2 : 1 or 1:2 ratio
i see how you got the ratio for the object ball but the cue ball isnt in the equation....if the cue ball moves to a different location its no longer a half ball hit......
3 quarts
look at the bottom 9ball example . distance to the long rail on the RHS is 2 diamonds and the distance from that spot to the pocket is 1/2 diamond. So the ratio is 1/2 : 2 or 1:4 ratio
same comment as above
hope this helps.