Is there a formula for stacking aspect ratios?

So one of my pet peeves back when I was into image editing is that in trying to stack images of different aspect ratios vertically such that they'd have the same width; or juxtapose them horizontally such that they'd have the same height; it was always a long, drawn-out game of guess and check to get it right, or at least close enough to right that I could crop it from there. (For the record, I'm referring to Cracked.com, where the combined width in question had to be equal to or fewer than 550 pixels, and the combined height equal to or fewer than 750, rather than outright shrinking one down to and only down to the width or height of the other. Cracked.com is no longer doing image editing contests, but now it's more a matter of curiosity.)

Speaking of guessing, I'm guessing there must be some mathematical formula that helps one figure this out more efficiently but I'm not even sure where to start on figuring out what that is. Does anyone have any ideas where to start on figuring this one out?