On the sizes of average of ratios

I once asked this silly question on an interview:

Which is larger, a micro-average or a macro-average?

Not knowing the answer for many years until recently when I sat down and worked it out, in fact in a more general sense, for non-averages as well, let a,b,c,d refer to any real number with d>=b>=0.

(a+c)/(b+d) <= (a/b + c/d)/2


a/b >= c/d

equality holds, and inequality inverts, in both simultaneously.

okay okay, all my friends laughing at me, I know this is silly little math exercise you did when you were in a diaper… whatever!!

The impact of this is, if one is to change from one average to another average half way through some testing period, while controlling the denominator of these inequalities, one would technically not be lying when one says the average has gone up. (not stating that the definition of average has changed)

Here is the mnemonic: Smaller Smaller Smaller Bigger: Ratio with the Smaller denominator being Smaller than ratio with the larger denominator means the Smaller average is Bigger than the bigger average,

TODO: generate general case averaging n ratios.