Nope, for a linear function f(x) = ax+b , f(u+v) = f(u)+f(v)
[quote=“Kimble, post:5, topic:725984”]
If you count the usual concept of fuel economy/consumption as “the same kind,” then there’s a counterexample:[ul][li]x miles per gallon = 112903/(480x) ≈ 235/x liters per 100 km[/ul]Converting mpg to km/L follows the normal rules.[/li][/QUOTE]
That’s because “fuel economy” in those two sets of units does not have the same dimensionality.
Dimensionality is what you get by reducing a unit of measurement to its dimensions. For example, length’s dimensionality is L whether you’re measuring it in inches, handspans, feet, miles, kilometers or angstroms; area is L[sup]2[/sup]; volume is L[sup]3[/sup], no matter what the unit. Time is t: speed’s dimensionality is Lt[sup]-1[/sup]. And so forth. Knowing how to analyze dimensionality is very helpful to see what kind of relationship there will be between different concepts and units: if the dimensionality is the same, then conversion will be in the form proposed by the OP.
Concepts such as resistance and conductance are related but they aren’t “the same” either from a dimensionality point of view or from the point of view of how they’re used. They’re inverses, so are their units.
If a = 2, b = 3, u = 5, v = 10:
f(x) = 2x+3
f(u+v) = 2*(5+10) + 3 = 215 + 3 = 33
f(u) + f(v) = 25+3 + 2*10+3 = 10 + 3 + 20 + 3 = 36
??!
You sure? Or is my 9th grade algebra that badly rusty already (hey, it’s possible).
f(x) = ax + b
f(u) = au + b
f(v) = av + b
f(u+v) = a(u+v) + b = au + av + b
f(u) + f(v) = (au + b) + (av + b) = au + av + 2b
N’est-ce pas?
ETA: Wow, some simulposting, eh wot?
Fourier transfoms? 
The pH scale, which measures the degree of a chemical solution’s acidity or alkalinity, is also a logarithmic scale.
Psychologists have also played around with a vaguely-defined scale to measure perceived pain levels. The unit of measure is called the dolor (derived from Latin, akin to the English “dolorous” and the given name “Dolores”). It’s vague because measurements depend upon subjects’ self-reported perceptions of various painful experiences, ranging from a pin-prick to getting burned nearly to death. IIRC, the scale was a kinds-sorta-somehow logarithmic scale, where each unit meant ten times more painful than the previous unit.
What about the Scoville scale, that is used to measure the perceived heat strength of chili peppers? From looking at some of the numbers (see table of values here), which run from 0 to 16,000,000,000) it looks like it isn’t logarithmic but probably ought to be.
I’ve been exercising, walking a few miles each day, and shedding excess pounds in the process. So, I’m converting miles to negative-pounds.
A common every-day conversion between two totally different measures, well-established in science and perfectly cromulent, is the conversion between units of distance and units of time, as for example when we say “I live about three hours from San Francisco”. This assumes some implied or agree-upon conversion factor (a measure of speed), and where for physical science purposes, the speed of light is a particularly useful conversion factor.
(Credit: I learned this at a fairly young age, when I was at a level of scientific sophistication where the above would be a fairly sophisticated insight indeed. I got it from a book by George Gamow.)
Of course you are right, serves me right for posting something without thinking about it.
There’s another reason you might add temperatures: To compute an average. In Surely You’re Joking, Mr. Feynman he has a chapter blasting grade-school math books for the stupidity of their word-problems, one of which called for adding up the temperatures of several stars. Then, immediately contradicting his own complaint, he notes that it might be useful if you wanted to get the average temperature of the stars.
There’s a street near here named Claribel. I picture that as a unit of measurement of the clarity of wine. 
Oh that’s good 
Repeating: hardness scales are essentially independent. They mean the same thing, but can’t be transformed, except by doing a complete analysis of the materials.
Another one might be Roughness. Roughness measures are essentially statistics. They all measure “roughness” but are completely different measures of roughness. You could be almost zero on one, and almost 100% on another.