You have two 50g containers of cream. One is 10% fat, and the other 20% fat. You combine them. What is the percentage of fat in the mixture?
If you answered D., you must be Ross McKitrick or Christopher Essex. From page 108 of Taken By Storm:
An example of something that behaves intensively would be the percent of milk fat in a coffee creamer. If you put two small containers of 10% coffee creamer together, you do not get 20% milk fat. The cream is still 10%, even if you have twice as much. In the same manner, if you have two identical boxes with the same energy and the same temperature, join them together. The resulting doubled box will have twice the energy, but it will not have twice the temperature. There is no amount of temperature; it measures the condition or state of the stuff in the box. …
For computing your average, why would you add up the cubes in linear form? … why not square the temperatures, or take them to the fourth power? … if you are averaging the kinetic energy of molecules, it makes sense to calculate the mean of the squares of the speeds, because energy, which goes as the square of the speed, is physically additive, while speeds themselves are not. Or, since the Stefan-Boltzmann law tells us that equilibrium radiative energy goes as the fourth power of temperature, why not raise the temperature to the fourth power before adding them up? …
With temperature, there is no basis on physical grounds to use a simple sum, some other sum or some other more complicated rule for averaging, because temperature is an intensive quantity.
OK, extensive quantities like mass and energy add when you combine things. Intensive quantities like temperature and fat percentage are the ratio of two extensive quantities. For fat percentage it’s the mass of fat divided by the total mass. So when you combine things intensive quantities don’t add, but the answer isn’t arbitrary. You just have to add up the extensive quantities and divide the totals. For the creamer example this turns out to be just your regular everyday weighted average where the weights are the mass of each creamer.
And if you remember your kinetic theory of gasses, you will have noticed how dodgy their justification for squaring temperatures was. The speed of a gas molecule is proportional to the square root of temperature, so adding up the squares of the speeds is just adding up the temperatures.