In general one can convert from one unit to another (of the same dimension) by using (($U$,$A$) and ($V$,$B$) being compatible unit pairs in the following)
$xU=(x\ast f_{U\rightarrow A}+o_{U\rightarrow A})A=yA$
Oftentimes the offset $o_{U\rightarrow A}$ will have a value of zero, which reduces the conversion to a simple multiplication with a factor $f_{U\rightarrow A}$. For some units like temperatures, however, we need that offset as well.
There are a few compound units in use, which are combinations of a temperature and some other unit like joule per kelvin or degree Celsius day.
My question is how would I convert those compound units unambiguously?
To be more specific: Assume we want to convert from a (compound) unit $UV$ to a unit $AB$ and are given the following conversion formulas:
- $xU=(x\ast f_{U\rightarrow A}+o_{U\rightarrow A})A$
- $xV=(x\ast f_{V\rightarrow B})B$
The conversion formula from $x UV = y AB$ seems to depend on the order of conversions:
If converting first from $U$ to $A$ and then from $V$ to $V$$B$ we end up with the following calculation
$xUV\\ =(x\ast f_{U\rightarrow A}+o_{U\rightarrow A})AV \\ =((x\ast f_{U\rightarrow A}+o_{U\rightarrow A})A\ast f_{V\rightarrow B})B \\ =(x\ast f_{U\rightarrow A}\ast f_{V\rightarrow B}+o_{U\rightarrow A}\ast f_{V\rightarrow B})AB$
Doing it the other way around, we end up with
$xUV\\ =(x\ast f_{V\rightarrow B})UB\\ =(x\ast f_{V\rightarrow B}\ast A\ast f_{U\rightarrow A}+o_{U\rightarrow A})B\\ =(x\ast f_{V\rightarrow B}\ast f_{U\rightarrow A}+o_{U\rightarrow A})AB$
So depending on the order of conversions we end up in results, which differ by
$\begin{array}{l}(x\ast f_{U\rightarrow A}\ast f_{V\rightarrow B}+o_{U\rightarrow A}\ast f_{V\rightarrow B})-(x\ast f_{V\rightarrow B}\ast f_{U\rightarrow A}+o_{U\rightarrow A})\\=o_{U\rightarrow A}\ast f_{V\rightarrow B}-o_{U\rightarrow A}\\=o_{U\rightarrow A}\ast(f_{V\rightarrow B}-1)\end{array}$
Can somebody explain to me, where I'm going wrong here or how this ambiguity is resolved in practisepractice?
