Conversion of compound temperature units 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 $B$ 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 practice?
 A: Note that those compound units are used to express rates of change:  change in energy per unit of temperature, or change in temperature per unit of time (day).  That is, $\Delta E/\Delta T$ and $\Delta T / \Delta t$.  The offset term cancels out when taking the differences.
A: Most of the time, the temperature quantity is a difference between two temperatures, not an absolute value. So, the offset is zero:
\begin{align}
\Delta T(F) &= T_2(F) - T_1(F) \\&= \left(\frac{9}{5}T_2(C) + 32\right) - \left(\frac{9}{5}T_1(C) + 32\right) \\&= \frac{9}{5}\left(T_2(C) - T_1(C)\right) \\&=\frac{9}{5}\Delta T(C)
\end{align}
This is true for quantities like entropy changes, heat capacities, and coefficients of thermal expansion.
An exception to this would be the universal gas law $PV = nk_BT.$ In this case, you need to use an absolute temperature scale where $T=0$ actually means absolute zero. So, your choices are Kelvin or Rankine, and the offset is zero, since zero Kelvin and zero Rankine mean the same thing.
