This post relates to the previous post Adding and Averaging Rates.

A rate is a fraction, though the denominator is often one (a unit rate). In general a rate could be symbolized as Δxy. And so the general addition of rates follows the general addition of fractions:

$\frac{\Delta&space;x_{1}}{\Delta&space;y_{1}}+\frac{\Delta&space;x_{2}}{\Delta&space;y_{2}}=&space;\frac{\Delta&space;x_{1}\Delta&space;y_{2}+\Delta&space;x_{2}\Delta&space;y_{1}}{\Delta&space;y_{1}\Delta&space;y_{2}}$

If, as is usual, the denominator is the independent variable, the denominator is the same for all rates, which simplifies the addition:

$\frac{\Delta&space;x_{1}}{\Delta&space;y}+\frac{\Delta&space;x_{2}}{\Delta&space;y}=&space;\frac{\Delta&space;x_{1}\Delta&space;y+\Delta&space;x_{2}\Delta&space;y}{(\Delta&space;y)^{2}}=&space;\frac{\Delta&space;x_{1}+\Delta&space;x_{2}}{\Delta&space;y}$

So the combined rate is the arithmetic sum of the two rates. Call this the arithmetic rate.

If, however, the numerator is the independent variable, the numerator is the same for all rates, which leads to a different result:

$\frac{\Delta&space;x}{\Delta&space;y_{1}}+\frac{\Delta&space;x}{\Delta&space;y_{2}}=&space;\Delta&space;x&space;\left&space;(&space;\frac{1}{\Delta&space;y_{1}}+\frac{1}{\Delta&space;y_{2}}&space;\right&space;)=&space;\frac{\Delta&space;x}{\left&space;((\Delta&space;y_{1})^{-1}+(\Delta&space;y_{2})^{-1}&space;\right&space;)^{-1}}=&space;\frac{\Delta&space;x}{\Delta&space;y_{1}\boxplus&space;\Delta&space;y_{2}}$

where the box plus designates harmonic addition. Call this the harmonic rate.

Compare this with the inverse of the converse rate with its independent variable in the denominator:

$\left&space;(\frac{\Delta&space;y_{1}}{\Delta&space;x}+\frac{\Delta&space;y_{2}}{\Delta&space;x}&space;\right&space;)^{-1}=&space;\left&space;(\frac{\Delta&space;y_{1}\Delta&space;x+\Delta&space;y_{2}\Delta&space;x}{(\Delta&space;x)^{2}}&space;\right&space;)^{-1}=&space;\frac{\Delta&space;x}{\Delta&space;y_{1}+\Delta&space;y_{2}}$

Or compare the inverse of the converse rate with its independent variable in the numerator:

$\left&space;(\frac{\Delta&space;y}{\Delta&space;x_{1}}+\frac{\Delta&space;y}{\Delta&space;x_{2}}&space;\right&space;)^{-1}=&space;\left&space;(\frac{\Delta&space;y}{\Delta&space;x_{1}&space;\boxplus&space;\Delta&space;x_{2}}&space;\right&space;)^{-1}=&space;\frac{\Delta&space;x_{1}&space;\boxplus&space;\Delta&space;x_{2}}{\Delta&space;y}$

So the combined rate is the harmonic sum of the two rates.