Addition of rates

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 Δx/Δy. And so the general addition of rates follows the general addition of fractions:

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

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

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

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:

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

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:

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

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

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

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

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