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 Δ*x*/Δ*y*. And so the general addition of rates follows the general addition of fractions:

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

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:

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:

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

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