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Adding and averaging rates

A rate is the quotient of two quantities with different, but related, units. A unit rate is a rate with a unit quantity, usually the denominator. A vector rate is a rate with a vector quantity, usually the numerator. Rates with the same units may be added, subtracted, and averaged.

Addition

Rates with the same units in their denominator are added using ordinary addition, which will be called arithmetic addition since addition takes place in the numerator. For example, if x1 and x2 are lengths, and t is a given time interval, then the time speed rates v1 and v2 are added by arithmetic addition:

v_{1}+v_{2}=\frac{x_{1}}{t}+\frac{x_{2}}{t} =\frac{x_{1}+x_{2}}{t} = v_{3}

If x1 and x2 are displacements, and t is a given time interval, then the time velocity rates v1 and v2 are added by arithmetic addition:

\mathbf{v}_{1}+\mathbf{v}_{2} =\frac{\mathbf{x}_{1}}{t}+\frac{\mathbf{x}_{2}}{t} =\frac{\mathbf{x}_{1}+\mathbf{x}_{2}}{t} = \mathbf{v}_{3}

where vector addition means addition of dimensions, i.e., parallelogram addition.

If t1 and t2 are time intervals, and x is an independent length, then the space pace rates w1 and w2 are added by arithmetic addition:

w_{1}+w_{2}=\frac{t_{1}}{x}+\frac{t_{2}}{x} =\frac{t_{1}+t_{2}}{x}=w_{3}

If t1 and t2 are dischronments, and x is an independent length, then the space lenticity rates w1 and w2 are added by arithmetic addition:

\mathbf{w}_{1}+\mathbf{w}_{2} =\frac{\mathbf{t}_{1}}{x}+\frac{\mathbf{t}_{2}}{x} =\frac{\mathbf{t}_{1}+\mathbf{t}_{2}}{t}=\mathbf{w}_{3}

Rates with the same units in their numerator are added using harmonic addition, also known as parallel addition, in which addition takes place in the denominator. For this reason it could be called denominator addition. If t1 and t2 are time intervals, and x is an independent length, then the space velocity rates u1 and u2 are added by harmonic addition (symbolized here by a circled plus):

u_{1} \oplus u_{2} =\left (u_{1}^{-1}+u_{2}^{-1} \right )^{-1} =\left (\frac{t_{1}}{x}+\frac{t_{2}}{x} \right )^{-1} =\left (\frac{t_{1}+t_{2}}{x} \right )^{-1} =\left (\frac{x}{t_{1}+t_{2}} \right )

If t1 and t2 are time intervals, and x is an independent length, then the space velocity rates u1 and u2 are added by harmonic addition:

\mathbf{u}_{1} \oplus \mathbf{u}_{2}=\mathbf{w}_{3}^{-1} =\left (\mathbf{w}_{1}+\mathbf{w}_{2} \right )^{-1} =\left (\mathbf{u}_{1}^{-1}+\mathbf{u}_{2}^{-1} \right )^{-1}

Averages

Rates with the same units in their denominator are averaged using the arithmetic mean. Rates with the same units in their numerator are averaged using the harmonic mean. For example, if x1 and x2 are lengths, and t is a given time interval, then the time speed rates v1 and v2 are averaged by the arithmetic mean:

\frac{{v}_{1}+{v}_{2}}{2} =\frac{{x}_{1}}{2t}+\frac{{x}_{2}}{2t} =\frac{{x}_{1}+{x}_{2}}{2t}=v_{3}

If x1 and x2 are displacements, and t is a given time interval, then the time velocity rates v1 and v2 are averaged by the arithmetic mean:

\frac{\mathbf{v}_{1}+\mathbf{v}_{2}}{2} =\frac{\mathbf{x}_{1}}{2t}+\frac{\mathbf{x}_{2}}{2t} =\frac{\mathbf{x}_{1}+\mathbf{x}_{2}}{2t} =\mathbf{v}_{3}

If t1 and t2 are time intervals, and x is a given length, then the space pace rates w1 and w2 are averaged by the arithmetic mean:

\frac{{w}_{1}+{w}_{2}}{2} =\frac{{t}_{1}}{2x}+\frac{{t}_{2}}{2x} =\frac{{t}_{1}+{t}_{2}}{2x}=w_{3}

If t1 and t2 are dischronments, and x is a given length, then the space lenticity rates w1 and w2 are averaged by the arithmetic mean:

\frac{\mathbf{w}_{1}+\mathbf{w}_{2}}{2} =\frac{\mathbf{t}_{1}}{2x}+\frac{\mathbf{t}_{2}}{2x} =\frac{\mathbf{t}_{1}+\mathbf{t}_{2}}{2x} =\mathbf{w}_{3}

If t1 and t2 are time intervals, and x is a given length, then the space speed rates u1 and u2 are averaged by the harmonic mean:

\frac{u_{1} \oplus u_{2}}{2} =\left (\frac{u_{1}^{-1}+u_{2}^{-1}}{2} \right )^{-1} =\left (\frac{t_{1}}{2x}+\frac{t_{1}}{2x} \right )^{-1} =\left (\frac{t_{1}+t_{2}}{2x} \right )^{-1} =\left (\frac{2x}{t_{1}+t_{2}} \right )

If t1 and t2 are dischronments, and x is a given length, then the space velocity rates u1 and u2 are averaged by the harmonic mean:

\frac{\mathbf{u}_{1} \oplus \mathbf{u}_{2}}{2} =\mathbf{w}_{3}^{-1} =\left (\frac{\mathbf{w}_{1}+\mathbf{w}_{2}}{2} \right )^{-1} =\left (\frac{\mathbf{u}_{1}^{-1}+\mathbf{u}_{2}^{-1}}{2} \right )^{-1}=\mathbf{u}_{3}

Another example would be electrical resistance, which equals voltage divided by current, and is connected in series with arithmetic addition or in parallel with harmonic addition. In a series circuit, the current across each resistor is the same, so resistance is added with arithmetic addition. On the other hand, resistors connected in parallel have the same voltage, so resistance is added with harmonic addition.

Conductance, the inverse of resistance, is connected in series with harmonic addition, or connected in parallel with arithmetic addition. Serial circuits are temporally sequential; parallel circuits are temporally parallel.

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