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 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 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 inverse speed (time 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 inverse velocity (time 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 different values in their denominators but 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 reciprocal addition. If t₁ and t₂ are time intervals, and x is an independent length, then the inverse pace (space speed) rates u₁ and u₂ are added by harmonic addition (symbolized here by a squared plus):

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

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

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

Averages

Rates with the same units in their denominator are averaged using the arithmetic mean. Rates with different values in their denominators but 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 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 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 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 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 inverse pace (space speed) rates u₁ and u₂ are averaged by the harmonic mean:

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

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

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

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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