This post is the latest in a series on rates.

A *rate* is a variable quantity measured with respect to a quantity determined independently. A rate is expressed as a ratio of the quantity measured and the independent quantity. A rate of change is a difference of quantities measured with respect to a difference of quantities determined independently. The denominator is the independent quantity.

The *converse rate* is the rate formed by interchanging the names, units, and dependencies of the quantities. If *x*/*y* is a rate with independent quantity *y*, then *y*/*x* is the converse rate with independent quantity *x*. If Δ*x*/Δ*y* is a rate of change, then Δ*y*/Δ*x* is the converse rate of change. Note that a rate and its converse are isomorphic.

The *inverse rate* is the rate formed by taking the reciprocal of the rate. If *x*/*y* is a rate, then 1/(*x*/*y*) is the inverse rate. If Δ*x*/Δ*y* is a rate of change, then 1/(Δ*x*/Δ*y*) is the inverse rate of change. Note that the independent quantity is the same for a rate and its inverse.

The *inverse converse rate* is the reciprocal of the converse rate, which equals the converse of the inverse rate. If *x*/*y* is a rate, then 1/(*y*/*x*) is the inverse converse rate. If Δ*x*/Δ*y* is a rate of change, then 1/(Δ*y*/Δ*x*) is the inverse converse rate of change. Note that the inverse converse rate has the same dependencies as the converse rate.

Rate | Converse Rate | Inverse Rate | Inverse Converse Rate |

x/y | y/x | 1/(x/y) | 1/(y/x) |

Δx/Δy | Δy/Δx | 1/(Δx/Δy) | 1/(Δy/Δx) |

For example:

The *speed* (or time speed) of a body is the length traversed per unit of independent time without regard to direction, Δ*x*/Δ* t*. The converse speed,

*pace*(or slowness), of a body is the traversal time per unit of independent length without regard to direction, Δ

*t*/Δ

*. The*

__x__*inverse speed*of a body is the reciprocal speed, 1/(Δ

*t*/Δ

*). The*

__x__*inverse pace*(or space speed) of a body is the reciprocal pace, 1/(Δ

*x*/Δ

*). The*

__t__*reverse speed*of a body is the negation of its speed, −Δ

*x*/Δ

*.*

__t__