Euclid wrote that “A ratio is a relation in respect of size between two magnitudes of the same kind.” Magnitudes of different kinds were not considered until Galileo. A ratio between magnitudes *a* and *b* is expressed as either *a* : *b* or *b* : *a*.

A quotient is the result of dividing a dividend (or numerator) by a divisor (or denominator). If the divisor is a unit (i.e., numerically one), and the units of measure are the same for the divisor and dividend, then the quotient is a measurement. If the divisor and dividend have different (though related) units of measure, the quotient is a rate. If additionally the divisor is a unit (i.e., numerically one), then the quotient is a unit rate.

Two different quotients can come from any ratio. The ratio *a* : *b* can be converted into *a*/*b* or as *b*/*a*. This is also true of a unit ratio or unit rate: the ratio *a* : 1 can be converted into *a*/1 or 1/*a*. Similarly, the ratio 1 : *b* can be converted into 1/*b* or *b*/1. The quotient *a*/1 is also called the quotient *a*. The quotient 1/*a* can be called the *inverse quotient*.

Because a rate is a ratio, this applies to rates, too. The unit rate *a*/1 is also called the rate *a*. The unit rate 1/*a* can be called the *inverse rate*. These two rates express the same ratio differently. Arithmetic operations on the unit rate *a*/1 are carried out the same as other arithmetic operations. But arithmetic operations on the inverse rate 1/*a* must follow reciprocal operations (see *here*).

The divisor (or denominator) of a quotient is an independent variable because its value can be set independently of the dividend (or numerator), which is a dependent variable. The dividend (or numerator) of an inverse quotient is a dependent variable because its value can be set independently of the divisor (or denominator). An independent variable can be set prior to an experiment, or it can be allowed to change independently of the experiment, i.e., as time measured by a clock.

A rate of change is “the ratio of the difference between values of a variable quantity at different times to the difference between the times; the change per unit [of elapsed] time” (*Collins Dictionary of Mathematics*, 2nd ed. 2002). Because time is measured by uniform motion, distance can be used instead, i.e.: a rate of change is the ratio of the difference between values of a variable quantity at different distances to the difference between the distances; the change per unit of elapsed distance.

For example, speed is the rate of change of a body’s distance per unit of elapsed time; pace is the rate of change of a body’s time per unit of elapsed distance.

There are also rates of vectors and scalars. The vector is the numerator, and the scalar is the denominator or a vector rate. For example, velocity is the rate of change of displacement per unit of elapsed time. Since displacement is a vector, velocity is a rate of a vector per a unit scalar. Again, since time is measured by uniform motion, distance can be used instead. For example, lenticity is the rate of change of dischronment per unit of elapsed distance.

The inverse rate of a vector and scalar has the vector in the denominator and the scalar in the numerator. For example, inverse velocity is the rate of elapsed or fixed distance per unit of measured chronation. Similarly, inverse lenticity is the rate of elapsed or fixed distance per unit of measured time.