Rate (mathematics)

For other uses, see Rate.

In mathematics, a rate is the ratio between two related quantities.[1] Often it is a rate of change. If the unit or quantity in respect of which something is changing is not specified, usually the rate is per unit of measurement. However, a rate of change can be specified per unit of time, or per unit of length or mass or another quantity. The most common type of rate is "per unit of time", such as speed, heart rate and flux. Ratios that have a non-time denominator include exchange rates, literacy rates and electric field (in volts/meter).

In describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate (for example a heart rate is expressed "beats per minute"). A rate defined using two numbers of the same units (such as tax rates) or counts (such as literacy rate) will result in a dimensionless quantity, which can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%) or fraction or as a multiple.

Often rate is a synonym of rhythm or frequency, a count per second (i.e., Hertz); e.g., radio frequencies or heart rate or sample rate.


Rates and ratios often vary with time, location, particular element (or subset) of a set of objects, etc. Thus they are often mathematical functions. For example, velocity v (distance traveled per unit time) of a transportation vehicle on a certain trip may be may be represented as a function of x (the distance traveled from the start of the trip) as v(x). Alternatively, one could express velocity as a function of time t from the start of the trip as v(t). Another representation of velocity on a trip is to partition the trip route into N segments and let vi be the constant velocity on segment i (v is a function of index i). Here each segment i, of the trip is a subset of the trip route.

A rate (or ratio) may often be thought of as a performance indicator, output-input ratio, benefit-cost ratio, all considered in the broad sense. For example, miles per hour in transportation is the output (or benefit) in terms of miles of travel, which one gets from spending an hour (a cost in time) of traveling (at this velocity).

A set of sequential indices i may be used to enumerate elements (or subsets) of a set of ratios under study. For example, in finance, one could define i by assigning consecutive integers to companies, to political subdivisions (such as states), to different investments, etc. The reason for using indices i, is so a set of ratios (i=0,N) can be used in an equation so as to calculate a function of the rates such as an average of a set of ratios. For example, the average velocity found from the set of vi's mentioned above. Finding averages may involve using weighted averages and possibly using the Harmonic mean.

A ratio r=a/b has both a numerator a and a denominator b. a and/orb may be a real number or integer. The inverse of a ratio r is 1/r = b/a.

Rate of change

See also: Derivative

Consider the case where the numerator of a rate is a function where happens to be the denominator of the rate . A rate of change of with respect to (where is incremented by ) can be formally defined in two ways:[2]

where f(x) is the function with respect to x over the interval from a to a+h. An instantaneous rate of change is equivalent to a derivative.

An example to contrast the differences between the average and instantaneous definitions: the speed of a car can be calculated:

  1. An average rate can be calculated using the total distance travelled between a and b, divided by the travel time
  2. An instantaneous rate can be determined by viewing a speedometer.

However these two formulas do not directly apply where either the range or the domain of is a set of integers or where there is no given formula (function) for finding the numerator of the ratio from its denominator.

Temporal rates

See also: Time derivative

In chemistry and physics:

Counts-per-time rates

Main article: Frequency

In computing:

Miscellaneous definitions:

Economics/finance rates/ratios

Other rates

See also


  1. See Webster's new international dictionary of the English language, second edition, unabridged. Merriam Webster Co. 2016. p.2065 definition 3. while this definition doesn't say "related" and while the ratio of two non-related quantities is technically a ratio, such a ratio has little (if any meaning). For example, what would be the utility of finding the ratio of such unrelated numbers as ratio of the weight ones residence to an integer selected at random between -10−9 and +109?
  2. Adams, Robert A. (1995). Calculus: A Complete Course (3rd ed.). Addison-Wesley Publishers Ltd. p. 129. ISBN 0-201-82823-5.
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