Rate (mathematics)

In mathematics, a rate is the ratio between two related quantities.^[1] If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable.

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.

Introduction

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 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 v_i 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 an 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 v_i'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

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

{\begin{aligned}{\mbox{Average rate of change}}&={\frac {f(a+h)-f(a)}{h}}\\{\mbox{Instantaneous rate of change}}&=\lim _{h\to 0}{\frac {f(a+h)-f(a)}{h}}\end{aligned}}

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 unit rates are average and instantaneous definitions: the speed of a car can be calculated:

An average rate can be calculated using the total distance travelled between a and b, divided by the travel time
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 $f()$ 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

In chemistry and physics:

Speed, being the distance covered per unit of time; e.g., miles per hour and meters per second
Acceleration, the rate of change in speed, or the change in speed per unit of time
Reaction rate, the speed at which chemical reactions occur
Volumetric flow rate, the volume of fluid which passes through a given surface per unit of time; e.g., cubic meters per second

Counts-per-time rates

Radioactive decay, the amount of radioactive material in which one nucleus decays per second, measured in Becquerels

In computing:

Bit rate, the number of bits that are conveyed or processed by a computer per unit of time
Symbol rate, the number of symbol changes (signalling events) made to the transmission medium per second
Sampling rate, the number of samples (signal measurements) per second

Miscellaneous definitions:

Rate of reinforcement, number of reinforcements per unit of time, usually per minute
Heart rate, usually measured in beats per minute

Economics/finance rates/ratios

Exchange rate, how much one currency is worth in terms of the other
Inflation rate, ratio of the change in the general price level during a year to the starting price level
Interest rate, the price a borrower pays for the use of money they do not own (ratio of payment to amount borrowed)
Price–earnings ratio, market price per share of stock divided by annual earnings per share
Rate of return, the ratio of money gained or lost on an investment relative to the amount of money invested
Tax rate, the tax amount divided by the taxable income
Unemployment rate, the ratio of the number of people who are unemployed to the number in the labor force
Wage Rate, the amount paid for working a given amount of time (or doing a standard amount of accomplished work) (ratio of payment to time)

Other rates

Birth rate, and mortality rate, the number of births or deaths scaled to the size of that population, per unit of time
Literacy rate, the proportion of the population over age fifteen that can read and write
Sex ratio or Gender ratio, the ratio of males to females in a population

References

^ 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 of ones residence to an integer selected at random between -10⁻⁹ and +10⁹?
^ Adams, Robert A. (1995). Calculus: A Complete Course (3rd ed.). Addison-Wesley Publishers Ltd. p. 129. ISBN 0-201-82823-5.

[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 of 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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