In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a rate is the
ratio
In mathematics, a ratio shows how many times one number contains another. For example, if there are eight oranges and six lemons in a bowl of fruit, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ...
between two related
quantities
Quantity or amount is a property that can exist as a multitude or magnitude, which illustrate discontinuity and continuity. Quantities can be compared in terms of "more", "less", or "equal", or by assigning a numerical value multiple of a unit ...
in different units. 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
Dependent and independent variables are variables in mathematical modeling, statistical modeling and experimental sciences. Dependent variables receive this name because, in an experiment, their values are studied under the supposition or demand ...
), then the numerator of the ratio expresses the corresponding ''rate of change'' in the other
(dependent) variable.
One common type of rate is "per unit of time", such as
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
,
heart rate
Heart rate (or pulse rate) is the frequency of the heartbeat measured by the number of contractions (beats) of the heart per minute (bpm). The heart rate can vary according to the body's physical needs, including the need to absorb oxygen and excr ...
and
flux
Flux describes any effect that appears to pass or travel (whether it actually moves or not) through a surface or substance. Flux is a concept in applied mathematics and vector calculus which has many applications to physics. For transport ph ...
. Ratios that have a non-time denominator include
exchange rate
In finance, an exchange rate is the rate at which one currency will be exchanged for another currency. Currencies are most commonly national currencies, but may be sub-national as in the case of Hong Kong or supra-national as in the case of ...
s,
literacy rate
Literacy in its broadest sense describes "particular ways of thinking about and doing reading and writing" with the purpose of understanding or expressing thoughts or ideas in written form in some specific context of use. In other words, huma ...
s, and
electric field
An electric field (sometimes E-field) is the physical field that surrounds electrically charged particles and exerts force on all other charged particles in the field, either attracting or repelling them. It also refers to the physical field fo ...
(in volts per 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
Heart rate (or pulse rate) is the frequency of the heartbeat measured by the number of contractions (beats) of the heart per minute (bpm). The heart rate can vary according to the body's physical needs, including the need to absorb oxygen and excr ...
is expressed "beats per minute"). A rate defined using two numbers of the same
units
Unit may refer to:
Arts and entertainment
* UNIT, a fictional military organization in the science fiction television series ''Doctor Who''
* Unit of action, a discrete piece of action (or beat) in a theatrical presentation
Music
* Unit (album), ...
(such as
tax rate
In a tax system, the tax rate is the ratio (usually expressed as a percentage) at which a business or person is taxed. There are several methods used to present a tax rate: statutory, average, marginal, and effective. These rates can also be p ...
s) or counts (such as
literacy rate
Literacy in its broadest sense describes "particular ways of thinking about and doing reading and writing" with the purpose of understanding or expressing thoughts or ideas in written form in some specific context of use. In other words, huma ...
) will result in a
dimensionless quantity
A dimensionless quantity (also known as a bare quantity, pure quantity, or scalar quantity as well as quantity of dimension one) is a quantity to which no physical dimension is assigned, with a corresponding SI unit of measurement of one (or 1) ...
, which can be expressed as a
percentage
In mathematics, a percentage (from la, per centum, "by a hundred") is a number or ratio expressed as a fraction of 100. It is often denoted using the percent sign, "%", although the abbreviations "pct.", "pct" and sometimes "pc" are also us ...
(for example, the global
literacy rate
Literacy in its broadest sense describes "particular ways of thinking about and doing reading and writing" with the purpose of understanding or expressing thoughts or ideas in written form in some specific context of use. In other words, huma ...
in 1998 was 80%),
fraction
A fraction (from la, fractus, "broken") represents a part of a whole or, more generally, any number of equal parts. When spoken in everyday English, a fraction describes how many parts of a certain size there are, for example, one-half, eight ...
, or
multiple.
Often ''rate'' is a synonym of
rhythm
Rhythm (from Greek , ''rhythmos'', "any regular recurring motion, symmetry") generally means a " movement marked by the regulated succession of strong and weak elements, or of opposite or different conditions". This general meaning of regular recu ...
or
frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, a count per second (i.e.,
hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
); e.g.,
radio frequencies
Radio frequency (RF) is the oscillation rate of an alternating electric current or voltage or of a magnetic, electric or electromagnetic field or mechanical system in the frequency range from around to around . This is roughly between the upper ...
,
heart rate
Heart rate (or pulse rate) is the frequency of the heartbeat measured by the number of contractions (beats) of the heart per minute (bpm). The heart rate can vary according to the body's physical needs, including the need to absorb oxygen and excr ...
s, or
sample rate
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples".
A sample is a value of the signal at a point in time and/or sp ...
s.
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
In mathematics, a function from a set to a set assigns to each element of exactly one element of .; the words map, mapping, transformation, correspondence, and operator are often used synonymously. The set is called the domain of the functi ...
.
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
In mathematics, the harmonic mean is one of several kinds of average, and in particular, one of the Pythagorean means. It is sometimes appropriate for situations when the average rate is desired.
The harmonic mean can be expressed as the recipro ...
.
A ratio r=a/b has both a numerator "a" and a denominator "b". The value of a and/or b may be a
real number
In mathematics, a real number is a number that can be used to measure a ''continuous'' one-dimensional quantity such as a distance, duration or temperature. Here, ''continuous'' means that values can have arbitrarily small variations. Every real ...
or
integer
An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...
. The
inverse of a ratio r is 1/r = b/a. A rate may be equivalently expressed as an inverse of its value if the ratio of its units are also inverse. For example, 5
mile
The mile, sometimes the international mile or statute mile to distinguish it from other miles, is a British imperial unit and United States customary unit of distance; both are based on the older English unit of length equal to 5,280 English ...
s (mi) per
kilowatt-hour
A kilowatt-hour (unit symbol: kW⋅h or kW h; commonly written as kWh) is a unit of energy: one kilowatt of power for one hour. In terms of SI derived units with special names, it equals 3.6 megajoules (MJ). Kilowatt-hours are a common bil ...
(kWh) corresponds to 1/5 kWh/mi (or 200
Wh/mi).
Rates are relevant to many aspects of everyday life. For example:
''How fast are you driving?'' The speed of car (often expressed in miles per hour) is a rate. ''What interest does your savings account pay you?'' The amount of interest paid per year is a rate.
Rate of change
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:
:
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
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
.
For example, the average
speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
of a car can be calculated using the total distance travelled between two points, divided by the travel time, while the instantaneous speed can be determined by viewing a
speedometer
A speedometer or speed meter is a gauge that measures and displays the instantaneous speed of a vehicle. Now universally fitted to motor vehicles, they started to be available as options in the early 20th century, and as standard equipment f ...
.
Temporal rates
In chemistry and physics:
*
Speed
In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude of the change of its position over time or the magnitude of the change of its position per unit of time; it is thus a scalar quanti ...
, the rate of change of
position
Position often refers to:
* Position (geometry), the spatial location (rather than orientation) of an entity
* Position, a job or occupation
Position may also refer to:
Games and recreation
* Position (poker), location relative to the dealer
* ...
, or the change of position per unit of time
*
Acceleration
In mechanics, acceleration is the rate of change of the velocity of an object with respect to time. Accelerations are vector quantities (in that they have magnitude and direction). The orientation of an object's acceleration is given by the ...
, the rate of change in speed, or the change in speed per unit of time
*
Power
Power most often refers to:
* Power (physics), meaning "rate of doing work"
** Engine power, the power put out by an engine
** Electric power
* Power (social and political), the ability to influence people or events
** Abusive power
Power may a ...
, the rate of doing
work
Work may refer to:
* Work (human activity), intentional activity people perform to support themselves, others, or the community
** Manual labour, physical work done by humans
** House work, housework, or homemaking
** Working animal, an animal tr ...
, or the amount of
energy
In physics, energy (from Ancient Greek: ἐνέργεια, ''enérgeia'', “activity”) is the quantitative property that is transferred to a body or to a physical system, recognizable in the performance of work and in the form of heat a ...
transferred per unit time
*
Frequency
Frequency is the number of occurrences of a repeating event per unit of time. It is also occasionally referred to as ''temporal frequency'' for clarity, and is distinct from ''angular frequency''. Frequency is measured in hertz (Hz) which is eq ...
, the number of occurrences of a repeating event per unit of time
**
Angular frequency
In physics, angular frequency "''ω''" (also referred to by the terms angular speed, circular frequency, orbital frequency, radian frequency, and pulsatance) is a scalar measure of rotation rate. It refers to the angular displacement per unit tim ...
and
rotation speed
Rotational frequency (also known as rotational speed or rate of rotation) of an object rotating around an axis is the frequency of rotation of the object. Its unit is revolution per minute (rpm), cycle per second (cps), etc.
The symbol for ...
, the number of
turns per unit of time
*
Reaction rate
The reaction rate or rate of reaction is the speed at which a chemical reaction takes place, defined as proportional to the increase in the concentration of a product per unit time and to the decrease in the concentration of a reactant per unit ...
, the speed at which chemical reactions occur
*
Volumetric flow rate
In physics and engineering, in particular fluid dynamics, the volumetric flow rate (also known as volume flow rate, or volume velocity) is the volume of fluid which passes per unit time; usually it is represented by the symbol (sometimes ). I ...
, the volume of fluid which passes through a given surface per unit of time; e.g.,
cubic meters per second
A cubic metre per second (m3s−1, m3/s, cumecs or cubic meter per second in American English) is the unit of volumetric flow rate in the International System of Units (SI) equal to that of a stere or cube with sides of in length exchanged or mo ...
Counts-per-time rates
*
Radioactive decay
Radioactive decay (also known as nuclear decay, radioactivity, radioactive disintegration, or nuclear disintegration) is the process by which an unstable atomic nucleus loses energy by radiation. A material containing unstable nuclei is consid ...
, the amount of radioactive material in which one nucleus decays per second, measured in
becquerel
The becquerel (; symbol: Bq) is the unit of radioactivity in the International System of Units (SI). One becquerel is defined as the activity of a quantity of radioactive material in which one nucleus decays per second. For applications relatin ...
s
In computing:
*
Bit rate
In telecommunications and computing, bit rate (bitrate or as a variable ''R'') is the number of bits that are conveyed or processed per unit of time.
The bit rate is expressed in the unit bit per second (symbol: bit/s), often in conjunction w ...
, the number of bits that are conveyed or processed by a computer per unit of time
*
Symbol rate
In a digitally modulated signal or a line code, symbol rate, modulation rate or baud rate is the number of symbol changes, waveform changes, or signaling events across the transmission medium per unit of time. The symbol rate is measured in ''b ...
, the number of symbol changes (signalling events) made to the transmission medium per second
*
Sampling rate
In signal processing, sampling is the reduction of a continuous-time signal to a discrete-time signal. A common example is the conversion of a sound wave to a sequence of "samples".
A sample is a value of the signal at a point in time and/or spac ...
, the number of samples (signal measurements) per second
Miscellaneous definitions:
*
Rate of reinforcement In behaviorism, rate of reinforcement is number of reinforcements per time, usually per minute. Symbol of this rate is usually ''Rf''. Its first major exponent was B.F. Skinner (1939). It is used in the Matching Law.
''Rf'' = ''# of reinforcements ...
, number of reinforcements per unit of time, usually per minute
*
Heart rate
Heart rate (or pulse rate) is the frequency of the heartbeat measured by the number of contractions (beats) of the heart per minute (bpm). The heart rate can vary according to the body's physical needs, including the need to absorb oxygen and excr ...
, usually measured in beats per minute
Economics/finance rates/ratios
*
Exchange rate
In finance, an exchange rate is the rate at which one currency will be exchanged for another currency. Currencies are most commonly national currencies, but may be sub-national as in the case of Hong Kong or supra-national as in the case of ...
, how much one currency is worth in terms of the other
*
Inflation rate
In economics, inflation is an increase in the general price level of goods and services in an economy. When the general price level rises, each unit of currency buys fewer goods and services; consequently, inflation corresponds to a reductio ...
, ratio of the change in the general price level during a year to the starting price level
*
Interest rate
An interest rate is the amount of interest due per period, as a proportion of the amount lent, deposited, or borrowed (called the principal sum). The total interest on an amount lent or borrowed depends on the principal sum, the interest rate, th ...
, the price a borrower pays for the use of money they do not own (ratio of payment to amount borrowed)
*
Price–earnings ratio
The price-earnings ratio, also known as P/E ratio, P/E, or PER, is the ratio of a company's share (stock) price to the company's earnings per share. The ratio is used for valuing companies and to find out whether they are overvalued or under ...
, market price per share of stock divided by annual earnings per share
*
Rate of return
In finance, return is a profit on an investment. It comprises any change in value of the investment, and/or cash flows (or securities, or other investments) which the investor receives from that investment, such as interest payments, coupons, ca ...
, the ratio of money gained or lost on an investment relative to the amount of money invested
*
Tax rate
In a tax system, the tax rate is the ratio (usually expressed as a percentage) at which a business or person is taxed. There are several methods used to present a tax rate: statutory, average, marginal, and effective. These rates can also be p ...
, the tax amount divided by the taxable income
*
Unemployment rate
Unemployment, according to the OECD (Organisation for Economic Co-operation and Development), is people above a specified age (usually 15) not being in paid employment or self-employment but currently available for work during the referen ...
, the ratio of the number of people who are unemployed to the number in the labor force
*
Wage rate
A wage is payment made by an employer to an employee for work done in a specific period of time. Some examples of wage payments include compensatory payments such as '' minimum wage'', ''prevailing wage'', and ''yearly bonuses,'' and remun ...
, 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
The birth rate for a given period is the total number of live human births per 1,000 population divided by the length of the period in years. The number of live births is normally taken from a universal registration system for births; populati ...
, and
mortality rate
Mortality rate, or death rate, is a measure of the number of deaths (in general, or due to a specific cause) in a particular population, scaled to the size of that population, per unit of time. Mortality rate is typically expressed in units of de ...
, the number of births or deaths scaled to the size of that population, per unit of time
*
Literacy rate
Literacy in its broadest sense describes "particular ways of thinking about and doing reading and writing" with the purpose of understanding or expressing thoughts or ideas in written form in some specific context of use. In other words, huma ...
, the proportion of the population over age fifteen that can read and write
*
Sex ratio
The sex ratio (or gender ratio) is usually defined as the ratio of males to females in a population. As explained by Fisher's principle, for evolutionary reasons this is typically about 1:1 in species which reproduce sexually. Many species devia ...
or gender ratio, the ratio of males to females in a population
See also
*
Derivative
In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. F ...
*
Gradient
In vector calculus, the gradient of a scalar-valued differentiable function of several variables is the vector field (or vector-valued function) \nabla f whose value at a point p is the "direction and rate of fastest increase". If the gradi ...
*
Hertz
The hertz (symbol: Hz) is the unit of frequency in the International System of Units (SI), equivalent to one event (or cycle) per second. The hertz is an SI derived unit whose expression in terms of SI base units is s−1, meaning that on ...
*
Slope
In mathematics, the slope or gradient of a line is a number that describes both the ''direction'' and the ''steepness'' of the line. Slope is often denoted by the letter ''m''; there is no clear answer to the question why the letter ''m'' is use ...
References
{{DEFAULTSORT:Rate (Mathematics)
Measurement
de:Rate