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In 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), 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 Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
,
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. Ratios that have a non-time denominator include exchange rates,
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 (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 rates) 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, which can be expressed as a percentage (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, 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 re ...
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 o ...
); 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 spa ...
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 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". 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 ...
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 languag ...
. 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 miles (mi) per kilowatt-hour (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 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: : \begin \mbox &= \frac\\ \mbox &= \lim_\frac \end 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. ...
. For example, the average
speed In everyday use and in kinematics, the speed (commonly referred to as ''v'') of an object is the magnitude Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
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.


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 Magnitude may refer to: Mathematics *Euclidean vector, a quantity defined by both its magnitude and its direction *Magnitude (ma ...
, 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 t ...
, 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 t ...
, 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 hea ...
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, the number of turns 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 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, 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, 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 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 s ...
, 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, 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, ...
, 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 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, cas ...
, 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 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 refere ...
, 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 d ...
, 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 d ...
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. ...
*
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 gr ...
*
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 o ...
*
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