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| In [[mathematics]], a '''rate''' is a [[ratio]] between two [[measurement]]s with different units.<ref>{{cite web |url=http://www.mathpropress.com/glossary/glossary.html#R |title=On-line Mathematics Dictionary |publisher=MathPro Press |date=January 14, 2006 |accessdate=2009-03-01}}</ref> If the unit or quantity in respect of which something is changing is not specified, usually the rate is ''per unit time''. However, a rate of change can be specified per unit [[time]], or per unit of [[length]] or [[mass]] or another quantity. The most common type of rate is "per unit time", such as [[speed]], [[heart rate]] and [[flux]]. Ratios that have a non-time denominator include [[exchange rate]]s, [[literacy rate]]s and [[electric flux]].
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| 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 of measurement|units]] (such as [[tax rate]]s) 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 (mathematics)|fraction]] or as a [[multiple (mathematics)|multiple]].
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| 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]].
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| ==Rate of change==
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| {{main|Derivative}}
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| A rate of change can be formally defined in two ways:<ref>{{cite book|last1=Adams |first1=Robert A. |title=Calculus: A Complete Course|edition=3rd |year=1995 |publisher=Addison-Wesley Publishers Ltd. |isbn=0-201-82823-5 |page=129}}</ref>
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| :<math>\begin{align}
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| \mbox{Average rate of change} &= \frac{f(a + h) - f(a)}{h}\\
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| \mbox{Instantaneous rate of change} &= \lim_{h \to 0}\frac{f(a + h) - f(a)}{h}
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| \end{align}</math>
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| 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]].
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| An example to contrast the differences between the ''average'' and ''instantaneous'' definitions: the [[speed]] of a car can be calculated:
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| #An average rate can be calculated using the total distance travelled between ''a'' and ''b'', divided by the travel time
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| #An instantaneous rate can be determined by viewing a [[speedometer]].
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| ==Terms based on rates==
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| In chemistry and physics:
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| *[[Speed]], being the distance covered per unit time; e.g., miles per hour and [[meters per second]]
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| :*[[Acceleration]], the rate of change in speed, or the change in speed per unit time | |
| *[[Radioactive decay]], the amount of radioactive material in which one nucleus decays per second, measured in [[Becquerel]]s
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| *[[Reaction rate]], the speed at which chemical reactions occur
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| *[[Volumetric flow rate]], the volume of fluid which passes through a given surface per unit time; e.g., [[cubic meters per second]]
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| In computing:
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| *[[Bit rate]], the number of bits that are conveyed or processed by a computer per unit of time
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| *[[Symbol rate]], the number of symbol changes (signalling events) made to the transmission medium per second
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| *[[Sampling rate]], the number of samples (signal measurements) per second
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| In finance:
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| *[[Interest rate]], the price a borrower pays for the use of money they do not own, usually expressed as a percentage rate over the period of one year; see also for related rates
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| *[[Exchange rate]], how much one currency is worth in terms of the other
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| *[[Inflation rate]], a measure of inflation change per year
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| *[[Rate of return]], the ratio of money gained or lost on an investment relative to the amount of money invested
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| *[[Tax rate]], the tax amount divided by the taxable income
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| Miscellaneous definitions:
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| *[[Rate of reinforcement]], number of reinforcements per time, usually per minute
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| *[[Heart rate]], usually measured in beats per minute
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| *[[Unemployment rate]], a ratio between those in the labor force to those who are unemployed
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| *[[Birth rate]] and [[mortality rate]], the number of births or deaths scaled to the size of that population, per unit time
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| *[[Literacy rate]], the proportion of the population over age fifteen that can read and write
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| ==References==
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| {{Reflist}}
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| ==See also==
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| *[[Derivative]]
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| *[[Flux]]
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| *[[Gradient]]
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| *[[Slope]]
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| {{DEFAULTSORT:Rate (Mathematics)}}
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| [[Category:Interest rates]]
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| [[Category:Measurement]]
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| [[de:Rate]]
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