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In mathematics, a rate is the quotient of two quantities in different units of measurement, often represented as a fraction.[1] If the divisor (or fraction denominator) in the rate is equal to one expressed as a single unit, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the dividend (the fraction numerator) of the rate expresses the corresponding rate of change in the other (dependent) variable. In some cases, it may be regarded as a change to a value, which is caused by a change of a value in respect to another value. For example, acceleration is a change in speed in respect to time

Temporal rate is a common type of rate ("per unit of time"), such as speed, heart rate, and flux.[2] In fact, often rate is a synonym of rhythm or frequency, a count per second (i.e., hertz); e.g., radio frequencies or sample rates. 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 as "beats per minute".

Rates that have a non-time divisor or denominator include exchange rates, literacy rates, and electric field (in volts per meter).

A rate defined using two numbers of the same units will result in a dimensionless quantity, also known as ratio or simply as a rate (such as tax rates) or counts (such as literacy rate). Dimensionless rates can be expressed as a percentage (for example, the global literacy rate in 1998 was 80%), fraction, or multiple.

Properties and examples

Further information: Ratio

Rates and ratios often vary with time, location, particular element (or subset) of a set of objects, etc. Thus they are often mathematical functions.

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 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 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". The value of a and b may be a real number or integer. 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 is 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 the 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:[3]

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.

For example, the average speed of a car can be calculated using the total distance traveled between two points, divided by the travel time. In contrast, the instantaneous velocity can be determined by viewing a speedometer.

Temporal rates

Main category: Temporal rates

See also: Time derivative

In chemistry and physics:

Counts-per-time rates

Main articles: Frequency and Aperiodic frequency

In computing:

Miscellaneous definitions:

Economics/finance rates/ratios

Other rates

See also


  1. ^ See Webster's New International Dictionary of the English Language, 2nd edition, Unabridged. Merriam Webster Co. 2016. p.2065 definition 3.
  2. ^ "IEC 60050 - Details for IEV number 112-03-18: "rate"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-13.
  3. ^ Adams, Robert A. (1995). Calculus: A Complete Course (3rd ed.). Addison-Wesley Publishers Ltd. p. 129. ISBN 0-201-82823-5.