Rate (mathematics)
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In
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
Rates that have a non-time divisor or denominator include
A rate defined using two numbers of the same units will result in a
Properties and examples
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,
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
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
In chemistry and physics:
- position, or the change of position per unit of time
- Acceleration, the rate of change in speed, or the change in speed per unit of time
- energytransferred per unit time
- Frequency, the number of occurrences of a repeating event per unit of time
- rotation speed, the number of turnsper unit of time
- Reaction rate, the speed at which chemical reactions occur
- 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 (signaling 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, the 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 the 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
See also
References
- ^ See Webster's New International Dictionary of the English Language, 2nd edition, Unabridged. Merriam Webster Co. 2016. p.2065 definition 3.
- ^ "IEC 60050 - Details for IEV number 112-03-18: "rate"". International Electrotechnical Vocabulary (in Japanese). Retrieved 2023-09-13.
- ISBN 0-201-82823-5.