Rate (mathematics)

Last updated November 13, 2023

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

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 $f$ 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:^[3]

{\begin{aligned}{\mbox{Average rate of change}}&={\frac {f(x+h)-f(x)}{h}}\\{\mbox{Instantaneous rate of change}}&=\lim _{h\to 0}{\frac {f(x+h)-f(x)}{h}}\end{aligned}}

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:

Speed, the rate of change of 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
Power, the rate of doing work, or the amount of energy transferred per unit time
Frequency, the number of occurrences of a repeating event per unit of time
- Angular frequency 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

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

Related Research Articles

Conversion of units is the conversion between different units of measurement for the same quantity, typically through multiplicative conversion factors which change the measured quantity value without changing its effects. Unit conversion is often easier within the metric or the SI than in others, due to the regular 10-base in all units and the prefixes that increase or decrease by 3 powers of 10 at a time.

<span class="mw-page-title-main">Frequency</span> Number of occurrences or cycles per unit time

Frequency, measured in hertz, 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 to distinguish it from spatial frequency. Ordinary frequency is related to angular frequency by a factor of 2 $π$ . The period is the interval of time between events, so the period is the reciprocal of the frequency: f = 1/T.

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.

In chemistry, the mole fraction or molar fraction, also called mole proportion or molar proportion, is a quantity defined as the ratio between the amount of a constituent substance, n_i, and the total amount of all constituents in a mixture, n_tot :

In physics, power is the amount of energy transferred or converted per unit time. In the International System of Units, the unit of power is the watt, equal to one joule per second. In older works, power is sometimes called activity. Power is a scalar quantity.

In everyday use and in kinematics, the speed 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 quantity. The average speed of an object in an interval of time is the distance travelled by the object divided by the duration of the interval; the instantaneous speed is the limit of the average speed as the duration of the time interval approaches zero. Speed is the magnitude of velocity, which indicates additionally the direction of motion.

In electronics, an analog-to-digital converter is a system that converts an analog signal, such as a sound picked up by a microphone or light entering a digital camera, into a digital signal. An ADC may also provide an isolated measurement such as an electronic device that converts an analog input voltage or current to a digital number representing the magnitude of the voltage or current. Typically the digital output is a two's complement binary number that is proportional to the input, but there are other possibilities.

In mathematics, a continued fraction is an expression obtained through an iterative process of representing a number as the sum of its integer part and the reciprocal of another number, then writing this other number as the sum of its integer part and another reciprocal, and so on. In a finite continued fraction, the iteration/recursion is terminated after finitely many steps by using an integer in lieu of another continued fraction. In contrast, an infinite continued fraction is an infinite expression. In either case, all integers in the sequence, other than the first, must be positive. The integers $are called the coefficients or terms of the continued fraction.$

$<span class="mw-page-title-main">Percentage</span> Number or ratio expressed as a fraction of 100$

In mathematics, a percentage 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 used. A percentage is a dimensionless number, primarily used for expressing proportions, but percent is nonetheless a unit of measurement in its orthography and usage.

<span class="mw-page-title-main">Ratio</span> Relationship between two numbers of the same kind

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. Similarly, the ratio of lemons to oranges is 6:8 and the ratio of oranges to the total amount of fruit is 8:14.

In probability theory, odds provide a measure of the likelihood of a particular outcome. They are calculated as the ratio of the number of events that produce that outcome to the number that do not. Odds are commonly used in gambling and statistics.

<span class="mw-page-title-main">Angular frequency</span> Rate of change of angle

In physics, angular frequency, also called angular speed and angular rate, is a scalar measure of the angle rate or the temporal rate of change of the phase argument of a sinusoidal waveform or sine function . Angular frequency is the magnitude of the pseudovector quantity angular velocity.

In mathematical finance, the Greeks are the quantities representing the sensitivity of the price of a derivative instrument such as an option to changes in one or more underlying parameters on which the value of an instrument or portfolio of financial instruments is dependent. The name is used because the most common of these sensitivities are denoted by Greek letters. Collectively these have also been called the risk sensitivities, risk measures or hedge parameters.

<span class="mw-page-title-main">Quotient</span> Mathematical result of division

In arithmetic, a quotient is a quantity produced by the division of two numbers. The quotient has widespread use throughout mathematics. It has two definitions: either the integer part of a division, or as a fraction or a ratio. For example, when dividing 20 by 3, the quotient is 6 in the first sense, and $in the second sense.$

In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be rational numbers; they may be taken in any field K. In this case, one speaks of a rational function and a rational fraction over K. The values of the variables may be taken in any field L containing K. Then the domain of the function is the set of the values of the variables for which the denominator is not zero, and the codomain is L.

$<span class="mw-page-title-main">Unit fraction</span> One over a whole number$

A unit fraction is a positive fraction with one as its numerator, 1/ $n$ . It is the multiplicative inverse (reciprocal) of the denominator of the fraction, which must be a positive natural number. Examples are 1/1, 1/2, 1/3, 1/4, 1/5, etc. When an object is divided into equal parts, each part is a unit fraction of the whole.

In finance, the Sharpe ratio measures the performance of an investment such as a security or portfolio compared to a risk-free asset, after adjusting for its risk. It is defined as the difference between the returns of the investment and the risk-free return, divided by the standard deviation of the investment returns. It represents the additional amount of return that an investor receives per unit of increase in risk.

A fraction 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-fifths, three-quarters. A common, vulgar, or simple fraction consists of an integer numerator, displayed above a line, and a non-zero integer denominator, displayed below that line. If these integers are positive, then the numerator represents a number of equal parts, and the denominator indicates how many of those parts make up a unit or a whole. For example, in the fraction 3/4, the numerator 3 indicates that the fraction represents 3 equal parts, and the denominator 4 indicates that 4 parts make up a whole. The picture to the right illustrates 3/4 of a cake.

In mathematics, a quadratic equation is a polynomial equation of the second degree. The general form is

In chemistry, the mass fraction of a substance within a mixture is the ratio $of the mass of that substance to the total mass of the mixture. Expressed as a formula, the mass fraction is:$

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.
↑ Adams, Robert A. (1995). Calculus: A Complete Course (3rd ed.). Addison-Wesley Publishers Ltd. p. 129. ISBN 0-201-82823-5.

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[1] See Webster's New International Dictionary of the English Language, 2nd edition, Unabridged. Merriam Webster Co. 2016. p.2065 definition 3.

[International_Electrotechnical_Vocabulary_x558-2] "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.

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