Rate (mathematics)

Last updated October 24, 2024

In mathematics, a rate is the quotient of two quantities, 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 velocity with 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 of the unit of measurement in which a quantity is expressed, typically through a multiplicative conversion factor that changes the unit without changing the quantity. This is also often loosely taken to include replacement of a quantity with a corresponding quantity that describes the same physical property.

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

Frequency, most often 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: T = 1/f.

In mathematics, the harmonic mean is a kind of average, one of the Pythagorean means.

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 :

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. Power is a scalar quantity.

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 non-negative 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 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 stopped 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 $are called the coefficients or terms of the continued fraction.$

Dimensionless quantities, or quantities of dimension one, are quantities implicitly defined in a manner that prevents their aggregation into units of measurement. Typically expressed as ratios that align with another system, these quantities do not necessitate explicitly defined units. For instance, alcohol by volume (ABV) represents a volumetric ratio; its value remains independent of the specific units of volume used, such as in milliliters per milliliter (mL/mL).

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

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">Angular velocity</span> Direction and rate of rotation

In physics, angular velocity, also known as angular frequency vector, is a pseudovector representation of how the angular position or orientation of an object changes with time, i.e. how quickly an object rotates around an axis of rotation and how fast the axis itself changes direction.

<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.

Delta-v, symbolized as $and pronounced deltah-vee, as used in spacecraft flight dynamics, is a measure of the impulse per unit of spacecraft mass that is needed to perform a maneuver such as launching from or landing on a planet or moon, or an in-space orbital maneuver. It is a scalar that has the units of speed. As used in this context, it is not the same as the physical change in velocity of said spacecraft.$

<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.

<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 a fraction or ratio. For example, when dividing 20 by 3, the quotient is 6 in the first sense and $in the second sense.$

<span class="mw-page-title-main">Rotational frequency</span> Number of rotations per unit time

Rotational frequency, also known as rotational speed or rate of rotation, is the frequency of rotation of an object around an axis. Its SI unit is the reciprocal seconds (s⁻¹); other common units of measurement include the hertz (Hz), cycles per second (cps), and revolutions per minute (rpm).

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.

<span class="mw-page-title-main">Velocity of money</span> Rate of money changing hands

The velocity of money measures the number of times that one unit of currency is used to purchase goods and services within a given time period. In other words, it's how many times money is changing hands. The concept relates the size of economic activity to a given money supply, and the speed of money exchange is one of the variables that determine inflation. The measure of the velocity of money is usually the ratio of the gross national product (GNP) to a country's money supply.

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.

Linear motion, also called rectilinear motion, is one-dimensional motion along a straight line, and can therefore be described mathematically using only one spatial dimension. The linear motion can be of two types: uniform linear motion, with constant velocity ; and non-uniform linear motion, with variable velocity. The motion of a particle along a line can be described by its position $, which varies with (time). An example of linear motion is an athlete running a 100-meter dash along a straight track.$

Velocity is the speed in combination with the direction of motion of an object. Velocity is a fundamental concept in kinematics, the branch of classical mechanics that describes the motion of bodies.

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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