# Rate (mathematics)

In mathematics, a rate is the ratio between two related quantities in different units.[1] If the denominator of the ratio is expressed as a single unit of one of these quantities, and if it is assumed that this quantity can be changed systematically (i.e., is an independent variable), then the numerator of the ratio expresses the corresponding rate of change in the other (dependent) variable.

The most common type of rate is "per unit of time", such as speed, heart rate and flux. Ratios that have a non-time denominator include exchange rates, literacy rates and electric field (in volts/meter).

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 (such as tax rates) 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 or as a multiple.

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.

## Introduction

Rates and ratios often vary with time, location, particular element (or subset) of a set of objects, etc. Thus they are often mathematical functions. For example, velocity v (distance tracity on segment i (v is a function of index i). Here each segment i, of the trip is a subset of the trip route.

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 i 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 so as 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 vi'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. a and/or b may be a real number or integer. The inverse of a ratio r is 1/r = b/a.

Rates occur in many areas of real life. For example: How fast are you driving? Miles per hour is a rate. What interest does your savings account pay you? Interest paid / year is a rate.

## Rate of change

Consider the case where the numerator ${\displaystyle f}$ of a rate is a function ${\displaystyle f(a)}$ where ${\displaystyle a}$ happens to be the denominator of the rate ${\displaystyle \delta f/\delta a}$. A rate of change of ${\displaystyle f}$ with respect to ${\displaystyle a}$ (where ${\displaystyle a}$ is incremented by ${\displaystyle h}$) can be formally defined in two ways:[2]

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

An example to contrast the differences between the unit rates are average and instantaneous definitions: the speed of a car can be calculated:

1. An average rate can be calculated using the total distance travelled between a and b, divided by the travel time
2. An instantaneous rate can be determined by viewing a speedometer.

However these two formulas do not directly apply where either the range or the domain of ${\displaystyle f()}$ is a set of integers or where there is no given formula (function) for finding the numerator of the ratio from its denominator.

## Temporal rates

In chemistry and physics:

### Counts-per-time rates

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 (signalling events) made to the transmission medium per second
• Sampling rate, the number of samples (signal measurements) per second

Miscellaneous definitions:

## Economics/finance rates/ratios

• Exchange rate, how much one currency is worth in terms of the other
• Inflation rate, 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 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

## References

1. ^ See Webster's new international dictionary of the English language, second edition, unabridged. Merriam Webster Co. 2016. p.2065 definition 3. while this definition doesn't say "related" and while the ratio of two non-related quantities is technically a ratio, such a ratio has little (if any meaning). For example, what would be the utility of finding the ratio of such unrelated numbers as ratio of the weight of ones residence to an integer selected at random between -10−9 and +109?
2. ^ Adams, Robert A. (1995). Calculus: A Complete Course (3rd ed.). Addison-Wesley Publishers Ltd. p. 129. ISBN 0-201-82823-5.
Derivative

The derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. For example, the derivative of the position of a moving object with respect to time is the object's velocity: this measures how quickly the position of the object changes when time advances.

The derivative of a function of a single variable at a chosen input value, when it exists, is the slope of the tangent line to the graph of the function at that point. The tangent line is the best linear approximation of the function near that input value. For this reason, the derivative is often described as the "instantaneous rate of change", the ratio of the instantaneous change in the dependent variable to that of the independent variable.

Derivatives may be generalized to functions of several real variables. In this generalization, the derivative is reinterpreted as a linear transformation whose graph is (after an appropriate translation) the best linear approximation to the graph of the original function. The Jacobian matrix is the matrix that represents this linear transformation with respect to the basis given by the choice of independent and dependent variables. It can be calculated in terms of the partial derivatives with respect to the independent variables. For a real-valued function of several variables, the Jacobian matrix reduces to the gradient vector.

The process of finding a derivative is called differentiation. The reverse process is called antidifferentiation. The fundamental theorem of calculus relates antidifferentiation with integration. Differentiation and integration constitute the two fundamental operations in single-variable calculus.

Harmonic mean

In mathematics, the harmonic mean (sometimes called the subcontrary mean) is one of several kinds of average, and in particular one of the Pythagorean means. Typically, it is appropriate for situations when the average of rates is desired.

The harmonic mean can be expressed as the reciprocal of the arithmetic mean of the reciprocals of the given set of observations. As a simple example, the harmonic mean of 1, 4, and 4 is

${\displaystyle \left({\frac {1^{-1}+4^{-1}+4^{-1}}{3}}\right)^{-1}={\frac {3}{{\frac {1}{1}}+{\frac {1}{4}}+{\frac {1}{4}}}}={\frac {3}{1.5}}=2\,.}$
Hua Luogeng

Hua Luogeng or Hua Loo-Keng (Chinese: 华罗庚; Wade–Giles: Hua Lo-keng; 12 November 1910 – 12 June 1985) was a Chinese mathematician and politician famous for his important contributions to number theory and for his role as the leader of mathematics research and education in the People's Republic of China. He was largely responsible for identifying and nurturing the renowned mathematician Chen Jingrun who proved Chen's theorem, the best known result on the Goldbach conjecture. In addition, Hua's later work on mathematical optimization and operations research made an enormous impact on China's economy. He was elected a foreign associate of the US National Academy of Sciences in 1982. He was elected a member of the standing Committee of the first to sixth National people's Congress, Vice-Chairman of the sixth National Committee of the Chinese People's Political Consultative Conference (April 1985) and Vice-Chairman of the China Democratic League (1979). He joined the Communist Party of China in 1979.

Hua did not receive a formal university education. Although awarded several honorary PhDs, he never got a formal degree from any university. In fact, his formal education only consisted of six years of primary school and three years of middle school. For that reason, Xiong Qinglai, after reading one of Hua's early papers, was amazed by Hua's mathematical talent, and in 1931 Xiong invited him to study mathematics at Tsinghua University.

Programme for International Student Assessment (2000 to 2012)

The Programme for International Student Assessment has had several runs before the most recent one in 2012. The first PISA assessment was carried out in 2000. The results of each period of assessment take about one year and a half to be analysed. First results were published in November 2001. The release of raw data and the publication of technical report and data handbook only took place in spring 2002. The triennial repeats follow a similar schedule; the process of seeing through a single PISA cycle, start-to-finish, always takes over four years. 470,000 15-year-old students representing 65 nations and territories participated in PISA 2009. An additional 50,000 students representing nine nations were tested in 2010.Every period of assessment focuses on one of the three competence fields of reading, math, science; but the two others are tested as well. After nine years, a full cycle is completed: after 2000, reading was again the main domain in 2009.

Ratio

In mathematics, a ratio is a relationship between two numbers indicating how many times the first number contains the second. For example, if a bowl of fruit contains eight oranges and six lemons, then the ratio of oranges to lemons is eight to six (that is, 8:6, which is equivalent to the ratio 4:3). Similarly, the ratio of lemons to oranges is 6:8 (or 3:4) and the ratio of oranges to the total amount of fruit is 8:14 (or 4:7).

The numbers in a ratio may be quantities of any kind, such as counts of persons or objects, or such as measurements of lengths, weights, time, etc. In most contexts both numbers are restricted to be positive.

A ratio may be specified either by giving both constituting numbers, written as "a to b" or "a:b", or by giving just the value of their quotient a/b, since the product of the quotient and the second number yields the first, as required by the above definition.

Consequently, a ratio may be considered as an ordered pair of numbers, as a fraction with the first number in the numerator and the second as denominator, or as the value denoted by this fraction. Ratios of counts, given by (non-zero) natural numbers, are rational numbers, and may sometimes be natural numbers. When two quantities are measured with the same unit, as is often the case, their ratio is a dimensionless number. A quotient of two quantities that are measured with different units is called a rate.

Time

Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past, through the present, to the future. Time is a component quantity of various measurements used to sequence events, to compare the duration of events or the intervals between them, and to quantify rates of change of quantities in material reality or in the conscious experience. Time is often referred to as a fourth dimension, along with three spatial dimensions.Time has long been an important subject of study in religion, philosophy, and science, but defining it in a manner applicable to all fields without circularity has consistently eluded scholars.

Nevertheless, diverse fields such as business, industry, sports, the sciences, and the performing arts all incorporate some notion of time into their respective measuring systems.Time in physics is unambiguously operationally defined as "what a clock reads". See Units of Time. Time is one of the seven fundamental physical quantities in both the International System of Units and International System of Quantities. Time is used to define other quantities – such as velocity – so defining time in terms of such quantities would result in circularity of definition. An operational definition of time, wherein one says that observing a certain number of repetitions of one or another standard cyclical event (such as the passage of a free-swinging pendulum) constitutes one standard unit such as the second, is highly useful in the conduct of both advanced experiments and everyday affairs of life. The operational definition leaves aside the question whether there is something called time, apart from the counting activity just mentioned, that flows and that can be measured. Investigations of a single continuum called spacetime bring questions about space into questions about time, questions that have their roots in the works of early students of natural philosophy.

Temporal measurement has occupied scientists and technologists, and was a prime motivation in navigation and astronomy. Periodic events and periodic motion have long served as standards for units of time. Examples include the apparent motion of the sun across the sky, the phases of the moon, the swing of a pendulum, and the beat of a heart. Currently, the international unit of time, the second, is defined by measuring the electronic transition frequency of caesium atoms (see below). Time is also of significant social importance, having economic value ("time is money") as well as personal value, due to an awareness of the limited time in each day and in human life spans.

University Laboratory High School (Urbana, Illinois)

The University of Illinois Laboratory High School, also known as Uni, was established in 1921 and is a laboratory school located on the engineering section of the University of Illinois campus in Urbana, Illinois. Its enrollment is approximately 300 students, spanning five years (the traditional grades 9–12, preceded by an 8th grade year known as the "subfreshman" year). The school is notable for the achievements of its alumni, including three Nobel laureates, and a Pulitzer Prize winner. In 2006 and 2008 it was recognized as a "public elite" school by Newsweek because of its students' high scores on the SAT. Before the recent change in the SAT's format, the average SAT score was 2045, and now varies from 1400 to 1600. The average ACT score is a 32.

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