ⓘ Rate of change, mathematics. In mathematics, a rate is the ratio between two related quantities in different units. If the denominator of the ratio is expressed ..


ⓘ Rate of change (mathematics)

In mathematics, a rate is the ratio between two related quantities in different units. 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, then the numerator of the ratio expresses the corresponding rate of change in the other 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.


1. 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 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. 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.


2. Rate of change

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

Average rate of change = f a + h − f a h Instantaneous rate of change = lim h → 0 f a + h − f a h {\displaystyle {\begin{aligned}{\mbox{Average rate of change}}&={\frac {fa+h-fa}{h}}\\{\mbox{Instantaneous rate of change}}&=\lim _{h\to 0}{\frac {fa+h-fa}{h}}\end{aligned}}}

where fx 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:

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

However these two formulas do not directly apply where either the range or the domain of f {\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.


3. Temporal rates

In chemistry and physics:

  • Acceleration, the rate of change in speed, or the change in speed per unit of time
  • Reaction rate, the speed at which chemical reactions occur
  • Speed, the rate of change of position, or the change of position per unit of time
  • Power, the rate of doing work, or the amount of energy transferred per unit time
  • Volumetric flow rate, the volume of fluid which passes through a given surface per unit of time; e.g., cubic meters per second
  • Frequency, the number of occurrences of a repeating event per unit of time

3.1. Temporal rates Counts-per-time rates

  • Radioactive decay, the amount of radioactive material in which one nucleus decays per second, measured in Becquerels

In computing:

  • Symbol rate, the number of symbol changes signalling events made to the transmission medium per second
  • Bit rate, the number of bits that are conveyed or processed by a computer per unit of time
  • Sampling rate, the number of samples signal measurements per second

Miscellaneous definitions:

  • Heart rate, usually measured in beats per minute
  • Rate of reinforcement, number of reinforcements per unit of time, usually per minute

4. Economics/finance rates/ratios

  • Interest rate, the price a borrower pays for the use of money they do not own ratio of payment to amount borrowed
  • Rate of return, the ratio of money gained or lost on an investment relative to the amount of money invested
  • 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
  • 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
  • Unemployment rate, the ratio of the number of people who are unemployed to the number in the labor force
  • Price–earnings ratio, market price per share of stock divided by annual earnings per share
  • Tax rate, the tax amount divided by the taxable income

5. Other rates

  • Sex ratio or Gender ratio, the ratio of males to females in a population
  • 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

  • Rate of change may refer to: Rate of change mathematics either average rate of change or instantaneous rate of change Instantaneous rate of change
  • quantity number theory structure algebra space geometry and change mathematical analysis It has no generally accepted definition. Mathematicians
  • another Rate mathematics a specific kind of ratio, in which two measurements are related to each other often with respect to time Rate particle
  • Lists of mathematics topics cover a variety of topics related to mathematics Some of these lists link to hundreds of articles some link only to a few
  • purposes they serve. In addition, as mathematics continues to be developed, these classification schemes must change as well to account for newly created
  • Mathematical finance, also known as quantitative finance and financial mathematics is a field of applied mathematics concerned with mathematical modeling
  • Mathematical economics is the application of mathematical methods to represent theories and analyze problems in economics. By convention, these applied
  • welfare impacts of government policies, social discount rate the basic mathematics are the same as discounted cash flow, but the cash value of human lives
  • The reaction rate or rate of reaction is the speed at which reactants are converted into products. For example, the oxidative rusting of iron under Earth s
  • Heart rate is the speed of the heartbeat measured by the number of contractions beats of the heart per minute bpm The heart rate can vary according