Rate (mathematics)

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In mathematics, a rate is the ratio between two related quantities[1] Often it is a rate of change. If the unit or quantity in respect of which something is changing is not specified, usually the rate is per unit of time. However, a rate of change can be specified per unit of time, or per unit of length or mass or another quantity. 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).




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 traveled per unit time) of a transportation vehicle on a certain trip may be may be represented as a function of x (the distance traveled from the start of the trip) as v(x). Alternatively, one could express velocity as a function of time t from the start of the trip as v(t). Another representation of velocity on a trip is to partition the trip route into N segments and let vi be the constant velocity 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 a performance indicator, 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 ratio r=a/b has both a numerator a and a denominator b. a and/orb may be a real number or integer. The inverse of a ratio r is 1/r = b/a.

Continuous 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:[2]

\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}


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

  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 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, a measure of inflation change per year
  • Interest rate, the price a borrower pays for the use of money they do not own
  • 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, a ratio between those in the labor force to those who are unemployed
  • Wage Rate, the amount paid for working a given amount of time (or doing a standard amount of accomplished work)

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


  1. See Webster's new international dictionary of the English language, second edition, unabridged. Merriam Webster Co. 1952. 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 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.<templatestyles src="Module:Citation/CS1/styles.css"></templatestyles>