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. One 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 describing the units of a rate, the word "per" is used to separate the units of the two measurements used to calculate the rate. A rate defined using two numbers of the same units or counts will result in a dimensionless quantity, which can be expressed as a percentage, fraction, or multiple. Often rate is a synonym of rhythm or frequency, a count per second ; e.g., radio frequencies, heart rates, or sample rates.
Introduction
Rates and ratios often vary with time, location, particular element of a set of objects, etc. Thus they are often mathematical functions. A rate 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 in terms of miles of travel, which one gets from spending an hour of traveling. A set of sequential indices i may be used to enumerate elements of a set of ratios under study. For example, in finance, one could define i by assigning consecutive integers to companies, to political subdivisions, to different investments, etc. The reason for using indices i, is so a set of ratios 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". The value of a and/or 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 are also inverse. For example, 5 miles/kWh is the same as 1/5 kWh/mile. 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 of a rate is a function where happens to be the denominator of the rate. A rate of change of with respect to can be formally defined in two ways: where f 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 travelled between two points, divided by the travel time, while the instantaneous speed can be determined by viewing a speedometer.
