Rate (mathematics)

In mathematics, a rate is the quotient of two quantities in different units of measurement, often represented as a fraction. 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. One common type of rate is "per unit of time", such as speed, heart rate, and flux. In fact, often rate is a synonym of rhythm or frequency, a count per second (i.e., hertz); e.g., radio frequencies, heart rates, 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. 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.