rate

Rate is a quantitative measure that expresses how one quantity changes in relation to another, typically per unit of time or per another base unit. A rate is a ratio with a meaningful reference quantity, so it often combines two magnitudes into a single comparative measure. Some rates are dimensionless when the two quantities match or are normalized, yielding percentages.

In mathematics, rate of change describes how a quantity varies with another. The average rate of change between two points is (ΔQ/Δt). The instantaneous rate is the derivative dQ/dt, which corresponds to the slope of the graph of Q versus t.

Common examples include speed = distance/time; flow rate = volume/time; chemical reaction rate = change in concentration/time; financial rates such as interest rate (percent per year), exchange rate (units of one currency per unit of another), tax rate (percent). Growth rate indicates percentage change per period; mortality rate is deaths per population per time.

Rates require units, and unit analysis is essential to interpret meaning. Rates can be per unit time, per unit distance, or per unit of any base quantity. They may be expressed as fractions, decimals, or percentages; rate normalization allows comparison across different scales.

Rates are used to model movement, consumption, economic performance, and risk. Understanding rate concepts helps compare performance, forecast trends, and convert between quantities.