Rate
A rate is a ratio that compares two quantities with different units, such as miles per hour or dollars per pound.
Formula
\text{rate} = \frac{\text{quantity A}}{\text{quantity B}} \text{ (different units)}
Definition
A rate is a special kind of ratio that compares two things with different units: speed (miles per hour), price (dollars per pound), and heart rate (beats per minute) are all rates, and the word "per" always signals one. Formally, a rate is a ratio $a/b$ where $a$ and $b$ carry different units, describing how one quantity changes per unit of another; a unit rate has a denominator of $1$ (like $65$ miles per $1$ hour), and rates can be simplified to unit rates for easier comparison. In physics, rates appear as derivatives (speed $= ds/dt$, acceleration $= dv/dt$, flow rate $= dV/dt$), and dimensional analysis uses the algebra of rates, treating conversion factors as rates equal to $1$, to verify formulas and convert units.
Example
A car traveling $150$ miles in $3$ hours has a rate of $150$ miles$/3$ hours $= 50$ miles per hour. Comparing prices as rates makes shopping decisions clear: Store A at $\$5.40$ for $3$ lbs of apples is $\$1.80/\text{lb}$, while Store B at $\$7.00$ for $4$ lbs is $\$1.75/\text{lb}$, so Store B is cheaper per pound. Unit conversion is just multiplying by rates equal to $1$: converting $60$ mph to m/s, $60 \times 1609.34/3600 \approx 26.8$ m/s, since each conversion factor (like $1609.34$ m per mile) is itself a rate equal to $1$.
Key Insight
Rates are everywhere in daily life, whenever you see "per" between two different units, you are looking at a rate, and rates let you compare quantities measured in completely different ways. They are the numerical core of almost every real-world comparison, fuel economy, nutrition labels, population density, inflation, and converting to a unit rate (per $1$ of the second quantity) is always the key to comparing them. Dimensional analysis exploits the fact that rates with equal numerator and denominator (like $1609.34$ m$/1$ mile $= 1$) can be multiplied freely without changing physical value, making unit conversion purely algebraic, one of the most powerful and underused tools in applied mathematics.