Rate
Fractions & DecimalsA 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.
Example
A car travels $150$ miles in $3$ hours. The rate is $150$ miles$/3$ hours $= 50$ miles per hour. The "per" always signals a rate - miles per hour, dollars per gallon, words per minute.
Key Insight
Rates are everywhere in daily life. When you see "per" between two different units, you are looking at a rate. Rates let you compare quantities that are measured in completely different ways.
Definition
A rate is a ratio $a/b$ where $a$ and $b$ have different units. Unlike a pure ratio, rates carry units and describe how one quantity changes per unit of another. A unit rate has a denominator of $1$ (e.g., $65$ miles per $1$ hour). Rates can be simplified to unit rates for easier comparison.
Example
Store A: $\$5.40$ for $3$ lbs of apples $= \$1.80/\text{lb}$. Store B: $\$7.00$ for $4$ lbs $= \$1.75/\text{lb}$. Store B is cheaper per pound. Comparing rates requires a common denominator unit ($1$ lb in this case).
Key Insight
Rates are the numerical core of almost every real-world comparison: fuel economy, nutrition labels, population density, and inflation are all rates. Converting to a unit rate (per 1 of the second quantity) is always the key to comparison.
Definition
A rate is a ratio of quantities with different dimensions (units). In physics, rates appear as derivatives: speed $= ds/dt$, acceleration $= dv/dt$, flow rate $= dV/dt$. In economics, a marginal rate is the derivative of one quantity with respect to another. Dimensional analysis (unit analysis) uses the algebra of rates to verify formulas and convert units.
Example
Unit conversion using rates: convert $60$ mph to m/s. $60 \text{ miles/hr} \times (1609.34 \text{ m/mile}) \times (1 \text{ hr}/3600 \text{ s}) = 60 \times 1609.34/3600 \text{ m/s} \approx 26.8 \text{ m/s}$. Each conversion factor is a rate equal to $1$ (same quantity, different units), so the value is unchanged.
Key Insight
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. This makes unit conversion purely algebraic and reduces errors - one of the most powerful and underused tools in applied mathematics.