Proportion
Fractions & DecimalsA proportion is an equation stating that two ratios are equal, used to solve problems where one quantity scales with another.
Formula
\frac{a}{b} = \frac{c}{d} \text{ (equivalently, } ad = bc\text{)}
Definition
A proportion says that two ratios are equal. It is an equation with a ratio on each side. Proportions help you solve problems like "if $3$ apples cost $\$1.50$, how much do $7$ apples cost?"
Example
$3$ apples$/\$1.50 = 7$ apples$/x$. Cross-multiply: $3x = 7 \times 1.50 = 10.50$. $x = \$3.50$. The proportion says the price-per-apple is the same in both cases.
Key Insight
Proportions are the math behind recipes, maps, models, and shopping comparisons. Whenever you scale something up or down while keeping the same ratio, you are using a proportion.
Definition
A proportion is the equation $a/b = c/d$, meaning the two ratios are equivalent. The cross-multiplication property states: $a/b = c/d$ iff $ad = bc$ (cross-products are equal). Given any three of the four values, the fourth can be found. The terms $a$ and $d$ are called the "extremes"; $b$ and $c$ are the "means." The product of extremes equals the product of means.
Example
A map uses scale $1$ inch $= 25$ miles. If two cities are $3.6$ inches apart on the map, what is the actual distance? $1/25 = 3.6/x$. Cross-multiply: $x = 25 \times 3.6 = 90$ miles.
Key Insight
The cross-multiplication rule is derived from multiplying both sides of $a/b = c/d$ by $bd$, giving $ad = bc$. This transforms a rational equation into a linear one - a powerful algebraic move that clears all fractions.
Definition
A proportion $a:b = c:d$ is equivalent to membership in the same equivalence class of the relation $\sim$ on $\mathbb{Q}$ defined by the cross-product. In category theory, proportions are related to morphisms in the category of ratios. Geometric proportion appears in the theory of similar triangles (AA similarity), where corresponding sides satisfy $a:b = c:d$. Harmonic proportion and geometric proportion have classical definitions in Greek mathematics.
Example
The geometric mean of $a$ and $b$ is the value $g$ such that $a/g = g/b$, i.e., $g^2 = ab$, so $g = \sqrt{ab}$. This is the value that forms a proportion $a:g = g:b$. For $a=4$, $b=9$: $g=6$; check $4/6 = 2/3$ and $6/9 = 2/3$.
Key Insight
Proportionality is the foundation of dimensional analysis, scaling laws, and similarity in geometry. Kepler's third law ($T^2$ proportional to $a^3$) and Newton's law of gravitation both assert proportional relationships. The concept of proportion is arguably the most broadly applied relationship in all of quantitative science.