Proportion
A 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, an equation with a ratio on each side, and it helps solve problems like "if $3$ apples cost $\$1.50$, how much do $7$ apples cost?" Formally, a proportion is the equation $a/b = c/d$, and the cross-multiplication property states $a/b = c/d$ iff $ad = bc$ (cross-products are equal, where $a$ and $d$ are the "extremes" and $b$ and $c$ are the "means"), so given any three of the four values, the fourth can be found. Proportion also has a geometric life: it underlies similar triangles (where corresponding sides satisfy $a:b = c:d$) and the geometric mean $g = \sqrt{ab}$, the value forming a proportion $a:g = g:b$.
Example
Solving $3$ apples$/\$1.50 = 7$ apples$/x$: cross-multiply to get $3x = 7 \times 1.50 = 10.50$, so $x = \$3.50$, the price per apple staying the same in both cases. A map using scale $1$ inch $= 25$ miles, with two cities $3.6$ inches apart, gives an actual distance of $25 \times 3.6 = 90$ miles by cross-multiplying $1/25 = 3.6/x$. The geometric mean of $a=4$ and $b=9$ is the value $g$ such that $a/g = g/b$, i.e. $g^2 = ab = 36$, so $g = 6$; check: $4/6 = 2/3$ and $6/9 = 2/3$.
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. The cross-multiplication rule comes from multiplying both sides of $a/b = c/d$ by $bd$, transforming a rational equation into a simpler linear one, a powerful algebraic move that clears all fractions at once. 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, making proportion arguably the most broadly applied relationship in all of quantitative science.