Ratio
A ratio is a comparison of two quantities showing how many times one value contains or is contained within the other.
Formula
a : b \text{ or } \frac{a}{b}
Definition
A ratio is a way to compare two amounts, telling you how much of one thing there is compared to another; it can be written with a colon ($3:4$), as a fraction ($3/4$), or with the word "to" ($3$ to $4$). Formally, a ratio $a:b$ compares two quantities $a$ and $b$ (with $b \neq 0$), expressing how many times $a$ "fits into" $b$ or the fractional relationship between them, and can compare part-to-part or part-to-whole; equivalent ratios come from multiplying both terms by the same nonzero constant. A ratio $a:b$ defines an element of $\mathbb{Q}$ (the rational $a/b$) for nonzero $b$, and the idea generalizes further: in projective geometry the cross-ratio of four collinear points is an invariant under projective transformations, and ratios also appear as eigenvalue ratios in spectral theory.
Example
If a bag has $3$ red marbles and $5$ blue marbles, the ratio of red to blue is $3:5$, for every $3$ red marbles there are $5$ blue. A recipe using $2$ cups flour to $3$ cups sugar has ratio $2:3$; scaling up by $4$ gives the equivalent ratio $8:12$, and a three-way ratio like $2:3:5$ splits a total of $10$ parts into fractions $2/10$, $3/10$, $5/10$. The golden ratio $\varphi = (1+\sqrt{5})/2$ satisfies $\varphi:1 = (1+\varphi):\varphi$, meaning a rectangle with sides in ratio $\varphi:1$ remains in the same ratio when a square is removed, a self-similar property behind its appearance in Fibonacci sequences and natural growth patterns.
Key Insight
Ratios are everywhere, recipes, maps, team scores, mixing paint, and unlike a plain subtraction comparison ("$3$ more"), a ratio shows the multiplicative relationship between quantities. A ratio $a:b$ and the fraction $a/b$ contain the same information, but ratios emphasize the relationship between two separate quantities while fractions emphasize part-of-a-whole, and understanding both interpretations is crucial for proportional reasoning. The projective line $\mathbb{P}^1(\mathbb{R})$ consists of equivalence classes of pairs $(a,b)$ with $(a,b) \sim (ka,kb)$ for $k \neq 0$, exactly the definition of ratio, revealing a deep connection between elementary fraction arithmetic and the geometry of perspective.