Equivalent Ratios
Equivalent ratios are ratios that express the same relationship between quantities, obtained by multiplying or dividing both terms by the same nonzero number.
Formula
a:b = (an):(bn) \text{ for any nonzero } n
Definition
Equivalent ratios are ratios that show the same relationship, just written with different numbers, created by multiplying or dividing both numbers by the same value. Formally, two ratios $a:b$ and $c:d$ are equivalent if $a/b = c/d$, equivalently if $ad = bc$; every ratio in a proportional relationship is equivalent to every other, and plotting equivalent pairs $(x,y)$ where $y/x = k$ traces out the line $y = kx$ through the origin. In $\mathbb{Q}$, equivalent ratios form equivalence classes $[a:b] = \{(na):(nb) : n \in \mathbb{Q}, n \neq 0\}$, exactly the equivalence classes of rational numbers; in projective space $\mathbb{P}^1$, these equivalent ratios $[a:b]$ become homogeneous coordinates, where the ratio itself is the meaningful quantity, not the individual values.
Example
$2:3$ is equivalent to $4:6$, $6:9$, and $10:15$; a recipe using $2$ cups flour to $3$ cups sugar can be doubled ($4:6$), tripled ($6:9$), or halved ($1:1.5$), all equivalent ratios describing the same "mix." A ratio table for $3:5$ lists $(3,5)$, $(6,10)$, $(9,15)$, $(12,20)$, $(15,25)$, every row sharing the same value of $y/x = 5/3$, and adding two rows together, like $(3+6, 5+10) = (9,15)$, lands on another row already in the table. In projective geometry, $[1:2]$, $[3:6]$, and $[-2:-4]$ are all the same projective point, the "line" $\mathbb{P}^1(\mathbb{R})$ being topologically a circle once the point at infinity $[1:0]$ is added.
Key Insight
Equivalent ratios are like equivalent fractions for comparisons between two things: they all describe the same "mix" or "rate" even though the actual numbers differ. Ratio tables are the gateway to understanding proportional relationships graphically, each row is a point on the line $y = kx$, and the ability to find any missing value using equivalent ratios is the foundation of all proportional reasoning in algebra. The equivalence-class perspective on ratios unifies fractions, ratios, proportions, unit rates, and projective coordinates into a single algebraic concept, the same conceptual leap that constructs $\mathbb{Q}$ from $\mathbb{Z}$, a foundational idea in abstract algebra that begins with the most elementary fraction facts.