Equivalent Ratios
Fractions & DecimalsEquivalent 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. You can create equivalent ratios by multiplying or dividing both numbers by the same value.
Example
$2:3$ is equivalent to $4:6$, $6:9$, and $10:15$. A recipe that uses $2$ cups of flour to $3$ cups of sugar can be doubled ($4:6$), tripled ($6:9$), or halved ($1:1.5$) - all equivalent ratios.
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 are different.
Definition
Two ratios $a:b$ and $c:d$ are equivalent if $a/b = c/d$, equivalently if $ad = bc$ (cross-products are equal). All ratios in a proportional relationship are equivalent. A ratio table lists many equivalent ratios for a given relationship, and plotting equivalent ratio pairs $(x, y)$ where $y/x = k$ produces the line $y = kx$ through the origin.
Example
Ratio table for $3:5$: $(3,5)$, $(6,10)$, $(9,15)$, $(12,20)$, $(15,25)$. All have $y/x = 5/3$. Adding a row: $(3+6, 5+10) = (9,15)$ - also in the table. Scaling: multiply any row by the same number to get another equivalent ratio.
Key Insight
Ratio tables are the gateway to understanding proportional relationships graphically. Each row is a point on the line $y = (5/3)x$. The ability to find any missing value using equivalent ratios is the foundation of all proportional reasoning in algebra.
Definition
Equivalent ratios form equivalence classes in $\mathbb{Q}$: the class $[a:b] = \{(na):(nb) : n \in \mathbb{Q}, n \neq 0\}$. These are exactly the equivalence classes of rational numbers. In projective space $\mathbb{P}^1$, equivalent ratios $[a:b]$ are homogeneous coordinates - the ratio itself is the meaningful quantity, not the individual values. A ratio table is a discrete sample from the line $y = kx$ in $\mathbb{R}^2$.
Example
In projective geometry, $[1:2]$ and $[3:6]$ and $[-2:-4]$ are all the same projective point. The "line" in projective space $\mathbb{P}^1(\mathbb{R})$ is the set of all equivalence classes of nonzero pairs, which is topologically a circle (the projective line). Adding the "point at infinity" $[1:0]$ compactifies the real line into $\mathbb{P}^1$.
Key Insight
The equivalence class perspective on ratios unifies fractions, ratios, proportions, unit rates, and projective coordinates into a single algebraic concept. The move from "ratios are numbers" to "ratios are equivalence classes" is 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.