Ratio
Fractions & DecimalsA 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. It tells you how much of one thing there is compared to another. You can write a ratio with a colon ($3:4$), as a fraction ($3/4$), or with the word "to" ($3$ to $4$).
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 marbles.
Key Insight
Ratios are everywhere: recipes, maps, team scores, and mixing paint all use ratios. Unlike regular subtraction comparisons ("$3$ more"), ratios show the multiplicative relationship between quantities.
Definition
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. Ratios can compare part-to-part or part-to-whole. Equivalent ratios are obtained by multiplying both terms by the same nonzero constant.
Example
A recipe uses $2$ cups flour to $3$ cups sugar: ratio $2:3$. Scaling up by $4$: $8:12$ (equivalent ratio). The ratio $2:3:5$ compares three quantities. Total parts $= 2+3+5=10$, so fractions are $2/10$, $3/10$, $5/10$.
Key Insight
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. Understanding both interpretations is crucial for proportional reasoning.
Definition
A ratio $a:b$ defines an element of $\mathbb{Q}$ (the rational $a/b$) for nonzero $b$. In projective geometry, the ratio (cross-ratio) of four collinear points is an invariant under projective transformations. In linear algebra, the ratio of two vector components is preserved under scalar multiplication, defining direction in projective space. Ratios also appear as eigenvalue ratios in spectral theory.
Example
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. This self-similar ratio property leads to its appearance in Fibonacci sequences and natural growth patterns.
Key Insight
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. Projective geometry thus emerges naturally from the idea of ratio, revealing a deep connection between elementary fraction arithmetic and the geometry of perspective.