A ratio table is a list of pairs, and you already know how to plot pairs as points (see Graphing Points in a Problem). A recipe uses 1 cup of rice for every 3 cups of water. Its equivalent ratios are the pairs $(1, 3)$, $(2, 6)$, $(3, 9)$ and $(4, 12)$.
Idea clave
When you graph equivalent ratios, the points lie on a straight line through $(0, 0)$. Each step of 1 to the right climbs the same amount: the unit rate.
Por qué funciona
Why through $(0, 0)$? Zero batches use 0 cups of rice and 0 cups of water. Why a straight line? Every extra cup of rice brings exactly 3 more cups of water, so every step to the right climbs by the same 3. A path that always climbs the same amount for the same step is a straight line.
Ejemplo resuelto: Use the line
The graph of a ratio goes through $(0, 0)$ and $(4, 10)$. What is $y$ when $x = 6$?
- The point $(4, 10)$ means the ratio $4 : 10$.
- Divide both by 2: $(2, 5)$ is on the line too.
- Multiply $(2, 5)$ by 3: $(6, 15)$.
Respuesta: $y = 15$.
Ejemplo resuelto: Find the point at $x = 1$
The point $(5, 8)$ is on the graph of a ratio. Which point on the line has $x = 1$?
- Divide both coordinates by 5: $5 \div 5 = 1$ and $8 \div 5 = 1.6$.
- The y-value at $x = 1$ is the unit rate: 1.6 for each 1.
Respuesta: $(1, 1.6)$.
Error común
Points that line up are not always a ratio. The pairs $(1, 4)$, $(2, 5)$, $(3, 6)$ also make a straight line, but each $y$ is $x + 3$: adding the same amount, not scaling. Their line crosses the y-axis at 3, not at 0. Quick check: divide $y$ by $x$ for every pair. A ratio gives the same answer each time ($\frac{3}{1} = \frac{6}{2} = \frac{9}{3}$); these give $4$, $2.5$ and $2$.