You have built ratio tables: multiply both amounts by the same number and the ratio stays the same. A table like that shows a proportional relationship. Many tables look steady but are not proportional. Here is how to tell.
Idea clave
Two quantities are proportional when $\frac{y}{x}$ is the same number for every pair in the table. That number is the constant of proportionality.
Por qué funciona
Proportional means "scale one amount and the other scales the same way." If 2 tickets cost \$30, then 4 tickets (twice as many) cost \$60 (twice as much). This only works when every $y$ is the same multiple of its $x$, and that is the same as saying $\frac{y}{x}$ never changes. Adding a fixed amount breaks it: with $y = x + 3$, going from $x = 1$ to $x = 2$ doubles $x$ but takes $y$ from 4 to 5.
Worked examples
Ejemplo resuelto: A proportional table
Is pay proportional to hours? $$\begin{array}{l|ccc} \text{Hours} & 2 & 3 & 5 \\ \hline \text{Pay (dollars)} & 30 & 45 & 75 \end{array}$$
- Divide each pay by its hours: $\frac{30}{2} = 15$, $\frac{45}{3} = 15$, $\frac{75}{5} = 15$.
- Every quotient is 15, so pay is proportional to hours: \$15 for each hour.
Respuesta: Yes, proportional
Ejemplo resuelto: Steady, but not proportional
Is $y$ proportional to $x$? $$\begin{array}{l|cccc} x & 1 & 2 & 3 & 4 \\ \hline y & 4 & 5 & 6 & 7 \end{array}$$
- Divide: $\frac{4}{1} = 4$, $\frac{5}{2} = 2.5$, $\frac{6}{3} = 2$, $\frac{7}{4} = 1.75$.
- The quotients change, so it is not proportional. Here $y$ is always $x + 3$: the table adds, it does not multiply.
Respuesta: No, not proportional
Error común
Calling a table proportional because its values go up by the same amount each time. Steady growth is not enough. Check: divide $y$ by $x$ in at least two columns (all of them, to be sure). Another check: does $x = 0$ go with $y = 0$? For $y = x + 3$ it would go with $3$, so it is not proportional.