In a proportional table, $\frac{y}{x}$ is the same for every pair. That fixed number is so useful that it has a name: the constant of proportionality, usually called $k$.
Idea clave
$k = \frac{y}{x}$ for any pair (with $x \ne 0$). It is the unit rate: the amount of $y$ that goes with 1 of $x$.
Por qué funciona
If every $y$ is $k$ times its $x$, then dividing $y$ by $x$ undoes the multiplying and leaves $k$. And when $x = 1$, $y = k \times 1 = k$, so $k$ is "how much for 1," the same unit rate you already know.
Worked examples
Ejemplo resuelto: From a table
$y$ is proportional to $x$. Find $k$. $$\begin{array}{l|ccc} x & 3 & 5 & 8 \\ \hline y & 12 & 20 & 32 \end{array}$$
- $k = \frac{y}{x}$: $\frac{12}{3} = 4$.
- Check another column: $\frac{32}{8} = 4$ ✓.
Respuesta: $k = 4$
Ejemplo resuelto: From a story
Rosa pays \$7.50 for 3 pounds of grapes. What is the constant of proportionality, in dollars per pound?
- Dollars per pound means dollars $\div$ pounds: $7.50 \div 3 = 2.50$.
- Each pound costs \$2.50, so $k = 2.5$.
Respuesta: $k = 2.5$ dollars per pound
Ejemplo resuelto: From a graph
A proportional graph passes through the origin and $(4, 10)$. Find $k$.
- Every point on the line has the same ratio, so use $(4, 10)$: $k = \frac{10}{4} = 2.5$.
- The line also passes through $(1, 2.5)$, the unit rate.
Respuesta: $k = 2.5$
$k$ does not have to be a whole number. It can be a fraction like $\frac{3}{4}$ or a decimal like $0.4$. It is still the amount of $y$ for each 1 of $x$.