Proportional relationships are relationships between two variables where their ratios are equivalent. Another way to think about them is that, in a proportional relationship, one variable is always a constant value times the other. That constant is know as the "constant of proportionality".
If the relationship between x and y is proportional, we can write a rule correlating them as
[tex]y=kx[/tex]
Where k is a constant.
From the table, we have the following values
[tex]\begin{gathered} y(2)=\frac{5}{2} \\ y(4)=5 \\ y(6)=\frac{15}{2} \\ y(12)=15 \end{gathered}[/tex]
If we substitute the first expression on our form, we have the following constant of proportionality
[tex]\begin{gathered} (\frac{5}{2})=k(2) \\ 2k=\frac{5}{2} \\ k=\frac{5}{2}\cdot\frac{1}{2} \\ k=\frac{5}{4} \end{gathered}[/tex]
If this is a proportional relationship, the constant of proportionality is 5/4. Let's check if this constant fits for the other values:
[tex]\begin{gathered} y(4)=\frac{5}{4}\cdot4=5 \\ y(6)=\frac{5}{4}\cdot6=\frac{15}{2} \\ y(12)=\frac{5}{4}\cdot12=15 \end{gathered}[/tex]
Since it fits, we have indeed a proportional relationship where 5/4 is the constant of proportionality.
