Question

Is the relationship between x and y proportional? If it is Yes, then what is the constant of proportionality? If it isn’t, then why?

Answer 1

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

`y=kx`

Where k is a constant.

From the table, we have the following values

`\begin{gathered} y(2)=(5)/(2) \\ y(4)=5 \\ y(6)=(15)/(2) \\ y(12)=15 \end{gathered}`

If we substitute the first expression on our form, we have the following constant of proportionality

`\begin{gathered} ((5)/(2))=k(2) \\ 2k=(5)/(2) \\ k=(5)/(2)\cdot(1)/(2) \\ k=(5)/(4) \end{gathered}`

If this is a proportional relationship, the constant of proportionality is 5/4. Let's check if this constant fits for the other values:

`\begin{gathered} y(4)=(5)/(4)\cdot4=5 \\ y(6)=(5)/(4)\cdot6=(15)/(2) \\ y(12)=(5)/(4)\cdot12=15 \end{gathered}`

Since it fits, we have indeed a proportional relationship where 5/4 is the constant of proportionality.