Skill: Recognize proportional relationships

A relationship is proportional when the same number connects the two quantities every time. This number is called the unit rate or constant of proportionality.

To check a table:

1. Divide each $y$-value by its matching $x$-value.
2. Compare the quotients.
3. If all the quotients are the same, the relationship is proportional.
4. Write the rule as $y=kx$, where $k$ is the constant ratio.

Example: Is the relationship in the table proportional?

$x$	$y$
2	5
4	10
6	15

Step 1: Find each ratio.

$$\frac{5}{2}=2.5$$ $$\frac{10}{4}=2.5$$ $$\frac{15}{6}=2.5$$

Step 2: Compare the ratios.

All three ratios are $2.5$, so the unit rate stays constant.

Step 3: Write the relationship.

Since $y$ is always $2.5$ times $x$, the equation is

$$y=2.5x$$

Therefore, the relationship is proportional. A proportional relationship can also be shown by a straight line that passes through $(0,0)$, the origin.