Skill: Determine a constant of proportionality

A proportional relationship has the form

$$y=kx$$

where $k$ is the constant of proportionality. To find $k$, divide the $y$-value by its corresponding $x$-value:

$$k=\frac{y}{x}$$

Example

A car travels at a constant speed. The table shows the distance $d$ traveled after $t$ hours.

Time, $t$ (hours)	Distance, $d$ (miles)
$2$	$130$
$4$	$260$
$6$	$390$

Find the constant of proportionality.

Step 1: Choose a pair of corresponding values.

Use $t=2$ hours and $d=130$ miles.

Step 2: Divide the output by the input.

$$k=\frac{d}{t}=\frac{130}{2}=65$$

Step 3: Include the units.

The constant of proportionality is

$$\boxed{65\text{ miles per hour}}$$

This means the car travels $65$ miles for every $1$ hour. The relationship can be written as

$$d=65t$$

Step 4: Check another pair of values.

$$\frac{260}{4}=65$$

The same value is obtained, so $65$ is the constant of proportionality. In a proportional relationship, the graph forms a straight line through $(0,0)$ because when the input is $0$, the output is also $0$.