Determine a constant of proportionality

In a proportional relationship, the constant of proportionality $k$ is the fixed multiplicative factor relating one quantity to another, so $y=kx$ and $k=y/x$ when $x\ne0$ . It can be determined from corresponding values in a table, a point on a graph, or an equation, and interpreted as the unit rate with appropriate units; proportional data form a line through the origin rather than an additive relationship. This scope excludes inverse and other more advanced generalized proportional relationships.

Detailed Explanation: 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 $0$ .

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