Recognize proportional relationships

A proportional relationship links two quantities so that the ratio of corresponding values remains constant; equivalently, one quantity is a constant multiple of the other, expressed as $y=kx$ , where $k$ is the constant of proportionality or unit rate. The relationship can be identified in tables, graphs, and equations: its graph is a straight line through the origin, distinguishing it from additive relationships and linear relationships with a nonzero intercept. This scope excludes inverse, nonlinear, and more abstract generalizations.

Detailed Explanation: Recognize proportional relationships

A proportional relationship means one quantity is always the same multiple of the other. It can be written as

y=kx

where $k$ is the constant of proportionality, or unit rate. To recognize one, divide corresponding values and check whether the ratio is always the same.

Example

The table shows the cost of notebooks.

Number of notebooks, $x$	Cost in dollars, $y$
1	3
2	6
4	12

Is the relationship proportional?

Step 1: Find the ratio for each pair.

\frac{y}{x}=\frac{3}{1}=3

\frac{y}{x}=\frac{6}{2}=3

\frac{y}{x}=\frac{12}{4}=3

Step 2: Check whether the ratios are constant.

All the ratios equal $3$ , so the relationship is proportional.

Step 3: Write the equation.

The constant of proportionality is $k=3$ , so

y=3x

This means each notebook costs 3 $. For example,$ 5$ notebooks would cost

y=3(5)=15

or 15$.

If the ratios had not been equal, the relationship would not be proportional. A proportional relationship also includes the point $(0,0)$ , because $y=3(0)=0$ .

Learn by doing: Recognize proportional relationships

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