Lesson 4: Describe Proportional Relationships: Constant of Proportionality

Property

The variables $y$ and $x$ are proportional if

Examples

• The cost $C$ for gallons $g$ of gas is proportional. If 5 gallons cost 20 dollars, the constant is $k = \frac{20}{5} = 4$. The equation is $C = 4g$.
• The number of words $w$ you type is proportional to the minutes $m$ you spend typing. If you type 240 words in 4 minutes, the constant is $k = \frac{240}{4} = 60$. The equation is $w = 60m$.
• The length in centimeters $c$ is proportional to the length in inches $i$. Since 1 inch is 2.54 cm, the constant of proportionality is $k = 2.54$. The equation is $c = 2.54i$.

Explanation

Proportional variables have a constant ratio. This means one variable is always a fixed multiple of the other. Think of it like a recipe: doubling the ingredients doubles the serving size. Their graph is a straight line through the origin (0,0).

Property

Proportional relationships can be represented by an equation of the form $y = kx$ or $y = rx$. In this equation:

• $x$ is the independent variable (input).
• $y$ is the dependent variable (output).
• $r$ (or $k$) is the constant of proportionality (the unit rate).

This equation shows that the output is always a constant multiple of the input.

Examples

• A machine prints 80 pages in 5 minutes. The unit rate is $r = \frac{80}{5} = 16$ pages per minute. The equation is $p = 16m$, where $p$ is pages and $m$ is minutes.
• The cost for apples is 2.50 dollars per pound. If $C$ is the total cost and $p$ is the number of pounds, the equation is $C = 2.5p$.
• A graph of a proportional relationship passes through $(4, 32)$. The unit rate is $\frac{32}{4}=8$. The equation representing this graph is $y = 8x$.

Explanation

An equation is like a powerful calculator for a proportional relationship. Once you find the constant rate ($r$), you can plug in any amount for $x$ to instantly find its corresponding amount $y$, without having to fill out a huge table.