Lesson 1: Proportional Relationships

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

The constant of proportionality, often called the unit rate ($r$), is the constant ratio in a proportional relationship.
If quantities $x$ and $y$ are proportional, their relationship can be described by the equation $y = rx$.
The constant $r$ can be found by calculating the ratio $r = \frac{y}{x}$ for any corresponding pair $(x, y)$ where $x \neq 0$. On a graph, the unit rate is represented by the point $(1, r)$.

Examples

• A car travels 180 miles in 3 hours. The constant of proportionality is $\frac{180}{3} = 60$ miles per hour. The equation is $d = 60h$.
• A graph of a proportional relationship between cost and pounds of bananas passes through the point $(1, 0.55)$. The constant of proportionality is 0.55 dollars per pound.
• A table shows that 4 movie tickets cost 52 dollars. The unit rate (constant of proportionality) is $\frac{52}{4} = 13$ dollars per ticket.

Explanation

The constant of proportionality is the 'secret multiplier' that connects the two quantities. It tells you how much of the second quantity you get for exactly one unit of the first quantity, like price per item or miles per hour.