Lesson 5: Compare Proportional Relationships

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.

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.