Topic 2

Unit Rate and Proportional Relationships

A relationship between two quantities is proportional if one quantity is a constant multiple of the other. Algebraically:

y = kx

where

k

is the constant of proportionality or unit rate. Key features:

Graph passes through the origin $(0,0)$ .
Slope equals $k$ . Each 1 unit increase in $x$ changes $y$ by $k$ .
From a table, $k = \dfrac{y}{x}$ is the same for all nonzero $x$ .

A relationship is not proportional if it has a nonzero intercept or if

\dfrac{y}{x}

is not constant. Any form

y = kx + b

with

b \ne 0

is not proportional.

Core Skills

Compute a unit rate by dividing per one unit of the denominator.
Test proportionality using equal ratios or a straight line through the origin.
Write and use $y = kx$ to predict unknown values and interpret slope as rate.
Convert among tables, equations, and graphs for the same proportional model.

Example 1: Finding a Unit Rate

A runner travels 9 miles in 72 minutes. Find miles per minute and miles per hour. Miles per minute:

\dfrac{9}{72} = \dfrac{1}{8} = 0.125

mi per min. Miles per hour: multiply by 60:

0.125 \times 60 = 7.5

mph. Model:

d = 7.5t

where

t

is hours.

Example 2: Proportional or Not

Decide if the relationship is proportional:

y = 4x - 3

. The intercept is

-3

. Since the graph does not pass through

(0,0)

, this is not proportional.

Example 3: From Table to Equation

\begin{array}{c|cccc} x & 2 & 5 & 8 & 10\\ \hline y & 6 & 15 & 24 & 30 \end{array}

Compute

\dfrac{y}{x}

\dfrac{6}{2}=3,\ \dfrac{15}{5}=3,\ \dfrac{24}{8}=3,\ \dfrac{30}{10}=3

. Constant ratio 3. Equation:

y = 3x

. Here

k=3

Example 4: Graph Interpretation

A graph of a line through

(0,0)

and

(4,10)

represents

y

versus

x

. Slope

k = \dfrac{10-0}{4-0} = \dfrac{10}{4} = 2.5

. Equation:

y = 2.5x

. The unit rate is 2.5 units of

y

per 1 unit of

x

Example 5: Context Prediction with $y = kx$

A fruit stand sells apples at a constant price. If 3 pounds cost $7.50, find the price for 8 pounds. Unit rate

k = \dfrac{7.50}{3} = 2.50

dollars per pound. Model

C = 2.5x

. For

x=8

C = 2.5 \cdot 8 = \$20

Key Takeaways

Proportional relationships have the form $y = kx$ and pass through the origin.
The constant $k$ is the unit rate and equals the slope of the graph.
Test proportionality by checking equal ratios or intercept at the origin.
Use $y = kx$ to convert, compare, and predict values in context.