Some common unit rates are distance per hour, cost per item, earnings per week, etc. Do you see how in each example the first quantity is related to one unit of the second quantity?

When we are looking at unit rates in tables and graphs, we want to know how much the dependent (y) variable will increase when the independent (x) variable is increased by one. The change in the values of y for every change in the values of x is referred to as the slope of the line. So the slope and unit rate represent the same thing.

In a given equation that shows the proportional relationship between two variables, y=kx, k is known as the constant of proportionality. Upon inspection we can notice that k is the slope of the equation and can be rewritten as y=mx, where m is the slope of the line.

Therefore for a given proportional relationship, the unit rate, slope, and constant of proportionality are all equal.

Examples

Example 1

The graph shows the amount of time it takes Kate to make beaded bracelets.

\text{Time (hours)}

\text{No. of bracelets made}

Find the slope of the line.

Worked Solution

Create a strategy

For a 1 unit increase in x on the graph, find the increase of y.

Apply the idea

The y-values increase by 5 every time x increases by 1.

The slope is 5.

Interpret the unit rate based on the slope of the line.

Worked Solution

Create a strategy

The slope of the graph is the unit rate.

Apply the idea

The slope 5 based on the graph means that 5 bracelets are made for every hour.

The unit rate is 5 bracelets per hour.

Example 2

Carl has kept a table of his reading habits which is shown below:

Number of weeks	12	24	36	48
Number of books read	20	40	60	80

Determine the unit rate of the number of books Carl reads for every week, rounding the answer in one decimal place.

Worked Solution

Create a strategy

Find the constant change in the y values for every change in the x values.

Apply the idea

Based on the table, for every 12 weeks (say between 24 and 12 weeks), Carl reads 20 books (40-20).

The unit rate is \dfrac{20}{12} books per week or 1.7 books per week.

Example 3

The number of cans of white and red paint needed to make 'flamingo pink' is represented by the equation y = 10 x where x is the number of cans of red paint and y is the number of cans of white paint needed.

Based on the equation, how many cans of white paint are needed for every can of red paint?

Worked Solution

Create a strategy

In the equation y=mx, m is the slope of the line. The slope is the unit rate.

Apply the idea

In the equation, y=10 x, the slope, m is 10.

This means that 10 cans of while paint are required to combine with 1 can of red paint to produce 'flamingo pink'.

Idea summary

A unit rate describes how many units of the first quantity corresponds to one unit of the second quantity.

When we are looking at unit rates on a graph, we want to know how much the dependent (y) variable will increase when the independent (x) variable is increased by one.

In a proportional relationship, the unit rate, slope, and constant of proportionality are all equal.

Outcomes

8.EE.B.5

Graph proportional relationships, interpreting the unit rate as the slope of the graph. Compare two different proportional relationships represented in different ways.

4.01 Unit rate and slope

Ideas

Unit rates from a graph, table or equation

Examples

Example 1

Example 2

Example 3

Outcomes

8.EE.B.5

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