We've already learnt about linear equations, which showed a relationship between two variables. Now we are going to look at a special kind of linear relationship called a proportional relationship.

Two quantities are said to be proportional if they vary in such a way that one is a constant multiple of the other. In other words, they always vary by the same constant.

For example, if you earn $\$18$$18 per hour, your earnings are directly proportional to the number of hours worked because $\text{earnings }=18\times\text{hours worked }$earnings =18×hours worked .

The equation for this relationship can be written as $y=18x$y=18x, where $y$y is the total earnings and $x$x is the number of hours worked.

The constant of proportionality

The constant of proportionality is the value that relates the two amounts. In the example above, the constant would be $18$18. Notice that this is exactly the same as saying the slope of the line is $18$18. A directly proportional relationship is a linear equation that has $b=0$b=0. In other words, it passes through the origin.

We can write a general equation for amounts that are directly proportional.

Equation for directly proportional relationships

$y=kx$y=kx

where $k$k is the constant of proportionality.

Once we solve the constant of proportionality, we can use it to answer other questions in this relationship.

Did you know?

Whilst the constant of proportionality can be any non-zero number, positive or negative, we generally only consider relationships with a positive value of k. That is, those with a positive rate of change.

Practice questions

Question 1

Consider the values in each table. Which of them could represent a directly proportional relationship between $x$x and $y$y?

$x$x $1$1 $3$3 $5$5 $7$7

$y$y $20$20 $16$16 $12$12 $8$8

A

$x$x $1$1 $5$5 $6$6 $20$20

$y$y $16$16 $12$12 $8$8 $4$4

B

$x$x $1$1 $2$2 $3$3 $4$4

$y$y $2$2 $8$8 $18$18 $32$32

C

$x$x $1$1 $2$2 $3$3 $4$4

$y$y $2$2 $4$4 $6$6 $8$8

D

$x$`x`	$1$1	$3$3	$5$5	$7$7
$y$`y`	$20$20	$16$16	$12$12	$8$8

$x$`x`	$1$1	$5$5	$6$6	$20$20
$y$`y`	$16$16	$12$12	$8$8	$4$4

$x$`x`	$1$1	$2$2	$3$3	$4$4
$y$`y`	$2$2	$8$8	$18$18	$32$32

$x$`x`	$1$1	$2$2	$3$3	$4$4
$y$`y`	$2$2	$4$4	$6$6	$8$8

Question 2

Consider the equation $P=70t$P=70t.

State the constant of proportionality.
Find the value of $P$P when $t=4$t=4.

Question 3

$William$ is making cups of fruit smoothie. The amount of bananas and strawberries he uses is shown in the proportion table.

Strawberries	$5$5	$10$10	$15$15	$20$20	$25$25
Bananas	$3.5$3.5	$7$7	$10.5$10.5	$14$14	$17.5$17.5

Sketch the graph of this proportional relationship.

Loading Graph...
What is the unit rate of this relationship?
Select ALL the statements that describe the proportional relationship.
For every $3.5$3.5 bananas $William$ uses, he adds $5$5 strawberries.
A
For every $5$5 bananas, $William$ uses $3.5$3.5 strawberries.
B
The unit rate of bananas in respect to strawberries is $\frac{7}{10}$710.
C
The unit rate of bananas in respect to strawberries is$\frac{10}{7}$107.
D

Outcomes

9.B3.5

Pose and solve problems involving rates, percentages, and proportions in various contexts, including contexts connected to real-life applications of data, measurement, geometry, linear relations, and financial literacy.

9.C3.1

Compare the shapes of graphs of linear and non-linear relations to describe their rates of change, to make connections to growing and shrinking patterns, and to make predictions.

6.05 Proportional relationships