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 the cost of some items is always five times the number of items, we can say that this is a proportional relationship because there is a constant multiple between the cost and the number of items, $5$5. We can write these proportional relationships as linear equations. The example above could be written as $y=5x$y=5x and again we can see that the coefficient of $x$x describes the constant of the proportional relationship.

We will learn more about the constant of proportionality and writing proportional relationships as equations later but now let's focus on determining whether relationships are proportional or not.

Remember!

A relationship is proportional if there is a constant multiple between the two variables, so a proportional relationship between $x$x and $y$y will look like $y=kx$y=kx, where $k$k is a constant. These will go through $\left(0,0\right)$(0,0).

Practice questions

Question 1

Look at the tables below. Determine whether each of them is showing a proportional relationship.

$1$1 $2$2 $3$3 $4$4 $5$5

$2$2 $4$4 $6$6 $8$8 $10$10

Proportional
A
Not proportional
B
$0$0 $0$0

$1$1 $7$7

$2$2 $14$14

$3$3 $6$6

$4$4 $28$28

Proportional
A
Not proportional
B

QUESTION 2

A $physiotherapist$ charges $\$55$$55 per patient she treats.

The table shows her weekly income in weeks where she treated $25$25 and $42$42 patients. Complete the table.

Number of patients seen in the week $12$12 $25$25 $32$32 $42$42 $51$51

Weekly income ( $dollars$ ) $\editable{}$ $1375$1375 $\editable{}$ $2310$2310 $\editable{}$
How much would she earn in a week where she treated $0$0 patients?
Is her weekly income proportional to the number of patients she sees in that week?
Yes
A
No
B

Number of patients seen in the week	$12$12	$25$25	$32$32	$42$42	$51$51
Weekly income ( $dollars$ )	$\editable{}$	$1375$1375	$\editable{}$	$2310$2310	$\editable{}$

QUESTION 3

Consider the points that have been plotted on the coordinate axes.

Loading Graph...

Five points are plotted in a coordinate plane as solid dots. In the illustration, the plotted points are $\left(1,4\right)$(1,4), $\left(2,8\right)$(2,8), $\left(3,12\right)$(3,12), $\left(4,16\right)$(4,16), and $\left(5,20\right)$(5,20) in sequential order. The coordinates of the points are not explicitly labeled or given.

Can a straight line be drawn through all the points?
Yes
A
No
B
As $x$x increases from $x=1$x=1 to $x=2$x=2, what is the increase in $y$y?
Is $y$y increasing at a constant rate?
Yes
A
No
B
What would $y$y equal when $x=0$x=0?
Do the values of $x$x and $y$y satisfy an equation of the form $y=kx$y=kx for some constant $k$k?
Yes
A
No
B
Is $y$y proportional to $x$x?
Yes
A
No
B

4.05 Identifying proportional relationships

Practice questions

Question 1

QUESTION 2

QUESTION 3

Outcomes

MA.8.AR.3.1

What is Mathspace

About Mathspace

$1$1	$2$2	$3$3	$4$4	$5$5
$2$2	$4$4	$6$6	$8$8	$10$10

$0$0	$0$0
$1$1	$7$7
$2$2	$14$14
$3$3	$6$6
$4$4	$28$28