Ratios and Rates

NZ Level 6 (NZC) Level 1 (NCEA)

Proportional Relationships

Lesson

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 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`=5`x` 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.

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

ANot proportional

BProportional

ANot proportional

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

ANot proportional

BProportional

ANot proportional

B

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

ANo

BYes

ANo

B

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

Loading Graph...

Can a straight line be drawn through all the points?

Yes

ANo

BYes

ANo

BAs $x$

`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

ANo

BYes

ANo

BWhat 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`=`k``x`for some constant $k$`k`?Yes

ANo

BYes

ANo

BIs $y$

`y`proportional to $x$`x`?Yes

ANo

BYes

ANo

B

Apply direct and inverse relationships with linear proportions

Apply numeric reasoning in solving problems