 Simple 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. We can also calculate the unit rate in a proportional relationship, which tells us how much the dependent variable will changes with a one unit increase in the independent variable.

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.

We can also compare proportional relationships to make judgements about rates of change.

#### Examples

##### Question 1

Consider the equation $y=7x$y=7x.

1. What is the slope of $y=7x$y=7x?

2. Select the graph that below that shows $y=7x$y=7x.

A

B

C

D

A

B

C

D

##### Question 2

Irene and Valentina are both making handmade birthday cards. Irene can make $8$8 cards every $13$13 minutes. Valentina can make $6$6 cards every $15$15 minutes.

1. Plot this information on the graph.

2. How can you tell who was quicker at making cards?

The steeper line relates to the faster maker

A

The shallower line relates to the faster maker

B

The steeper line relates to the faster maker

A

The shallower line relates to the faster maker

B

##### Question 3

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

 Strawberries Bananas $4$4 $8$8 $12$12 $16$16 $20$20 $5.5$5.5 $11$11 $16.5$16.5 $22$22 $27.5$27.5
1. Graph this proportional relationship.

2. What is the unit rate of this relationship?

3. Select ALL the statements that describe the proportional relationship.

For every $5.5$5.5 bananas Oliver uses, he adds $4$4 strawberries.

A

The unit rate of bananas in respect to strawberries is $\frac{11}{8}$118.

B

For every $4$4 bananas, Oliver uses $5.5$5.5 strawberries.

C

The unit rate of bananas in respect to strawberries is$\frac{8}{11}$811.

D

For every $5.5$5.5 bananas Oliver uses, he adds $4$4 strawberries.

A

The unit rate of bananas in respect to strawberries is $\frac{11}{8}$118.

B

For every $4$4 bananas, Oliver uses $5.5$5.5 strawberries.

C

The unit rate of bananas in respect to strawberries is$\frac{8}{11}$811.

D

We'll learn later about direct proportional relationships and inverse proportional relationships.

### Outcomes

#### 9D.AG3.03

Describe the meaning of the slope and y-intercept for a linear relation arising from a realistic situation and describe a situation that could be modelled by a given linear equation