We have already learned about 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. That constant is called the constant of proportionality.

Proportional equations

As you recall, proportional relationships can also be written as linear equations that, when graphed, will pass through the origin.

Remember!

A relationship is proportional if there is a constant multiple between the two variables.
Proportional relationships can be written generally in the form: $y=kx$y=kx , where $k$k is the constant of proportionality
The graph of a proportional relationship will always pass through the origin $\left(0,0\right)$(0,0).

Now we are going to look at some consumer applications of proportional reasoning.

Worked example

question 1

The cost of t-shirts at our favorite store in the mall is always five times the number of shirts that we purchase (without tax). Determine whether this represents a proportional relationship and if so, find the constant of proportionality and express the relationship as a linear equation.

Think: They told us that the cost of t-shirts at this store is always five times the number of items. In this case we can say that this is a proportional relationship because there is a constant multiple between the cost and the number of items.

Do: We can write these proportional relationships as linear equations. This 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. So, $5$5 is the constant of proportionality.

Let's try some practice questions.

Practice questions

question 2

$Frank$ serves $8$8 cups of coffee every $9$9 minutes.

Using $y$y for the number of cups of coffee, and $x$x for the amount of minutes that have passed, write an equation where $y$y is the subject that represents this proportional relationship.

question 3

The amount of white and red paint needed to make 'flamingo pink' is shown in the graph.

Loading Graph...

A coordinate plane displays a straight line starting at the origin $\left(0,0\right)$(0,0) and passing through point $\left(10,100\right)$(10,100), indicating a proportional relationship between the amount of white paint (x) and the amount of red paint (y) needed to make 'flamingo pink'. Both axes are marked with even intervals and numbered from 0 to 10 on the x-axis and from 0 to $500$500 on the y-axis. It's important to note that while point $\left(10,100\right)$(10,100) is mentioned in the description, it is not explicitly provided and was inferred from the graph.

Let $x$x represent the amount of white paint and $y$y represent the amount of red paint needed. What is the equation of this line where $y$y is the subject?
What does the equation of the line tell you?
$1$1 can of red paint requires $10$10 cans of white paint to make the perfect flamingo pink.
A
For every $10$10 cans of red paint, you need to use $100$100 cans of white paint.
B
$1$1 can of white paint requires $10$10 cans of red paint to make the perfect flamingo pink.
C
For every $10$10 cans of white paint, you need $10$10 cans of red paint.
D

question 4

$James$ wants to buy cereal, and sees that a $500$500 gram box is priced at $\$6.05$$6.05.

What is the price of the cereal per $100$100 grams? Give your answer to the nearest $cent$ .
Form an equation relating $y$y (the cost of the cereal) to $x$x (the weight of the cereal box in grams).
$James$ sees that a $750$750 gram box of the same cereal is priced at $\$8.18$$8.18. Opting for the larger box, what is the saving per $100$100 grams?

Give your answer to the nearest $cent$ .

Outcomes

MA.7.AR.3.1

Apply previous understanding of percentages and ratios to solve multi-step real-world percent problems.

MA.7.AR.3.2

Apply previous understanding of ratios to solve real-world problems involving proportions.

4.11 Consumer applications of proportional reasoning

Keeping it in proportion