Ratios and Rates

NZ Level 6 (NZC) Level 1 (NCEA)

Proportional Relationships as Equations

Lesson

We've already learnt about proportional relationships, where two quantities vary in such a way that one is a constant multiple of the other. In other words, they always vary by the same constant.

Proportional relationships can also be written as linear equations.

Remember!

Proportional relationships can be written generally in the form:

$y=kx$`y`=`k``x`

where $k$`k` is the constant of proportionality

Notice how this general form of proportional relationships looks very similar to the $y=mx+b$`y`=`m``x`+`b` form we use to solve equations. However, one of the features of proportional relationships is that, when graphed, they pass through the origin $\left(0,0\right)$(0,0), which is why there is no $b$`b` term.

So, we can solve proportional relationships like equations to make judgments about the relationship between two variables.

Let's look through some examples now.

Frank serves $2$2 cups of coffee every $4$4 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 that represents this proportional relationship.

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

a) Let $x$`x` represent the amount of white paint and y represent the amount of red paint needed. What is the equation of this line?

b) What does the equation of the line tell you?

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

a) What is the unit price of the cereal per $100$100 grams? Give your answer to the nearest cent.

b) Form an equation relating $y$`y` (the cost of the cereal) to $x$`x` (the weight of the cereal box in grams).

c) 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.

Apply direct and inverse relationships with linear proportions

Apply numeric reasoning in solving problems