4.07 Identifying the constant of proportionality
We have already learned about proportional relationships, where two variables vary in such a way that one is a constant positive multiple of the other. In other words, they always vary by the same constant. We call this constant the constant of proportionality.
Proportional relationships are always in the form $y=kx$y=kx. We know that $k$k represents the multiplicative factor. However, it also represents the constant of proportionality. When we graph these relationships, they produce straight lines with positive slopes that always pass through the origin, $\left(0,0\right)$(0,0).
Apples: $5$5 for $\$3$$3
For example, let's say a shop is selling apples for $\$3$$3.
We know that five apples will cost $\$3$$3, ten apples will cost $\$6$$6, fifteen apples will cost $\$9$$9 and so on.
We know that the price will increase at a constant rate.
We could graph these two variables in a table.
| Number of apples ($x$x) | $0$0 | $5$5 | $10$10 | $15$15 |
| Cost ($y$y) | $0$0 | $3$3 | $6$6 | $9$9 |
Since we know $5$5 apples cost $\$3$$3, we can work out how much one apple costs:
$3\div5=0.6$3÷5=0.6
This means that each apple costs $60$60 cents and we can say that this is the constant of proportionality.
Further, we can write this as an equation: $y=0.6x$y=0.6x.
The constant of proportionality is always positive.
Since proportional relationships are in the form $y=kx$y=kx, we can also calculate the constant of proportionality ($k$k) by rearranging this equation and we find:
$k=\frac{y}{x}$k=yx
Practice questions
Question 2
In the following proportionality table, the second row is obtained by multiplying the top row by the constant of proportionality. Complete the table and find that constant.
Complete the table:
$7$7 $8$8 $10$10 $\editable{}$ $\editable{}$ $72$72 $\editable{}$ $117$117 What is the constant of proportionality?
question 3
Fred is making batches of bread rolls. He knows he can make $60$60 bread rolls in $10$10 hours, and $120$120 bread rolls in $20$20 hours.
What is the constant of proportionality?
question 4
The graph shown represents the amount of money Kenneth earns.
Identify the constant of proportionality of the graphed line.