Proportionality
This is a term that simply seeks to identify the relationship between variables. The type of
proportionality helps us to understand how the variables relate to each other. There are three
general types:
•
•
•
Linear (simple)
Direct
Indirect/Inverse
Linear Proportionality
With a linear relationship, there is a regular increase or decrease of one variable with respect to
the other. Put another way, the two variables depend on each other BUT NOT EXCLUSIVELY.
Example
Let Mr. Barrett’s height be represented by ‘y’ and your height be represented by ‘x’. Further let
us say that Mr. Barrett’s height can be calculated by multiplying your height by 1.5 then adding
0.5 metres.
Putting the above into equation form we would get:
𝑦 = 1.5𝑥 + 0.5
which you should recognise as a form of the equation of a straight line:
𝑦 = 𝑚𝑥 + 𝑐
What that equation tells us is that as your height increases or decreases, Mr. Barrett’s height
would do the same. In the event that you cease to exist however, Mr. Barrett will still have a
height. What would be the value of his height in this case?
The simple proportionality relationship can be represented graphically as shown below:
𝑦2
𝑦1
𝑚=
c
𝑥1
𝑥2
𝑦2 − 𝑦1
𝑥2 − 𝑥1
Direct Proportionality
With direct proportionality the two variables depend on each other EXCLUSIVELY. One can not
exist without the other.
Example
Similar to the example above, let’s now say that the height of Mr. Barrett is simply 1.5 times
your height. In this case if there is no you, Mr. Barrett also ceases to exist. The graph in this case
will look as shown below.
𝑦2
𝑦1
𝑚=
𝑥1
𝑦2 − 𝑦1
𝑥2 − 𝑥1
𝑥2
A graph of direct proportionality is a straight line which passes through the origin. In other
words, the intercept is equal to zero. A direct proportional relationship is expressed in the
following manner:
𝑦∝𝑥
this is a relationship
This statement only tells us the nature of the relationship. It can’t be used to actually solve for
values An equation is required to do this. To change a relationship to an equation a constant of
proportionality is required. The above statement becomes:
𝑦 = 𝑚𝑥
m – constant of proportionality
Inverse/Indirect Proportionality
Two variables are inversely proportional when as one increases the other decreases. This is
represented graphically below.
This relationship is expressed in the following way:
𝑦∝
1
𝑥
As an equation it is:
𝑦=𝑘×
1 𝑘
=
𝑥 𝑥