﻿ Direct and Inverse Variation
GETTING BIGGER? GETTING SMALLER?
(Direct and Inverse Variation)
PRACTICE (online exercises and printable worksheets)

Bigger means farther away from zero and smaller means closer to zero.
(This is discussed in more detail in a future section.)

Suppose that $\,y = 2x\,$.
When $\,x\,$ gets bigger, $\,y\,$ gets bigger.
When $\,y\,$ gets bigger, $\,x\,$ gets bigger.
In this type of relationship, $\,x\,$ and $\,y\,$ ‘follow each other’ in size:
when one gets bigger, so does the other. When one gets smaller, so does the other.

This kind of relationship between two variables is called direct variation:
if there is a nonzero number $\,k\,$ for which $\,y = kx\,$, then we say that ‘$\,y\,$ varies directly as $\,x\,$’.

Now suppose that $\,y = \frac{2}{x}\,$.
When $\,x\,$ gets bigger, $\,y\,$ gets smaller.
When $\,x\,$ gets smaller, $\,y\,$ gets bigger.
In this type of relationship, $\,x\,$ and $\,y\,$ have sizes that go in different directions:
when one gets bigger, the other gets smaller. When one gets smaller, the other gets bigger.

This kind of relationship between two variables is called inverse variation:
if there is a nonzero number $\,k\,$ for which $\displaystyle \,y = \frac{k}{x}\,$, then we say that ‘$\,y\,$ varies inversely as $\,x\,$’.

EXAMPLES:
Question: Consider the formula $\,PV = nRT\,$.
As $\,T\,$ gets bigger, what happens to $\,V\,\,$?
(Assume all other variables are held constant.)
Solution: $V\,$ gets bigger
There is a direct relationship between $\,T\,$ and $\,V\,$.
As $\,T\,$ gets bigger, so does $\,V\,$.
Intuition: Both variables are ‘upstairs’ on opposite sides of the equation.
Question: Consider the formula $\,P = \frac{nRT}{V}\,$.
As $\,P\,$ gets bigger, what happens to $\,V\,\,$?
(Assume all other variables are held constant.)
Solution: $V\,$ gets smaller
There is an inverse relationship between $\,P\,$ and $\,V\,$.
As $\,P\,$ gets bigger, $\,V\,$ gets smaller.
Intuition: One variable is ‘upstairs’ and the other ‘downstairs’ on opposite sides of the equation.
Master the ideas from this section