The prior lesson
Getting Bigger? Getting Smaller?
introduces the concepts of direct and inverse variation.
Study it first, being sure to clickclickclick several exercises at the bottom to check your understanding.
This current lesson builds on these prior concepts.
The following are equivalent:
When you are told that ‘$y\,$ is proportional to $x\,$’ then direct variation is being described.
To emphasize this fact, the extra word ‘directly’ can be inserted:
‘$y\,$ is directly proportional to $\,x$’.
Recall that
variables that are directly proportional ‘follow each other in size’:
when one gets bigger (farther away from zero), so does the other;
when one gets smaller (closer to zero), so does the other.
Notice: if $\,y\,$ is proportional to $\,x\,$, then $\,x\,$ is proportional to $\,y$.
Thus, we can simply say ‘$x\,$ and $\,y$ are proportional’ or
‘$y\,$ and $\,x$ are proportional’.
Why is this? Study the following list of equivalent sentences:
$y\,$ is proportional to $\,x\,$  given; assumed to be true 
$y = kx\,$, for $\,k\ne 0$  an equivalent statement of direct variation; see the list above 
$x = \frac 1k y\,$, for $\,\frac 1k\ne 0$  multiplication property of equality (divide both sides
by $\,k\ne 0\,$); also, $\,k\,$ is nonzero if and only if its reciprocal is nonzero 
$x\,$ is proportional to $\,y\,$  an equivalent statement of direct variation; see the list above 
Also notice:
Next, we talk about inverse variation:
The following are equivalent:
Recall that
variables that are inversely proportional have sizes that ‘go in opposite directions’:
when one gets bigger, the other gets smaller;
when one gets smaller, the other gets bigger.
Similar to the argument above: if $\,y\,$ is inversely proportional to $\,x\,$, then $\,x\,$ is inversely proportional to $\,y$.
Thus, we can simply say ‘$x\,$ and $\,y$ are inversely proportional’ or
‘$y\,$ and $\,x$ are inversely proportional’.
Also notice:
Sometimes it is necessary to talk about a relationship between more than two variables:
The following are equivalent:
By combining phrases, a wide variety of relationships can be described between variables.
For example:
On this exercise, you will not key in your answer. However, you can check to see if your answer is correct. 
PROBLEM TYPES:
