If two quantities <em>a</em> and <em>b</em> are directly proportional to one another, then both <em>a</em> and <em>b</em> change in the <u>same direction</u>.
For example, if the amount/volume of coffee I have in a cup is directly proportional to the total volume of the cup, then the larger the cup gets, the more coffee I can fill it with. On the other hand, if the cup gets smaller, the amount of coffee it can hold also gets smaller.
Both <em>a</em> and <em>b</em> don't necessarily change at the same rate, though, which means one of these needs to get appropriately scaled by some factor <em>k</em>, so that
<em>a</em> = <em>kb</em>
Continuing the coffee example: Suppose I only fill my cup halfway. If the total volume of the cup is <em>a</em>, then the amount of coffee I pour in is <em>b</em> = 1/2 <em>a</em>. So the relationship of these two quantities is governed by the equation,
<em>a</em> = 2<em>b</em>
and in this case, <em>k</em> = 2.
If instead <em>a</em> and <em>b</em> are inversely proportional, this means as one quantity changes, the other changes in the <u>opposite direction</u>.
Suppose I have two cups that I use for coffee that can both hold the same amount <em>k</em> if they are filled completely, but one cup is taller than the other. In order for both cups to be able to hold the same volume of coffee, the taller cup must have a thinner profile.
The cups are cylindrical, so that their volumes are equal to the products of the area of their base <em>a</em> and their height <em>b</em>. If one cup is twice as tall as the other, then for the smaller cup we could have
<em>ab</em> = <em>k</em>
and for the taller one,
<em>a</em> (2<em>b</em>) = <em>k</em>
But in order to get the same volume, the quantity <em>a</em> for the shorter cup must be cut in half to preserve equality:
(1/2 <em>a</em>) (2<em>b</em>) = <em>ab</em> = <em>k</em>
So one quantity is doubled (increases), while the other gets halved (decreases).
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