<em>Direct variation </em>is some relationship between two variables where scaling one variable by a certain factor scales the other variable by that same factor. For example, in this problem, we're told that "a varies directly as b." What this means is that if we were to double a, b would also double; if we were to triple a, b would triple, and in general, anything we multiply a by we would also multiply b by.
<em>Inverse variation</em> has the opposite effect. If two variables vary inversely, scaling one by some factor scales the other by <em>the reciprocal of that factor</em>. If a varied inversely as c, doubling a would <em>cut </em><em>c </em><em>in half; </em>tripling a would scale c by 1/3, and so on. In this problem, though, a varies inversely as <em>the square of </em><em>c</em>. This means that if we scaled c by 4, a would by scaled by 1/4², or 1/16.
Our question now becomes: what are b and c being scaled by in this problem, and what does that do to a?
To find the scaling factors of b and c, we can take the ratios of their scaled values to their original values. We find that b has been scaled by 4/9, and c has been scaled by 8/6 = 4/3.
For a first step, since a and b vary directly, we can simply scale a's original value of 7 by 4/9 to get a scald value of 7 * (4/9) = 28/9. Next, since a varies inversely as the square of c, we need to scale a by <em>the reciprocal of (4/3)². </em>(4/3)² = 16/9, and the reciprocal of 16/9 is 9/16. Scaling 28/9 by 9/16, we find the final scaled value of a to be
28/9 × 9/16 = 28/16 = 7/4
