Suppose we have some positive quantites $P$ and $Q$ which depend on some choices that we make, and we want to show that some choice makes the quotient $P/Q$ fall below some cool bound.
One idea is to make our choices randomly in some way and show that $P/Q$ is small on average. That is, we can use the trivial bound $$ \min P/Q \leq \mathbf{E}[P/Q]. $$ But this inequality is unlikely to be of much use, because we still have to compute $P/Q$ for a random choice. A much more useful inequality arises as follows. Observe that $$ \mathbf{E}[P - Q\mathbf{E}[P]/\mathbf{E}[Q]] = 0, $$ whence $$ \min P/Q \leq \mathbf{E}[P]/\mathbf{E}[Q]. $$ This inequality is much more likely to be useful because now we can compute expectations first and then take the quotient. Moreover, in some cases this will be even give a better bound than the other inequality.
I'm not looking so much for a detailed explanation of what's going on in this specific inequality, but rather for general intuition. Is this just a trick? How can other tricks like this be anticipated?
(Setting: In proving the discrete Cheeger inequality, $P$ is the number of edges coming out of a subset of a graph and $Q$ is the minimum of the size of the subset and the size of its complement, but this question is about general technique and not this specific problem.)
