problem: Given two real numbers $p>1$ and $q>1$ so that $\frac{1}{p} + \frac{1}{q} = 1$, what is the value of $(p-1)(q-1)$?
solution: Multiplying both sides of the given equation by $p$, we see that $1+\frac{p}{q} = p \Rightarrow \frac{p}{q} = p-1 \Rightarrow p = q(p-1)$. But then $(p-1)(q-1) = q(p-1) - (p-1) = p -p +1 = \boxed{1}$.