Given two real numbers $p>1$ and $q>1$ so that $\frac{1}{p} + \frac{1}{q} = 1$ and $pq = 4$, what is $q$?
I notice that the given equations involve fractions and reciprocals, so I wonder if there is a way to simplify them by multiplying or dividing both sides by something. I also notice that the second equation, $pq = 4$, implies that $q = \frac{4}{p}$, so I can substitute that into the first equation and get $\frac{1}{p} + \frac{p}{4} = 1$. Now I have one equation with one unknown, $p$, so I can try to solve for it. To clear the fractions, I multiply both sides by $4p$ and get $4 + p^2 = 4p$. Rearranging the terms, I get $p^2 - 4p + 4 = 0$. This is a quadratic equation, so I can try to factor it or use the quadratic formula. Factoring seems easier in this case, since I can see that $(p-2)^2 = 0$ is equivalent to the equation. Therefore, $p = 2$ is the only solution. To find $q$, I use the equation $q = \frac{4}{p}$ and plug in $p = 2$. I get $q = \frac{4}{2} = 2$. 