Problem Statement
(a) First, determine two positive integers $x$ and $y$ with $\dfrac{2x + 11y}{3x + 4y} = 1$.
Now, let $u$ and $v$ be two positive rational numbers with $u<v$. If we write $u$ and $v$ as fractions $u = \dfrac{a}{b}$ and $v = \dfrac{c}{d}$, not necessarily in lowest terms and with $a$, $b$, $c$, $d$ positive integers, then the fraction $\dfrac{a + c}{b + d}$ is called a “mediant” of $u$ and $v$. Since $u$ and $v$ can be written in many different forms, there are many different mediants of $u$ and $v$. In (a), you showed that 1 is a mediant of $\dfrac{2}{3}$ and $\dfrac{11}{4}$.
(b) Prove that the average of $u$ and $v$, namely $\dfrac{1}{2} (u+v)$, is a mediant of $u$ and $v$.
(c) Prove that every mediant, $m$, of $u$ and $v$ satsifies $u < m < v$.