For consecutive Farey fractions we have $$\frac{a}{b}, \frac{a}{b} + frac{c}{d} the mediant is obtained via a "simple's man addition":$$ \frac{c}{d} frac{p}{q} = \frac{a+c}{b+d} $$which since \frac{a}{b},\frac{c}{d} are consecutive if and only if det\begin{pmatrix}a & c\\ b & d\end{pmatrix} = 1 also turns out to be the rule of invariance of the determinant when you add a column to another column. 1 [made Community Wiki] For consecutive Farey fractions we have$$ \frac{a}{b} + \frac{c}{d} = \frac{a+c}{b+d}  which since $\frac{a}{b},\frac{c}{d}$ are consecutive if and only if $det\begin{pmatrix}a & c\\ b & d\end{pmatrix} = 1$ also turns out to be the rule of invariance of the determinant when you add a column to another column.