The mediant of two fractions

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The mediant of the fractions $\frac{a}{b}$ and $\frac{c}{d}$ is $\frac{a+c}{b+d}$ (it is how the Stern–Brocot tree is built). Compute the mediant of $\frac{2}{5}$ and $\frac{3}{7}$ and check that it lies between the two.

Solution

$\frac{2+3}{5+7} = \frac{5}{12}.$ Compare with the starting fractions: $\frac{2}{5}=0.400$, $\ \frac{5}{12}\approx 0.4167$, $\ \frac{3}{7}\approx 0.4286$. Indeed $\frac{2}{5} < \boxed{\frac{5}{12}} < \frac{3}{7}.$ The mediant always falls between the two fractions (it is not their average, but it always has this “in-between” property).