Prove: $\frac{a+c}{b+d}$ lies between $\frac{a}{b}$ and $\frac{c}{d}$ (for positive $a$, $b$, $c$, $d$) I am looking for proof that, if you take any two different fractions and add the numerators together then the denominators together, the answer will always be a fraction that lies between the two original fractions.
Would be grateful for any suggestions!

Hint $\ $ The mediant $(a\!+\!b)/(c\!+\!d)\,$ is the slope of the diagonal of the parallelogram with vector sides $(b,a),\ (d,c).\:$ But the slope of the diagonal lies between the slopes of the sides.
