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## 'Mediant' printed from http://nrich.maths.org/

If you take two tests and get $a$ marks out of a maximum $b$ in the
first, scoring ${a\over b}$, and you do better with $c$ marks out
of $d$ in the second, scoring ${c\over d}$, you expect the result
of combining the marks for the two tests, the mediant ${{a+c}\over
{b+d}}$, to lie between the results for the two tests separately.
If 0 < ${a\over b}$ < ${c\over d} $ prove, with reference to
the diagram, that $${a\over b} < {{a+c}\over {b+d}} < {c\over
d}.$$