If you take two tests and get a marks out of a maximum b in the first, scoring ^a/_b, and you do better with c marks out of d in the second, scoring ^c/_d , you expect the result of combining the marks for the two tests, the mediant

a+c b+d

, to lie between the results for the two tests separately. If 0 < ^a/_b < ^c/_d prove, with reference to the diagram, that

a b < a+c b+d < c d .