If  $\displaystyle\frac{a_{1}}{b_{1}},\,\ldots,\,\frac{a_{n}}{b_{n}}$  are any real fractions with positive denominators and

are the least and the greatest of the fractions, then

The equality signs are valid if and only if all fractions are equal; in this case one has

Proof.  Set  $\displaystyle q_{1}:=\frac{a_{1}}{b_{1}}$,  …,  $\displaystyle q_{n}:=\frac{a_{n}}{b_{n}}$.  Then we have  $a_{1}\!+\!\ldots\!+\!a_{n}=b_{1}q_{1}\!+\!\ldots\!+\!b_{n}q_{n}$,  which apparently has the lower bound  $(b_{1}\!+\cdots+\!b_{n})m$  and the upper bound  $(b_{1}\!+\ldots+\!b_{n})M$.  Dividing the three last expressions by the sum  $b_{1}\!+\ldots+\!b_{n}$  yields the asserted double inequality (1).