Mediant and arithmetic mean of several fractions

Question

Suppose there are several fractions $\frac{a_1}{b_1}, \ldots, \frac{a_n}{b_n}$ and $0 < \frac{a_i}{b_i} \leq 1$ for $1 \leq i \leq n$

Define the Mediant of the above fractions

$M = \frac{a_1 + \cdots + a_n}{b_1 + \cdots + b_n}$

and the arithmetic mean of the fractions

$A = \displaystyle \frac{1}{n} \sum_{i=1}^n \frac{a_i}{b_i}$

(1) Is it always true that $A \geq M$? If yes, please provide a standard reference. If not, please provide a counterexample.

(2) What happens to $A$ and $M$ if $a_i = b_i$ in $n \alpha$ cases and $a_i = 0.1b_i$ in $n(1-\alpha)$ cases where $\alpha = 0.7, 0.8, 0.9$? Can we say something at least qualitatively?

Answer 1

Most definitely no. Try $1\over2$ and ${999\over1000}\approx1$. Then $A\approx{0.75}$ and $M\approx1$. Or maybe you want them the other way around? Then try $1\over2$ and $999\over1\,000\,000$.

See that? A fraction with huge numerator and denominator may appear anywhere on $(0,1)$; it will dominate the mediant, but not the average.

Answer 2

Take the two fractions $\frac{1}{2}$ and $\frac{3}{4}$. Then $A=\frac{5}{8}\approx 0.63$ and $M=\frac{2}{3}\approx 0.67$, so $A<M$.