7. It is known that the fraction $\frac{a}{b}$ is less than the fraction $\frac{c}{d}$ and $b>d>0$. Determine which is smaller: the arithmetic mean of these two fractions or the fraction $\frac{a+c}{b+d}$.
Answer: The fraction $\frac{a+c}{b+d}$ is smaller. Solution. $\frac{a}{b}<\frac{c}{d} \Leftrightarrow a d<b c \Leftrightarrow a d(b-d)<$ $b c(b-d) \Leftrightarrow a b d+b c d<b^{2} c+a d^{2} \Leftrightarrow 2 a b d+2 b c d<a b d+b c d+b^{2} c+a d^{2}$ $\Leftrightarrow 2(a+c) b d<(a d+b c)(b+d) \Leftrightarrow \frac{a+c}{b+d}<\frac{a d+b c}{2 b d} \Leftrightarrow \frac{a+c}{b+d}<\frac{1}{2}\left(\frac{a}{b}+\frac{c}{d}\right)$.