a)\(\frac{a}{b}< \frac{c}{d}\Rightarrow\frac{a}{b}.bd< \frac{c}{d}.bd\Rightarrow ad< cb\)(đpcm)
b)Ta có:
ad<cd=>ab+ad<ab+cd
=>a(b+d)<b(b+d)
=>\(\frac{a\left(b+d\right)}{b\left(b+d\right)}< \frac{b\left(a+c\right)}{b\left(b+d\right)}\)
=>\(\frac{a}{b}< \frac{a+c}{b+d}\)(1)
ad<bc=>ad+cd<bc+cd
=>d(a+c)<c(b+d)
=>\(\frac{d\left(a+c\right)}{d\left(b+d\right)}< \frac{c\left(b+d\right)}{d\left(b+d\right)}\)
=>\(\frac{a+c}{b+d}< \frac{c}{d}\)(2)
Từ (1) và (2) => \(\frac{a}{b}< \frac{a+c}{b+d}< \frac{c}{d}\)(đpcm)