Question

Cho a, b, c > 0 và \(\dfrac{a}{b}\) < 1
CMR: \(\dfrac{a}{b}\) < \(\dfrac{a+c}{b+c}\)

Answer 1

\(\dfrac{a}{b}=\dfrac{a\left(b+c\right)}{b\left(b+c\right)}=\dfrac{ab+ac}{b^2+bc}\)

\(\dfrac{a+c}{b+c}=\dfrac{b\left(a+c\right)}{b\left(b+c\right)}=\dfrac{ba+bc}{b^2+bc}\)

Do \(ab=ba;ac< bc\) do \(\dfrac{a}{b}< 1\) hay \(a< b\)

\(\Rightarrow ab+ac< bc+ba\)

\(Vậy\) \(\dfrac{a}{b}< \dfrac{a+c}{b+c}\) \(\left(đpcm\right)\)