Question

Cho a,b,c,d thuộc n^*  thỏa  mãn a/b <c/d

CMR :2019a+c/2019b+d<c/d

Answer 1

Vì \(\frac{a}{b}< \frac{c}{d}\Rightarrow ad< bc\)

Quy đồng mẫu hai phân số \(\frac{2019a+c}{2019b+d}\) và \(\frac{c}{d}\)

\(\frac{2019a+c}{2019b+d}=\frac{d\left(2019a+c\right)}{d\left(2019b+d\right)}=\frac{2019ad+cd}{2019bd+d^2}\)

\(\frac{c}{d}=\frac{c\left(2019b+d\right)}{d\left(2019b+d\right)}=\frac{2019bc+2019cd}{2019bd+d^2}\)

Vì ad < bc nên 2019ad + cd < 2019bc + 2019cd => \(\frac{2019a+c}{2019b+d}< \frac{c}{d}\)(đpcm)