Question

[Nguyen Hai Bang]

Chứng minh rằng nếu a/b<c/d(b, d>0) thì: a/b<a+c/b+d<c/d

Answer 1

[Le Thi Khanh Huyen]

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

Có:

• \(\frac{ab+ad}{b\left(b+d\right)}< \frac{ab+bc}{b\left(b+d\right)}\)

\(\Rightarrow\frac{a\left(b+d\right)}{b\left(b+d\right)}< \frac{b\left(a+c\right)}{b\left(b+d\right)}\)

\(\Rightarrow\frac{a}{b}< \frac{a+c}{b+d}\left(1\right)\)

• \(\frac{ad+cd}{d\left(b+d\right)}< \frac{bc+cd}{d\left(b+d\right)}\)

\(\Rightarrow\frac{d\left(a+c\right)}{d\left(b+d\right)}< \frac{c\left(b+d\right)}{d\left(b+d\right)}\)

\(\Rightarrow\frac{a+c}{b+d}< \frac{c}{d}\left(2\right)\)

\(\left(1\right)\left(2\right)\Rightarrow\frac{a}{b}< \frac{a+c}{b+d}< \frac{c}{d}\)