Question

Cho a, b, c, d > 0 thỏa mãn \(\frac{a}{b}< \frac{c}{d}\) 

Chứng minh rằng \(\frac{a}{b}< \frac{a+c}{b+d}< \frac{c}{d}\)

Answer 1

\(\frac{a}{b}< \frac{c}{d}\)

\(\Leftrightarrow ad< bc\)

\(\Leftrightarrow ad+ab< bc+ab\)

\(\Leftrightarrow a\left(b+d\right)< b\left(a+c\right)\)

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

\(\frac{a}{b}< \frac{c}{d}\)

\(\Leftrightarrow ad< bc\)

\(\Leftrightarrow ad+cd< bc+cd\)

\(\Leftrightarrow d\left(a+c\right)< c\left(b+d\right)\)

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

Từ (1) ; (2) \(\Rightarrow\frac{a}{b}< \frac{a+c}{b+d}< \frac{c}{d}\) (đpcm)