Question

Cho \(\frac{a}{b}\)>\(\frac{c}{d}\)(b;d\(\in\)N*)

CMR:\(\frac{c}{d}\)<\(\frac{a+c}{b+d}\)<\(\frac{a}{b}\)

Answer 1

Có: \(\frac{a}{b}< \frac{c}{d}\Leftrightarrow ad< bc\)

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

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

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

Có: \(\frac{a}{b}< \frac{c}{d}\Leftrightarrow ad< bc\)

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

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

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

Từ (1) và (2) suy ra: \(\frac{a}{b}< \frac{a+c}{b+d}< \frac{c}{d}\)