Question

Chứng minh rằng nếu \(\frac{a}{b}< \frac{c}{d}\left(b>0,d>0\right)\)thì\(\frac{a}{b}< \frac{a+c}{b+d}< \frac{c}{d}\)

Answer 1

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

\(\Rightarrow ad< bc\)

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

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

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

Lại có : ad < bc

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

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

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

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