Question

[Lưu Nhật Khánh Ly]

Cho a/b <c/d Chứng minh rằng : \(\frac{a}{b}<\frac{a+c}{b+d}<\frac{c}{d}\)  (b,d>0)

Answer 1

[LxP nGuyỄn hÒAnG vŨ]

có a/b <c/d=>a.d<b.c
=>ad+ab<bc+ab(cộng 2 vế với ab)
=>a(b+d)<b(a+c)
=>a/b<a+c/b+d(đpcm)

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li-ke cho mình nhé bnLưu Nhật Khánh Ly

Answer 2

[Trần Đức Thắng]

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

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

\(\frac{a}{b}=\frac{a\left(b+d\right)}{b\left(b+d\right)}=\frac{ab+ad}{b\left(b+d\right)}\)

\(\frac{a+c}{b+d}=\frac{b\left(a+c\right)}{b\left(b+d\right)}=\frac{ab+bc}{b\left(b+d\right)}\)

ad < bc => ab +ad  < ab + bc 

=> \(\frac{a}{b}<\frac{a+c}{b+d}\)

CM tương tự \(\frac{a+c}{b+d}<\frac{c}{d}\)