Question

Cho  \(\frac{a}{b}=\frac{c}{d}\)

Chứng minh  \(\frac{a}{b-a}=\frac{c}{d-c}\)

Biết  \(\left(a;b;c;d\ne0\right)\)

Answer 1

Đặt \(\frac{a}{b}=\frac{c}{d}=k\)

\(\Rightarrow\hept{\begin{cases}a=bk\\c=dk\end{cases}}\)

Ta có :

\(\frac{a}{b-a}=\frac{bk}{b-bk}=\frac{bk}{b.\left(1-k\right)}=\frac{k}{1-k}\left(1\right)\)

\(\frac{c}{d-c}=\frac{dk}{d-dk}=\frac{dk}{d\left(1-k\right)}=\frac{k}{1-k}\left(2\right)\)

Từ 1 và 2 

=> \(\frac{a}{b-a}=\frac{c}{d-c}\left(ĐPCM\right)\)

Answer 2

Ta có: a/b=c/d => b/a=d/c => b/a-1=d/c-1 hay b/a-a/a=d/c-c/c => b-a/a=d-c/c => a/b-a=c/d-c

Vậy a/b-a=c/d-c