\(\dfrac{a}{b}=\dfrac{c}{d}\)
\(\Rightarrow ad=bc\)
\(\Rightarrow ac+bc=ad+bc\)
\(\Rightarrow c.\left(a+b\right)=a.\left(c+d\right)\)
\(\Rightarrow\dfrac{a}{a+b}=\dfrac{c}{c+d}\)
\(\dfrac{a}{b}=\dfrac{c}{d}\)
\(\Rightarrow\dfrac{b}{a}=\dfrac{d}{c}\)
\(\Rightarrow\dfrac{b}{a}+1=\dfrac{d}{c}+1\)
\(\Rightarrow\dfrac{b+a}{a}=\dfrac{d+c}{c}\)
\(\Rightarrow\dfrac{a}{a+b}=\dfrac{c}{c+d}\left(đpcm\right)\)
\(\dfrac{a}{b}=\dfrac{c}{d}\)
\(\Leftrightarrow\dfrac{a}{b}+1=\dfrac{c}{d}+1\)
\(\Leftrightarrow\dfrac{a}{b}+\dfrac{b}{b}=\dfrac{c}{d}+\dfrac{d}{d}\)
\(\Leftrightarrow\dfrac{a+b}{b}=\dfrac{c+d}{d}\)
\(\Rightarrowđpcm\)
Còn cách nữa :v
Đặt:
\(\dfrac{a}{b}=\dfrac{c}{d}=k\)
\(\Rightarrow\left\{{}\begin{matrix}a=bk\\c=dk\end{matrix}\right.\)
\(\dfrac{a}{a+b}=\dfrac{bk}{bk+b}=\dfrac{bk}{b\left(k+1\right)}=\dfrac{k}{k+1}\)
\(\dfrac{c}{c+d}=\dfrac{dk}{dk+d}=\dfrac{dk}{d\left(k+1\right)}=\dfrac{k}{k+1}\)
\(\Rightarrow\dfrac{a}{a+b}=\dfrac{c}{c+d}\Rightarrowđpcm\)
