Đáp án:
\[\begin{array}{l}
\text{Đặt}:\dfrac{a}{b} = \dfrac{c}{d} = k\\
\Rightarrow \left\{ \begin{array}{l}
a = b.k\\
c = d.k
\end{array} \right.\\
1)\dfrac{{a - b}}{{a + b}} = \dfrac{{b.k - b}}{{bk + b}} = \dfrac{{b.\left( {k - 1} \right)}}{{b.\left( {k + 1} \right)}} = \dfrac{{k - 1}}{{k + 1}}\\
\dfrac{{c - d}}{{c + d}} = \dfrac{{dk - d}}{{dk + d}} = \dfrac{{d\left( {k - 1} \right)}}{{d\left( {k + 1} \right)}} = \dfrac{{k - 1}}{{k + 1}}\\
\Rightarrow \dfrac{{a - b}}{{a + b}} = \dfrac{{c - d}}{{c + d}}\\
2)\dfrac{a}{{a + b}} = \dfrac{{bk}}{{bk + b}} = \dfrac{k}{{k + 1}}\\
\dfrac{c}{{c + d}} = \dfrac{{dk}}{{dk + d}} = \dfrac{k}{{k + 1}}\\
\Rightarrow \dfrac{a}{{a + b}} = \dfrac{c}{{c + d}}\\
3)\dfrac{{a - b}}{a} = \dfrac{a}{a} - \dfrac{b}{a} = 1 - \dfrac{b}{a}\\
\dfrac{{c - d}}{c} = 1 - \dfrac{d}{c}\\
Do:\dfrac{a}{b} = \dfrac{c}{d} \Rightarrow \dfrac{b}{a} = \dfrac{d}{c}\\
\Rightarrow 1 - \dfrac{b}{a} = 1 - \dfrac{d}{c}\\
\Rightarrow \dfrac{{a - b}}{a} = \dfrac{{c - d}}{c}\\
4)\dfrac{{3a + 5b}}{{3a - 5b}} = \dfrac{{3.bk + 5b}}{{3bk - 5b}} = \dfrac{{3b + 5}}{{3b - 5}}\\
\dfrac{{3c + 5d}}{{3c - 5d}} = \dfrac{{3b + 5}}{{3b - 5}}\\
\Rightarrow \dfrac{{3a + 5b}}{{3a - 5b}} = \dfrac{{3c + 5d}}{{3c - 5d}}
\end{array}\]
Câu 5 tương tự câu 4
