Bài 1: Cho tỉ lệ thức: $\frac{a}{b}$ = $\frac{c}{d}$. Chứng minh: a) $\frac{a}{a + b}$ = $\frac{c}{c + d}$ b) $\frac{a + c}{a}$ = $\frac{b

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1. Đáp án:

   $\begin{array}{l}
   Dat:\dfrac{a}{b} = \dfrac{c}{d} = k\\
    \Rightarrow \left\{ \begin{array}{l}
   a = b.k\\
   c = d.k
   \end{array} \right.\\
   a)\dfrac{a}{{a + b}} = \dfrac{{b.k}}{{b.k + b}} = \dfrac{{b.k}}{{b\left( {k + 1} \right)}} = \dfrac{k}{{k + 1}}\\
   \dfrac{c}{{c + d}} = \dfrac{{d.k}}{{d.k + d}} = \dfrac{{d.k}}{{d\left( {k + 1} \right)}} = \dfrac{k}{{k + 1}}\\
   Vay\,\dfrac{a}{{a + b}} = \dfrac{c}{{c + d}}\\
   b)\dfrac{{a + c}}{a} = \dfrac{a}{a} + \dfrac{c}{a} = 1 + \dfrac{c}{a}\\
   \dfrac{{b + d}}{b} = \dfrac{b}{b} + \dfrac{d}{b} = 1 + \dfrac{d}{b}\\
   Do:\dfrac{a}{b} = \dfrac{c}{d}\\
    \Rightarrow \dfrac{c}{a} = \dfrac{d}{b}\\
    \Rightarrow 1 + \dfrac{c}{a} = 1 + \dfrac{d}{b}\\
    \Rightarrow \dfrac{{a + c}}{a} = \dfrac{{b + d}}{d}\\
   c)\dfrac{{a – b}}{{c – d}} = \dfrac{{bk – b}}{{dk – d}} = \dfrac{{b\left( {k – 1} \right)}}{{d\left( {k – 1} \right)}} = \dfrac{b}{d}\\
   \dfrac{{a + c}}{{b + d}} = \dfrac{{bk + dk}}{{b + d}} = \dfrac{{k\left( {b + d} \right)}}{{b + d}} = k\\
   Do:\dfrac{b}{d} \ne k\\
    \Rightarrow \dfrac{{a – b}}{{c – d}} \ne \dfrac{{a + c}}{{b + d}}
   \end{array}$

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