[tex]\displaystyle\bf\\\boxed{\bf~TEOREMA~:~\frac{a}{b} = \frac{c}{d} \Leftrightarrow ad=bc}~.\\\\------------------\\\\\frac{2x}{5y} =0,4,~transformam~pe~0,4~in~fractie~ordinara.\\\\\boxed{\bf 0,4=\frac{4}{10}=\frac{2}{5}}~.\\\\\frac{2x}{5y}=\frac{2}{5} \Leftrightarrow 5\cdot2x=5\cdot2y~~\bigg{|}:(2\cdot5) \implies \boxed{\bf x=y }~.\\asadar,~valoarea~raportului~\frac{6y}{5x+y},~este~:~\frac{6y}{5y+y} = \frac{6y}{6y} = \boxed{\bf 1}~.[/tex]
[tex]\displaystyle\bf\\daca~"4=",~din~enunt~reprezenta~"4-"~sau~"4+",~atunci~avem~:~\\\\1.~~\frac{4x+6y}{5x+y} = \frac{10x}{6x} = \frac{10}{6} = \boxed{\bf\frac{5}{3}}~.\\\\2.~~\frac{4x-6y}{5x+y} = -\frac{2x}{6x}=-\frac{2}{6}=\boxed{\bf-\frac{1}{3}}~.\\\\daca~"4="~nu~reprezenta~nimic,~vezi~prima~varianta.[/tex]
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[tex]\boxed{\bf \dfrac{a}{b} = \dfrac{c}{d} \Leftrightarrow a\cdot d = b\cdot c}[/tex]
[tex]\bf \dfrac{2x}{5y} = \dfrac{4}{10}= \dfrac{\not4}{\not10}= \dfrac{2}{5} \implies 5\cdot 2x = 2\cdot 5y \implies 10x = 10y\:\:\Big|:10 \implies\boxed{\bf x = y}[/tex]
[tex]\bf \dfrac{4x-6y}{5x+y} \Leftrightarrow \dfrac{4x-6x}{5x+x}= \dfrac{-2x}{6x} =\boxed{\bf -\dfrac{1}{3} }[/tex]
[tex]\bf \dfrac{4x+6y}{5x+y} \Leftrightarrow \dfrac{4x+6x}{5x+x}= \dfrac{10x}{6x} =\boxed{\bf \dfrac{5}{3} }[/tex]
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