Question

Daca x/y=2/3 sa se calculeze 7x/7x+y =

Answer 1

O variantă:

$\dfrac{x}{y} = \dfrac{2}{3}\\ \\ \dfrac{7x}{7x+y} = \dfrac{u}{v}\Big|^{-1} \Rightarrow \dfrac{7x+y}{7x} = \dfrac{v}{u} \Rightarrow \dfrac{7x}{7x}+\dfrac{y}{7x} = \dfrac{v}{u} \Rightarrow \\ \\ \Rightarrow 1 +\dfrac{y}{7x} = \dfrac{v}{u} \overset{(*)}{\Rightarrow} \\ \\ \boxed{\dfrac{x}{y} = \dfrac{2}{3}\Big|\cdot 7 \Rightarrow \dfrac{7x}{y} = \dfrac{14}{3}\Big|^{-1} \Rightarrow \dfrac{y}{7x}= \dfrac{3}{14} }$

$\overset{(*)}{\Rightarrow }1 +\dfrac{3}{14} = \dfrac{v}{u} \Rightarrow \dfrac{14+3}{14} = \dfrac{v}{u} \Rightarrow \dfrac{17}{14} = \dfrac{v}{u} \Big|^{-1} \Rightarrow\dfrac{14}{17} = \dfrac{u}{v} \Rightarrow \\ \\ \Rightarrow \boxed{\boxed{\dfrac{7x}{7x+y} = \dfrac{14}{17}}}$

Altă variantă:

$\dfrac{x}{y} = \dfrac{2}{3} \Rightarrow x = \dfrac{2}{3}\cdot y \Rightarrow x = \dfrac{2y}{3} \\ \\ \dfrac{7x}{7x+y} = \dfrac{7\cdot \dfrac{2y}{3}}{7\cdot \dfrac{2y}{3}+y} = \dfrac{\dfrac{14y}{3}}{\dfrac{14y}{3}+\dfrac{3y}{3}} = \dfrac{\dfrac{14y}{3}}{\dfrac{14y+3y}{3}} = \dfrac{\dfrac{14y}{3}}{\dfrac{17y}{3}} = \dfrac{14y}{3}\cdot \dfrac{3}{17y} = \\ \\ = \dfrac{14}{17}$