Question

Se X=0,22222...e Y=011111..., as frações geratrizes de X e Y são, respectivamente

(A) 1/2 e 1/1
(B) 2/9 e 1/9
(C) 2/1 e 1/2
(D) 2/10 1/10

Answer 1

$\mathrm{\mathbf{(I)}\ x=0,222...=0,\bar{2}}\\ \mathrm{\mathbf{(II)}\ 10x=2,222...=2,\bar{2}}\\ \mathrm{Subtraindo\ (I)\ de\ (II)\ \to\ 10x-x=2,\bar{2}-0,\bar{2}\ \to\ 9x=2\ \to\ \boxed{\mathrm{x=\dfrac{2}{9}}}}$

$\mathrm{\mathbf{(I)}\ y=0,111...=0,\bar{1}}\\ \mathrm{\mathbf{(II)}\ 10y=1,111...=1,\bar{1}}\\ \mathrm{Subtraindo\ (I)\ de\ (II)\ \to\ 10y-y=1,\bar{1}-0,\bar{1}\ \to\ 9y=1\ \to\ \boxed{\mathrm{y=\dfrac{1}{9}}}}$

$\therefore$

$\textbf{Resposta:}\ \mathrm{(B)\ \dfrac{2}{9}\ \ e\ \ \dfrac{1}{9}}$