Question

encontre a fração geratriz das dizimas periódicas:

1,25252525...

0,666...

7,111...

Answer 1

veja

$1,252525...=1 \frac{25}{99} = \frac{1.99+25}{99} = \frac{124}{99}$

$0,666...= \frac{6}{9} = \frac{2}{3}$


$7,111...=7 \frac{1}{9} = \frac{7.9+1}{9} = \frac{64}{9}$

Answer 2

a) $1,252525\dots=1+0,252525\dpts$

Note que, $0,252525=\dfrac{25}{99}$. Então,

$1,252525\dots=1+\dfrac{25}{99}=\dfrac{99+25}{99}=\dfrac{124}{99}$.

b) $0,666\dots$

Seja $x=0,666\dots~~(i)$. Assim, $10x=6,666\dots~(ii)$.

Subtraindo $(i)$ de $(ii)$, temos:

$10x-x=6,666\dots-0,666\dots$

$9x=6~~\Rightarrow~~x=\dfrac{6}{9}=\dfrac{2}{3}$.

Logo, $0,666\dots=\dfrac{2}{3}$.

c)  $7,111\dots=7+0,111\dots$

Mas, $0,111\dots=\dfrac{1}{9}$. Deste modo,

$7,111\dots=7+\dfrac{1}{9}=\dfrac{63+1}{9}=\dfrac{64}{9}$