Question

Para cada dizima periodica encontre a fração geratriz
1) 0,222...
2) 0,1212...
3) 0,303030...
4) 0,4848...
5) 1,666...
6) 2,333...
7) 0,1222..
8) 0,01222..
9) 2,3636...
10) 1,0555...

Answer 1

$1)~~0,222\dots$

$x=0,222\dots$

$10x=2,222\dots$

$10x-x=2,222\dots-0,222\dots \iff 9x=2 \iff \boxed{x=\dfrac{2}{9}}$


$2)~~0,1212\dots$

$x=0,1212\dots$

$100x=12,1212\dots$

$100x-x=12,1212\dots-0,1212\dots \iff 99x=12 \iff \boxed{x=\dfrac{12}{99}}$


$3)~~0,303030\dots$

$x=0,303030\dots$

$100x=30,303030\dots$

$100x-x=30,3030\dots-0,3030\dots \iff 99x=30$
 
$x=\dfrac{30}{99} \iff \boxed{x=\dfrac{10}{33}}$


$4)~~0,4848\dots$

$x=0,4848\dots$

$100x=48,4848\dots$

$100x-x=48,4848\dots-0,4848\dots \iff 99x=48$

 $x=\dfrac{48}{99} \iff \boxed{x=\dfrac{16}{33}}$


$5)~~1,666\dots$

$x=1,666\dots$

$10x=16,666\dots$

$10x-x=16,666\dots-1,666\dots \iff 9x=15 \iff x=\dfrac{15}{9} \iff \boxed{x=\dfrac{5}{3}}$


$6)~~2,333\dots$

$x=2,333\dots$

$10x=23,333\dots$

$10x-x=23,333\dots-2,333\dots \iff 9x=21 \iff x=\dfrac{21}{9} \iff \boxed{x=\dfrac{7}{3}}$


$7)~~0,1222\dots$

$x=0,1222\dots$

$100x=12,222\dots$

$10x=1,222\dots$

$100x-10x=12,222\dots-1,222\dots \iff 90x=11 \iff \boxed{x=\dfrac{11}{90}}$

$8) 0,01222\dots$

$1000x=12,222\dots$

$100x=1,222\dots$

$1000x-100x=12,222\dots-1,222\dots \iff 900x=11 \iff \boxed{x=\dfrac{11}{900}}$


$9)~~2,3636\dots$

$x=2,3636\dots$

$100x=236,3636\dots$

$100x-x=236,3636\dots-2,3636\dots \iff 99x=234$

$x=\dfrac{234}{99} \iff \boxed{x=\dfrac{78}{33}}$


$10)~~1,0555\dots$

$x=1,0555\dots$

$100x=105,555\dots$

$10x=10,555\dots$

$100x-10x=105,555\dots-10,555\dots \iff 90x=95$
 
$x=\dfrac{95}{90} \iff \boxed{x=\dfrac{19}{18}}$