Question

Determine a fração que gerou as seguintes dízimas periódicas ( DEMONSTRE TODOS OS CÁLCULOS E SIMPLIFIQUE, SEM POSSÍVEL ) A) 45,1212 B) 81,7373 C)4,0666 D) 83,124124 E) 6,120120 F)0,0505

Answer 1

$A)\\ x=45,\overline{12}\\100x=4\ 512,\overline{12}\\\\ 100x-x=4\ 512,12-45,12\\99x=4467\\x=\frac{4467^{\div3}}{99^{\div3}}=\ \textgreater \ \boxed{x=\frac{1489}{33}}\\\\\\ B)\\ x=81,\overline{73}\\ 100x=8\ 173,\overline{73}\\\\ 100x-x=8173,73-81,73\\ 99x=8092\\\\\boxed{x=\frac{8092}{99}}\\\\\\ C)\\ x=4,0\overline{6}\\ 10x=40,\overline{6}\\ 100x=406,\overline{6}\\\\ 100x-10x=406,6-40,6\\90x=366\\\\ x= \frac{366^{\div3}}{90^{\div3}} =\ \textgreater \ \boxed{x=\frac{122}{30}}$

$D)\\ x=83,\overline{124}\\ 1000x=83\ 124,\overline{124}\\\\ 1000x-x=83\ 124,124-83,124\\ 999x=83041\\\\\boxed{x= \frac{83\ 041}{999} }\\\\\\ E)\\ x=6,\overline{120}\\ 1000x=6\ 120,\overline{120}\\\\ 1000x-x=6120,120-6,120\\ 999x=6114\\\\ x= \frac{6114^{\div3}}{999^{\div3}} =\ \textgreater \ \boxed{x= \frac{2038}{333} }\\\\\\ F)\\ x=0,\overline{05}\\ 100x=5,\overline{05}\\\\ 100x-x=5,05-0,05\\ 99x=5\\\\\boxed{x= \frac{5}{99} }$