Question

Encontre as frações geratrizes das dízimas periódicas a seguir:
(a) 3,777...
(b) 0,2555...
(c) -12,181818...
(d) 4,01313...

Answer 1

$\Large\boxed{\begin{array}{l}\rm a)\\\sf x=3,777\dotsc\cdot10\\\sf 10x=37,777\dotsc\\-\underline{\begin{cases}\sf 10x=37,777\dotsc\\\sf x=3,777\dotsc\end{cases}}\\\sf 9x=34\\\sf x=\dfrac{34}{9}\checkmark\end{array}}$

$\Large\boxed{\begin{array}{l}\rm b)\\\sf x=0,2555\dotsc\cdot10\\\sf 10x=2,555\dotsc\cdot10\\\sf 100x=25,555\dotsc\\-\underline{\begin{cases}\sf100x=25,555\dotsc\\\sf 10x=2,555\dotsc\end{cases}}\\\sf 90x=23\\\sf x=\dfrac{23}{90}\checkmark\end{array}}$

$\Large\boxed{\begin{array}{l}\rm c)\\\sf x=12,181818\dots\cdot100\\\sf 100x=1218,181818\dotsc\\-\underline{\begin{cases}\sf 100x=1218,181818\dotsc\\\sf x=12,181818\dotsc\end{cases}}\\\sf 99x=1206\\\sf x=\dfrac{1206\div9}{99\div9}\\\\\sf x=\dfrac{134}{11}\\\sf -12,181818\dotsc=-\dfrac{134}{11}\checkmark\end{array}}$

$\Large\boxed{\begin{array}{l}\rm d)~\sf x=4,01313\dotsc\cdot10\\\sf 10x=40,131313\dotsc\cdot100\\\sf 1000x=4013,131313\dotsc\\-\underline{\begin{cases}\sf 1000x=4013,131313\dotsc\\\sf 10x=40,131313\dotsc\end{cases}}\\\sf 990x=3973\\\sf x=\dfrac{3973}{990}\checkmark\end{array}}$