Question

qual a fração geratriz de 0,212121 de 3,777 de 5,111 de 6,353535 e de 1,234234

Answer 1

0,212121

$Seja\;\;x=0.212121\ldots\\\\ 100x = 21.212121\ldots \iff 100x = 21+x\\\\ 100x-x = 21\\ 99x = 21 \implies x = \dfrac{21}{99}$



3,777...

x = 3,777...
10x = 37,777....

$10x - x = 37.777\ldots - 3.777\dots\\\\ 9x = 34\\\\ x = \dfrac{34}{9}$


5.111...

x = 5.111....
10x = 51.111...

$10x - x = 51.111\ldots - 5.111\dots\\\\ 9x = 46\\\\ x = \dfrac{46}{9}$



6.353535...

x = 6.353535...
100x = 635.353535...

$100x - x = 635.3535\ldots - 6.353535\dots\\\\ 99x = 629\\\\ x = \dfrac{629}{99}$



1.234234...

x = 1.234234
1000x = 1234.234234...

$1000x - x = 1234.234234\ldots - 1.234234\dots\\\\ 999x = 1233\\\\ x = \dfrac{1233}{999}$