Question

Determine os valores das dízimas periódicas:

a) 25,362525... ( )

b) 31,423636... ( )

c) 0,0564242... ( )​

Answer 1

a)

$x=25.362525\dots\\ 100x=2536.2525\dots \\ 10000x=253525.2525\dots\\\\ 10000x-100x=253525.2525\dots -2536.2525\dots\\\\ 9900x=251089\ \therefore\ \boxed{x=\dfrac{251089}{9900}}$

b)

$x=31.423636\dots\\ 100x=3142.3636\dots\\ 10000x=314236.3636\dots\\\\ 10000x-100x=314236.3636\dots-3142.3636\dots\\\\ 9900x=311094\ \therefore\ \boxed{x=\dfrac{311094}{9900}}$

c)

$x=0.0564242\dots\\ 1000x=56.4242\dots\\ 100000x=5642.4242\dots\\\\ 100000x-1000x=5642.4242\dots-56.4242\dots\\\\ 99000x=5586\ \therefore\ \boxed{x=\dfrac{5586}{99000}}$