Question

26. Escreva em forma de fração irredutível as dízi-
mas periódicas
a) 0.272727 ... =
b) 17555.. =. c) 0,291 com traço em cima.​

Answer 1

a)

$a = 0.272727... \\ \\ 100a = 27.272727... \\ \\ 100a - a = 27.2727... - 0.2727... \\ \\ 99a = 27 \\ \\ \frac{99a}{99} = \frac{27}{99} \\ \\ a = \frac{27}{99} = \frac{27 \div 9}{99 \div 9} = \frac{3}{11} \\ \\ a = \frac{3}{11}$

b)

$b = 0.17555... \\ \\ 100b = 17.555... \\ 1000b = 175.555... \\ \\ \\ 1000b - 100b = 175.555... - 17.555... \\ \\ 900b = 158 \\ \\ \frac{900b}{900} = \frac{158}{900} \\ \\ b = \frac{158}{900} = \frac{158 \div 2}{900 \div 2} = \frac{79}{450} \\ \\ b = \frac{79}{450}$

c)

$c = 0.291291... \\ \\ 1000c = 291.291291... \\ \\ 1000c - c = 291.291... - 0.291... \\ \\ 999c = 291 \\ \\ c = \frac{291}{999} = \frac{291 \div 3}{999 \div 3} = \frac{97}{333}$