Question

encontre a fração geratriz de cada dízima periódica


a)0,2... b)1,16... c)0,125​...

Answer 1

a)

$f = 0.2222... \\ 10f = 2.2222... \\ \\ 10f - f = (2.2222...) - (0.2222...) \\ 9f = 2 \\ \\ \frac{9f}{9} = \frac{2}{9} \\ \\ f = \frac{2}{9}$

b)

$g = 1.1616... \\ 100g = 116.1616... \\ \\ 100g - g = (116.1616...) - (1.1616...) \\ 99g = 115 \\ \\ \frac{99g}{99} = \frac{115}{99} \\ \\ g = \frac{115}{99}$

c)

$k = 0.125125... \\ 1000k = 125.125125... \\ \\ 1000k - k = 125 \\ 999k = 125 \\ \\ \frac{999k}{999} = \frac{125}{999} \\ \\ k = \frac{125}{999}$