Question

1) Transforme as dízimas periódicas simples em fração geratriz:

a) 0,222222...
b) 0,252525...
c) 0,363636...
d) 0,476476476...
e) 0,41666...​

Answer 1

Resposta:

Explicação passo-a-passo:

a) x = 0,222...

   10x = 2,222...

10x - x = 2,222... - 0,222... => 9x = 2 => x = 9/2

b) x = 0,2525...

   100x = 25,2525...

100x - x = 25,25... - 0,2525... => 99x = 25 => x = 25/99

c) x = 0,3636...

100x = 36,3636...

100x - x = 36,3636... - 0,3636... => 99x = 36 => x = 36/99 = 4/11

d) x = 0,476476...

1000x = 476,476476...

1000x - x = 476,476476... - 0,476476... => 999x = 476 => x= 476/999

Answer 2

Explicação passo-a-passo:

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a)Temos a seguinte dízima periódica:

$x=\text{0,2222...}$

Multiplicando a dízima por 10 e por 1:

$10x=\text{2,2222...}\\x=\text{0,2222...}$

Subtraindo:

$10x-x=\text{2,2222...}-\text{0,2222...}\\\\9x=2\\\\x=\dfrac{2}{9}\\\\\boxed{\boxed{\text{0,2222...}= \dfrac{2}{9}}}$

b)Temos a seguinte dízima periódica:

$x=\text{0,2525...}$

Multiplicando a dízima por 100 e por 1:

$100x=\text{25,2525...}\\x=\text{0,2525...}$

Subtraindo:

$100x-x=\text{25,2525...}-\text{0,2525...}\\\\99x=25\\\\x=\dfrac{25}{99} \\\\\boxed{\boxed{\text{0,2525...}=\dfrac{25}{99}}}$

c)Temos a seguinte dízima periódica:

$x=\text{0,3636...}$

Multiplicando a dízima por 100 e por 1:

$100x=\text{36,3636...}\\x=\text{0,3636...}$

Subtraindo:

$100x-x=\text{36,3636...}-\text{0,3636...}\\\\99x=36\\\\x=\dfrac{36}{99}=\dfrac{6}{11}\\\\\boxed{\boxed{\text{0,3636...}= \dfrac{6}{11}}}$

d)Temos a seguinte dízima periódica:

$x=\text{0,476476...}$

Multiplicando a dízima por 1000 e por 1:

$1000x=\text{476,476476...}\\x=\text{0,476476...}$

Subtraindo:

$1000x-x=\text{476,476476...}-\text{0,476476...}\\\\999x=476\\\\x=\dfrac{476}{999} \\\\\boxed{\boxed{\text{0,476476...}=\dfrac{476}{999}}}$

e)Temos a seguinte dízima periódica:

$x=\text{0,41666...}$

Multiplicando a dízima por 1000 e por 100:

$1000x=\text{416,666...}\\100x=\text{41,666...}$

Subtraindo:

$1000x-100x=\text{416,666...}-\text{41,666...}\\\\900x=375\\\\x=\dfrac{375}{900}=\dfrac{125}{300}=\dfrac{5}{12} \\\\\boxed{\boxed{\text{0,41666...}=\dfrac{5}{12}}}$