Question

1-Encontre a fração geratriz de:
a) 4,6767...
b) 1,4646...
c) 0,315315...
d) 0,8888...

Answer 1

Explicação passo-a-passo:

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a)Temos a seguinte dízima periódica:

$x=\text{4,6767...}$

Multiplicando a dízima por 100 e por 1:

$100x=\text{467,6767...}\\x=\text{4,6767...}$

Subtraindo:

$100x-x=\text{467,6767...}-\text{4,6767...}\\\\99x=463\\\\x=\dfrac{463}{99}\\\\\boxed{\boxed{\text{4,6767...}= \dfrac{463}{99}}}$

b)Temos a seguinte dízima periódica:

$x=\text{1,4646...}$

Multiplicando a dízima por 100 e por 1:

$100x=\text{146,4646...}\\x=\text{1,4646...}$

Subtraindo:

$100x-x=\text{146,4646...}-\text{1,4646...}\\\\99x=145\\\\x=\dfrac{145}{99}\\\\\boxed{\boxed{\text{1,4646...}= \dfrac{145}{99}}}$

c)Temos a seguinte dízima periódica:

$x=\text{0,315315...}$

Multiplicando a dízima por 1000 e por 1:

$1000x=\text{315,315315...}\\x=\text{0,315315...}$

Subtraindo:

$1000x-x=\text{315,315315...}-\text{0,315315...}\\\\999x=315\\\\x=\dfrac{315}{999}=\dfrac{35}{111}\\\\\boxed{\boxed{\text{0,315315...}= \dfrac{35}{111}}}$

d)Temos a seguinte dízima periódica:

$x=\text{0,8888...}$

Multiplicando a dízima por 10 e por 1:

$10x=\text{8,8888...}\\x=\text{0,8888...}$

Subtraindo:

$10x-x=\text{8,8888...}-\text{0,8888...}\\\\9x=8\\\\x=\dfrac{8}{9}\\\\\boxed{\boxed{\text{0,8888...}=\dfrac{8}{9}}}$