Question

Determine a fracão geratriz da dizima de 4,7232323....

Answer 1

x = 4,72323...
10x = 47,2323...
1.000x = 4723,2323...

1.000x - 10x = 4723,2323... - 47,2323...
990x = 4.676
x = 4.676/990
x = 2.338/495

Answer 2

Temos o seguinte:

$4,7232323... = \dfrac{47,232323...}{10} = \dfrac{47}{10} + 0,0232323...$

Assim:

$0,0232323... = \frac{23}{1000} + \frac{23}{100000} ... = \sum \limits^\infty_{k=0} { \frac{23}{100^{k + 1}*10} } = \sum \limits^\infty_{k=0} { \frac{23}{10^{2k+2}*10} } \\ \\ 0,0232323... =\sum \limits^\infty_{k=0} { \frac{23}{10^{2k+3}} }$

Logo:

$\frac{47}{10} + \sum \limits^\infty_{k=0} { \frac{23}{10^{2k+3}} }$

Agora calculando a somatória, usando soma infinita de P.G, com razão q=1/100:

$S = \frac{a_{1}}{1- q} = \dfrac{ \frac{23}{10^3} }{1 - \frac{1}{100} }} = \dfrac{23}{10^3} * \dfrac{10^2}{99} = \boxed{\dfrac{23}{990} }$

Assim temos:

$4,7232323...= \frac{47}{10} + \frac{23}{990} = \dfrac{47*990 + 10*23}{990*10} = \dfrac{46530 + 230}{9900} = \dfrac{46760}{9900} \\ \\ 4,7232323...= \dfrac{4676}{990} = \boxed{\dfrac{2338}{495} }$