A) \[\frac{419}{990}\]
B) \[\frac{419}{999}\]
C) \[\frac{417}{990}\]
D) \[\frac{417}{999}\]
Correct Answer: A
Solution :
We have \[0.\,4\overset{\,\,\,.\,\,\,.\,\,\,}{\mathop{23\,}}\,=0.4232323.......\] \[=0.4+0.023+0.00023+0.0000023+........\infty \] \[=\frac{4}{10}+\frac{23}{{{10}^{3}}}+\frac{23}{{{10}^{5}}}+\frac{23}{{{10}^{7}}}........\infty \] \[=\frac{4}{10}+\frac{23}{{{10}^{3}}}\left[ 1+\frac{1}{{{10}^{2}}}+\frac{1}{{{10}^{4}}}+..........\infty \right]\] \[=\frac{4}{10}+\frac{23}{1000}\left( \frac{1}{1-\frac{1}{{{10}^{2}}}} \right)=\frac{4}{10}+\frac{23}{990}=\frac{419}{990}\].
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