What is 0.09 repeating as a fraction?

Hint: Careful while converting the given problem to fraction. The fraction that are given in the problem is not repeating. Don’t convert 0.09 into fraction. We need to convert the given repeating decimal number that is 0.090909….. into fraction. Here we are going to use geometric progression to solve this.

Complete step-by-step answer:
We denote the repeating decimal by brackets or by a horizontal bar over the decimal.
That is \[0.(09)\] or \[0.\overline {09} \] .
Now we have \[0.(09)\] , it can be written as
 \[0.(09) = 0 + 0.09 + 0.0009 + 0.000009 + - - - \]
That is the fraction can be written as 0 plus the sum of an infinite geometric sequence, for which first term as \[{a_1} = 0.09\] and common ratio \[r = \dfrac{{{a_2}}}{{{a_1}}} = \dfrac{{0.0009}}{{0.09}} = 0.01\] .
We have a formula to calculate the infinite sum of all terms
 \[s = \dfrac{{{a_1}}}{{1 - r}}\]
Substituting we have,
 \[s = \dfrac{{0.09}}{{1 - 0.01}}\]
 \[ = \dfrac{{0.09}}{{0.99}}\]
Multiply numerator and denominator by 100, we have:
 \[ \Rightarrow 0.(09) = \dfrac{9}{{99}}\]
Or
 \[ \Rightarrow 0.(09) = \dfrac{1}{{11}}\] is the required answer.
So, the correct answer is “ \[\dfrac{1}{{11}}\] ”.

Note: We use the geometric series concept to convert the decimal number into a fraction. we use the infinite sum of all terms \[s = \dfrac{{{a_1}}}{{1 - r}}\] , hence by substituting the all the values to the formula we convert the decimal number. While simplification we use simple arithmetic operations.