When 0.090909… is converted into a fraction, then the result is

Let x=0.090909… --- (Equation 1)
Since two digits (09) are repeating continuously, multiply both sides by 100:
$$100x=9.090909\dots$ --- (Equation 2)

Subtract Equation 1 from Equation 2:

100x−x=(9.090909…)−(0.090909…)
99x=9
x=
99
9
​

Simplify the fraction by dividing the numerator and denominator by their greatest common divisor (9):

x=
11
1
​