Elementary and Intermediate Algebra: Concepts & Applications (6th Edition)

Because each digit of $0.1\bar 1$ is exactly one-third of the corresponding digit in $0.3\bar 3$, $0.1\bar 1$ is exactly 1/3 of $0.3\bar 3$, so $0.1\bar 1=(1/3)*(1/3)=1/9$.