The decimal representation of $\frac{1}{11}$ is $0.09090909 \ldots$

Another way to write this decimal representation is $0 . \overline{09}$.

Similarly, $0 . \overline{125}$ represents the number $0.125125125 \ldots$

The decimal representation of $\frac{1}{7}$ is $0 . \overline{142857}$.

In the decimal representation of $\frac{1}{7}$, the 100th digit to the right of the decimal is
(A) 1
(B) 4
(C) 2
(D) 8
(E) 5
The digits to the right of the decimal place in the decimal represenation of $\frac{1}{7}$ occur in blocks of 6 , repeating the block of digits 142857 .

Since $16 \times 6=96$, then the 96th digit to the right of the decimal place is the last in one of these blocks; that is, the 96th digit is 7 .

This means that the 97 th digit is 1 , the 98 th digit is 4 , the 99 th digit is 2 , and the 100th digit is 8 .