Firts, Let 
be the repeated decimal that we are trying to convert , so 
equation (1)
Next, lets find how many digits are repeating:
It is pretty cleat that 90 is repeating, and 90 has two digits. So we are going to multiply our equation by 100 to move the decimal point two places:
(2)
Subtract equation (1) from equation (2):
Solve for
We can conclude that 10.9090909091... expressed as a rational number, i<span>n the form pq where p and q are positive integers with no common factors, is </span>
.
Next, lets find how many digits are repeating:
It is pretty cleat that 90 is repeating, and 90 has two digits. So we are going to multiply our equation by 100 to move the decimal point two places:
Subtract equation (1) from equation (2):
Solve for
We can conclude that 10.9090909091... expressed as a rational number, i<span>n the form pq where p and q are positive integers with no common factors, is </span>