Express 10. as a rational number, in the form pq where p and q are positive integers with no common factors.

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, in the form pq where p and q are positive integers with no common factors, is .