Question

Express 10.9090909091... as a rational number, in the form pq where p and q are positive integers with no common factors.

Answer 1

Firts, Let $x$ be the repeated decimal that we are trying to convert , so $x=10.9090909090$ 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:
$100x=1090.90909090$ (2)

Subtract equation (1) from equation (2):
$100x-x=1090.90909090-10.90909090$
$99x=1080$

Solve for $x$
$x= \frac{1080}{99}$

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 $\frac{1080}{99}$.