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= (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
`(1080)/(99)`.