Question

How do you find the repeating decimal 0.45 with 45 repeated as a fraction?

Answer 1

`0.bar(45) = 5/11`

Explanation:

First, in case you have not encountered it before, note that you can indicate a repeating decimal by placing a bar over the repeating pattern:

`0.bar(45) = 0.45454545...`

With that notation, see what happens when we multiply `0.bar(45)` by `(100-1)`:

`(100-1)0.bar(45) = 100*0.bar(45) - 1*0.bar(45)`

`color(white)((100-1)0.bar(45)) = 45.bar(45) - 0.bar(45)`

`color(white)((100-1)0.bar(45)) = 45`

The `100` shifted our original decimal representation `2` places to the left - the length of the repeating pattern.

Then subtracting the original cancelled out the repeating tail.

Next, divide both ends by `(100-1)` and simplify to find:

`0.bar(45) = 45/(100-1) = 45/99 = (5*color(red)(cancel(color(black)(9))))/(11*color(red)(cancel(color(black)(9)))) = 5/11`