Q:  How do I express o.403403403403403 etc as a fraction?

A:  Okay, here is the “trick” to this:

Let x = .403403403403…..

Now, how many repeating digits are there?  3, right?  “403” just repeats, so that makes 3 repeating digits.  So, we multiply everything by 1000.  (the number of zeroes should match the number of repeating digits… if there were 4 repeating digits, we would mutlply by 10000….)

So, 3 repeating digits means multiply by 1000:

(1)  x = .403403403403…..

1000*x = 1000*(.403403403403…..)

(2)  1000x = 403.403403403….

Now, subtract equation (1) from equation (2):

1000x = 403.403403403….

–      x = .403403403403…..

999x = 403

Notice all the decimals cancelled out????

So, 999x = 403

x = 403/999

And, x = .403403403403….

So, .403403403403…. = 403/999

Moral of the story?? Whatever is repeating is divided by the same number of 9’s!

.333333333… = 3/9

.44444444… = 4/9

.919191919191… = 91/99

.101101101101… = 101/999

Make sense?