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?
Thank you so much! 🙂
This has really helped.
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