\(\displaystyle \frac{0.93 - 0.90} {0.90}\)

I'm told to find the result as a decimal, to four decimal places, the convert it to a fraction. Ok. Got the first part, the answer is 0.0333. Then I convert it to a fraction. Got that, too: \(\displaystyle \frac{333} {10000}\)

The only problem is, the answer in my book says that the correct fraction is \(\displaystyle \frac{1} {30}\). At face value \(\displaystyle \frac{333} {10000}\) and \(\displaystyle \frac{1} {30}\) are not equivalent fractions. And yet they both equal 0.0333! So how is \(\displaystyle \frac{333} {10000}\) the wrong answer?

I decided to go about it another way---by using equivalent fractions in the original equation instead of decimals, i.e., (93/100 - 90/100) / (90/100); this time I did indeed come up with \(\displaystyle \frac{1} {30}\)!

But I'm still not getting why it is that way. \(\displaystyle \frac{1} {30}\)is exactly equal to 0.0333.... But when you try to then turn that 0.0333 back into a fraction, the resulting fraction is \(\displaystyle \frac{333} {10000}\), which is NOT equal to \(\displaystyle \frac{1} {30}\). And yet... it is equal to the decimal form of \(\displaystyle \frac{1} {30}\).

How can both of those fractions equal 0.0333, and yet... they don't equal each other?

I assume the answer has something to do with the nature of the repeating 3 at the end of the number, but I can't figure out exactly what role it plays in it. There's got to be a practical difference. Suppose I had a piece of string that were 88 "units" long, and I wanted to cut off 1/30 of it. Well I'd then cut off 2.933333 units... but if I were to cut off 333/10000 units of string, then I'd cut off 2.9304 units of the string. So one string would indeed be longer than the other. And yet somehow they would actually be the same?

If anyone can help me understand this, I'd really appreciate it.

Thanks!