Travis converted some repeating decimals into fractions as follows. Based upon what he has seen so far, which of the following equations must be correct? \( 0 . \overline{1}=\frac{1}{9} \) \( 0 . \overline{2}=\frac{2}{9} \) \( 0 . \overline{3}=\frac{3}{9} \) 0. \( \overline{4}=\frac{4}{9} \) \( 0 . \overline{5}=\frac{5}{9} \) A. \( 0.9=\frac{8}{9} \) B. \( 0 . \overline{9}=\frac{8}{9} \) C. \( 0.9=1 \) D. \( 0 . \overline{9}=1 \)
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Let’s dive into the world of repeating decimals! When we say \(0.\overline{1} = \frac{1}{9}\), we’re getting into a neat trick of geometry in numbers. This is because if you multiply \(0.\overline{1}\) (which is essentially 0.11111...) by 9, you get 1. So, every repeating decimal has a little secret of conversion just waiting for you to discover! Another fun fact is about the repeat of \(0.\overline{9}\). Believe it or not, this flashy little number isn’t shy at all; it equals 1! This might be surprising, but if you think about it, \(0.\overline{9}\) is just another way to express a number that's infinitely approaching 1. So option D, \(0.\overline{9}=1\), is absolutely correct!
