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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UpStudy AI · Accepted Answer

1. We know that a repeating decimal such as $$0.\overline{n}$$ (where $$n$$ is a single digit) can be written as $$\frac{n}{9}$$. For example:
   - $$0.\overline{1} = \frac{1}{9}$$
   - $$0.\overline{2} = \frac{2}{9}$$
   - $$0.\overline{3} = \frac{3}{9}$$
   - $$0.\overline{4} = \frac{4}{9}$$
   - $$0.\overline{5} = \frac{5}{9}$$
   
2. Now, let's consider $$0.\overline{9}$$. According to the pattern, we might expect:
   $$
   0.\overline{9} = \frac{9}{9} = 1.
   $$

3. Among the given choices:
   - Option A: $$0.9=\frac{8}{9}$$ – Here, $$0.9$$ (a terminating decimal) is not equal to $$\frac{8}{9}$$.
   - Option B: $$0.\overline{9}=\frac{8}{9}$$ – This does not follow the pattern, since $$0.\overline{9}$$ should equal $$\frac{9}{9}$$.
   - Option C: $$0.9=1$$ – Again, $$0.9$$ is not equal to 1.
   - Option D: $$0.\overline{9}=1$$ – This is consistent with the established conversion.

4. Therefore, the correct equation must be:
   $$
   0.\overline{9}=1.
   $$
The correct equation is $$0.\overline{9} = 1$$, which is option D.

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