What is $6 \div 0.\overline{6}$?
First let's convert $0.\overline{6}$ to a fraction. Let $p=0.\overline{6}$ and multiply both sides of this equation by 10 to obtain $10p=6.\overline{6}$. Subtracting the left-hand sides $10p$ and $p$ as well as the right-hand sides $6.\overline{6}$ and $0.\overline{6}$ of these two equations gives $9p=6$, which implies $p=2/3$. We divide $6$ by $2/3$ to get $$6 \div \frac{2}{3} = \cancelto{3}{6}\hspace{1mm} \cdot \frac{3}{\cancel{2}} = \boxed{9}.$$
9
Level 4
Prealgebra
Problem: What is $6 \div 0.\overline{6}$?
Solution:<|START_OF_LATENT|><|PRIOR_PREFIX|>To solve the problem, we first need to understand the value of $0.\overline{6}$. This notation represents a repeating decimal, which can be expressed as a fraction. 

Let \( x = 0.\overline{6} \). Then, multiplying both sides by 10 gives us \( 10x = 6.\overline{6} \). By subtracting the original equation from this new equation, we have:

\[
10x - x = 6.\overline{6} - 0.\overline{6}
\]
\[
9x = 6
\]

Dividing both sides by 9 yields:

\[
x = \frac{6}{9} = \frac{2}{3}
\]

Now we can substitute this value back into the original problem. We need to compute \( 6 \div 0.\overline{6} \), which is equivalent to \( 6 \div \frac{2}{3} \). Dividing by a fraction is the same as multiplying by its reciprocal:

\[
6 \div \frac{2}{3} = 6 \times \frac{3}{2}
\]

Calculating this gives:

\[
6 \times \frac{3}{2} = \frac{18}{2} = 9
\]

Thus, the final answer is \( 9 \).<|END_OF_LATENT|> The final answer is $9$.