Question

Rewrite as a simplified fraction.

\[\large{0.\overline{80} = {?}}\]

Answer 1

[AI-generated answer — Bot GPT 3.5]

To rewrite the decimal $0.\overline{80}$ as a fraction, we need to figure out what the repeating block of digits is. We can do this by subtracting a finite version of the decimal from the infinite version of the decimal.

Let $n = 0.\overline{80}$. Multiply both sides of this equation by $100$ to get $100n = 80.\overline{80}$.

Subtracting these two equations, we have:

\begin{align*}
(100n) - n &= (80.\overline{80}) - (0.\overline{80})\\
99n &= 80 \\
n &= \frac{80}{99}
\end{align*}

Therefore, the simplified fraction representation of $0.\overline{80}$ is $\boxed{\frac{80}{99}}$.