Convert 0.09¯¯¯¯¯ to a rational number in simplest form. Show your work. (1 point)

Answer 1

Let $x=0.09\bar{9}$, meaning $x=0.09999999\dots$.

To eliminate the repeating decimal, we can multiply $x$ by 10:

$$10x=0.99999999\dots$$

Next, we can set up a second equation for $x$:

$$x=0.09999999\dots$$

Now, we can subtract the first equation from the second:

$$10x-x=0.99999999\dots -0.09999999\dots$$

This simplifies to:

$$9x=0.9$$

Next, we can solve for $x$:

$$x=\frac{0.9}{9}$$ $$x=\frac{9}{90}$$

Now, we simplify $\frac{9}{90}$:

$$\frac{9}{90}=\frac{1}{10}$$

Thus, the rational number in simplest form that represents $0.09\bar{9}$ is

$$\boxed{{\displaystyle \frac{1}{10}}}$$