Question

Convert 0.09¯¯¯¯¯ to a rational number in simplest form. Show your work.

Answer 1

To convert the repeating decimal \( 0.09\overline{9} \) into a rational number, we can follow these steps.

Let \( x = 0.09\overline{9} \). This means that \( x = 0.099999...\).

To isolate the repeating part, we can multiply \( x \) by a power of 10 that moves the decimal point to the right, placing the repeating part as the leading part.

1. **Multiply by 10**:
\[
10x = 0.99999...
\]

2. **Now, create another equation by multiplying \( x \) by 100**:
\[
100x = 9.99999...
\]

Now, we have two equations:
- \( 10x = 0.99999... \) (1)
- \( 100x = 9.99999... \) (2)

Next, we can subtract equation (1) from equation (2).

3. **Subtract the two equations**:
\[
100x - 10x = 9.99999... - 0.99999...
\]
\[
90x = 9
\]

4. **Solve for \( x \)**:
\[
x = \frac{9}{90}
\]

5. **Simplify the fraction**:
\[
x = \frac{1}{10}
\]

Thus, the rational number in simplest form that represents \( 0.09\overline{9} \) is \(\frac{1}{10}\).