Question

Use long division to find a repeated decimal equivalency for the fraction. Include all of your work as part of your final answer.


8/9

Answer 1

To find the decimal equivalency of the fraction \( \frac{8}{9} \) using long division, we will divide 8 by 9.

### Step 1: Set up the long division
We will divide 8 by 9. Since 8 is less than 9, we will write 8 as 8.000000 (adding decimal places for the division).

### Step 2: Perform the long division
1. **Divide**: 9 goes into 8 zero times. We write 0 above the line and add a decimal point.
\[
0.
\]
We now look at 8.0 (or 80 after we bring down the decimal).

2. **Multiply**: 0 times 9 is 0.
\[
9 \times 0 = 0
\]
We write this 0 below 8:
\[
8.0
\]
Subtract:
\[
8 - 0 = 8
\]

3. **Bring down the next digit**: We bring down the next 0 (from 8.0), making it 80.

4. **Again divide**: Now, how many times does 9 go into 80? It goes in 8 times. We write 8 above the line:
\[
0.8
\]
We now multiply:
\[
9 \times 8 = 72
\]
Writing this below 80:
\[
80 - 72 = 8
\]

5. **Repeat**: We bring down another 0, making it again 80, and then repeat the steps.

Each time, we find that:
\[
9 \text{ goes into } 80 \text{ (again) } 8 \text{ times, leading to } 72.
\]
We can see that we will continually return to 8 remaining, continually producing another 8 in the division.

### Conclusion
Since the steps above repeat, we find that the decimal representation of \( \frac{8}{9} \) is:
\[
0.\overline{8}
\]
This notation \( \overline{8} \) indicates that the digit 8 repeats indefinitely.

**Final answer:**
\[
\frac{8}{9} = 0.\overline{8}
\]