Convert each of the following into a vulgar fraction :
`0.bar8`

Answer

To convert the repeating decimal \( 0.\overline{8} \) into a vulgar fraction, we can follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 0.\overline{8} \).

### Step 2: Multiply by 10
Since the decimal repeats every one digit, we multiply both sides of the equation by 10 to shift the decimal point one place to the right:
\[ 10x = 8.\overline{8} \]

### Step 3: Set up the subtraction
Now we have two equations:
1. \( x = 0.\overline{8} \)
2. \( 10x = 8.\overline{8} \)

Next, we subtract the first equation from the second:
\[ 10x - x = 8.\overline{8} - 0.\overline{8} \]

### Step 4: Simplify the subtraction
This simplifies to:
\[ 9x = 8 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 9 to solve for \( x \):
\[ x = \frac{8}{9} \]

### Conclusion
Thus, the repeating decimal \( 0.\overline{8} \) can be expressed as the vulgar fraction:
\[ 0.\overline{8} = \frac{8}{9} \]

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