Convert each of the following into a vulgar fraction :
`0.1bar(43)`

Answer

To convert the repeating decimal \(0.1\overline{43}\) into a vulgar fraction, follow these steps:

### Step 1: Define the repeating decimal
Let \( x = 0.1\overline{43} \).

### Step 2: Eliminate the repeating part
Since the repeating part "43" has two digits, we will multiply \( x \) by 1000 (to shift the decimal point three places to the right) and by 10 (to shift it one place to the right):
\[ 10x = 1.43\overline{43} \]
\[ 1000x = 143.43\overline{43} \]

### Step 3: Set up the equations
Now we have two equations:
1. \( 10x = 1.43\overline{43} \) (Equation 1)
2. \( 1000x = 143.43\overline{43} \) (Equation 2)

### Step 4: Subtract the equations
Subtract Equation 1 from Equation 2 to eliminate the repeating part:
\[ 1000x - 10x = 143.43\overline{43} - 1.43\overline{43} \]
This simplifies to:
\[ 990x = 142 \]

### Step 5: Solve for \( x \)
Now, divide both sides by 990:
\[ x = \frac{142}{990} \]

### Step 6: Simplify the fraction
To simplify \( \frac{142}{990} \), we find the greatest common divisor (GCD) of 142 and 990. The GCD is 2:
\[ x = \frac{142 \div 2}{990 \div 2} = \frac{71}{495} \]

### Final Answer
Thus, the repeating decimal \( 0.1\overline{43} \) can be expressed as the vulgar fraction:
\[ \frac{71}{495} \]

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