Convert each of the following into a vulgar fraction :
`0.12bar(34)`

Answer

To convert the decimal \(0.12\overline{34}\) into a vulgar fraction, we can follow these steps:

### Step 1: Define the Decimal
Let \( x = 0.12\overline{34} \). This means \( x = 0.1234343434...\) where \(34\) is the repeating part.

### Step 2: Eliminate the Repeating Part
Since the repeating part consists of 2 digits (34), we multiply \( x \) by \( 100 \) to shift the decimal point two places to the right:
\[ 100x = 12.34343434... \]

### Step 3: Set Up the Equation
Now we have two equations:
1. \( x = 0.12343434...\)
2. \( 100x = 12.34343434...\)

### Step 4: Subtract the First Equation from the Second
Subtract the first equation from the second:
\[ 100x - x = 12.34343434... - 0.12343434... \]
This simplifies to:
\[ 99x = 12.22 \]

### Step 5: Convert the Decimal to a Fraction
Now, we need to convert \( 12.22 \) into a fraction. We can express \( 12.22 \) as:
\[ 12.22 = \frac{1222}{100} \]
Thus, we have:
\[ 99x = \frac{1222}{100} \]

### Step 6: Solve for \( x \)
Now, divide both sides by \( 99 \):
\[ x = \frac{1222}{9900} \]

### Step 7: Simplify the Fraction
To simplify \( \frac{1222}{9900} \), we can find the greatest common divisor (GCD) of \( 1222 \) and \( 9900 \). The GCD is \( 2 \):
\[ x = \frac{1222 \div 2}{9900 \div 2} = \frac{611}{4950} \]

### Final Answer
Thus, the decimal \( 0.12\overline{34} \) can be expressed as the vulgar fraction:
\[ \frac{611}{4950} \]

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