The vulgar fraction of `0.39bar39` is:

Options

A. `(15)/(33)`
B. `(11)/(39)`
C. `(17)/(39)`
D. `(13)/(33)`

Answer

To convert the repeating decimal \(0.39\overline{39}\) into a vulgar fraction, we can follow these steps:

### Step 1: Define the repeating decimal
Let \(x = 0.39\overline{39}\). This means \(x = 0.39393939...\).

### Step 2: Multiply by a power of 10
To eliminate the repeating part, we multiply \(x\) by \(100\) (since the repeating part has two digits):
\[ 100x = 39.39393939... \]
This can be considered as equation (1).

### Step 3: Multiply by a larger power of 10
Next, we multiply \(x\) by \(10000\) to shift the decimal point further:
\[ 10000x = 3939.39393939... \]
This can be considered as equation (2).

### Step 4: Subtract the two equations
Now we subtract equation (1) from equation (2):
\[ 10000x - 100x = 3939.39393939... - 39.39393939... \]
This simplifies to:
\[ 9900x = 3900 \]

### Step 5: Solve for \(x\)
Now, we can solve for \(x\):
\[ x = \frac{3900}{9900} \]

### Step 6: Simplify the fraction
To simplify \(\frac{3900}{9900}\), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is \(300\):
\[ x = \frac{3900 \div 300}{9900 \div 300} = \frac{13}{33} \]

### Conclusion
Thus, the vulgar fraction of \(0.39\overline{39}\) is:
\[ \frac{13}{33} \]

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