To convert the repeating decimal \( 0.14\overline{14} \) into a fraction, we can follow these steps:
Let \( x = 0.14\overline{14} \).
First, multiply \( x \) by 100 to shift the decimal point two places to the right:
\[ 100x = 14.14\overline{14} \]
Next, multiply \( x \) by 10000 to shift the decimal point four places to the right:
\[ 10000x = 1414.14\overline{14} \]
Now, subtract the first equation from the second:
\[ 10000x - 100x = 1414.14\overline{14} - 14.14\overline{14} \]
This simplifies to:
\[ 9900x = 1400 \]
Now, solving for \( x \):
\[ x = \frac{1400}{9900} \]
Next, we can simplify the fraction:
\[ \frac{1400 \div 100}{9900 \div 100} = \frac{14}{99} \]
Thus, the repeating decimal \( 0.14\overline{14} \) as a fraction is \( \frac{14}{99} \).
Now based on your options regarding \( 0.14\overline{14} \):
- \( \frac{7}{50} \)
- \( \frac{14}{101} \)
- \( \frac{7}{500} \)
- \( \frac{14}{99} \)
None of the provided options match \( \frac{14}{99} \).
If you meant \( 0.14\) (which is not repeating), that would be a different case. Since we were processing \( 0.14\overline{14} \), the correct answer is not present in your responses. However, if it was \( 0.14\) alone, the answer would be \( \frac{14}{100} = \frac{7}{50}\).
Please clarify if you were asking about a different repeating decimal.