To convert the repeating decimal \( 0.123123123\ldots \) to a fraction, we can use the following method:
Let \( x = 0.123123123\ldots \).
Since the repeating portion is "123" which has 3 digits, we multiply both sides of the equation by \( 1000 \) to move the decimal point three places to the right:
\[ 1000x = 123.123123123\ldots \]
Now we have two equations:
- \( x = 0.123123123\ldots \)
- \( 1000x = 123.123123123\ldots \)
Now, we can subtract the first equation from the second:
\[ 1000x - x = 123.123123123\ldots - 0.123123123\ldots \]
This simplifies to:
\[ 999x = 123 \]
Now, solve for \( x \):
\[ x = \frac{123}{999} \]
To simplify \( \frac{123}{999} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3:
\[ x = \frac{123 \div 3}{999 \div 3} = \frac{41}{333} \]
Thus, the repeating decimal \( 0.123123123\ldots \) can be expressed as the fraction \( \frac{41}{333} \).
When checking the options given in your question, we can see that none of the provided options seem correct since the simplified answer \( \frac{41}{333} \) does not match any of them directly.
Therefore, the correct fraction for the repeating decimal \( 0.123123123\ldots \) is:
\[ \frac{41}{333} \]
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