The decimal \( 0.123123123\ldots \) is a repeating decimal. To convert it to a fraction, we can set it equal to a variable:
Let \( x = 0.123123123\ldots \)
To eliminate the repeating part, we can multiply both sides by 1000 (since the repeating part has three digits):
\[ 1000x = 123.123123123\ldots \]
Now we have two equations:
- \( x = 0.123123123\ldots \)
- \( 1000x = 123.123123123\ldots \)
Next, we can subtract the first equation from the second:
\[ 1000x - x = 123.123123123\ldots - 0.123123123\ldots \]
This simplifies to:
\[ 999x = 123 \]
Now, solving for \( x \):
\[ x = \frac{123}{999} \]
We can simplify \( \frac{123}{999} \):
Both 123 and 999 can be divided by 3:
\[ \frac{123 \div 3}{999 \div 3} = \frac{41}{333} \]
Thus, the fraction representation of \( 0.123123123\ldots \) is:
\[ \frac{41}{333} \]
However, based on your provided options, the closest answer representing this repeating decimal is:
123 over 999.
This is technically not fully simplified, but accurately represents the fraction form of the repeating decimal.