To convert the repeating decimal \( 0.123123123\ldots \) into a fraction, we can use the technique for handling repeating decimals.
We can set \( x = 0.123123123\ldots \).
To eliminate the repeating portion, we can multiply by 1000 (since the repeating part is 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, we can solve for \( x \):
\[ x = \frac{123}{999} \]
Next, we simplify the fraction \( \frac{123}{999} \). The greatest common divisor (GCD) of 123 and 999 is 3:
\[ \frac{123 \div 3}{999 \div 3} = \frac{41}{333} \]
However, since this fraction is not one of the provided answer options, we should evaluate the original format given:
Among the answer options provided, \( \frac{123}{999} \) matches the first option.
Thus, the answer is:
123 over 999
1 answer