To convert the repeating decimal \( 0.123123123... \) to a fraction, we can use the following steps:
Let \( x = 0.123123123... \).
Since the repeating part "123" has 3 digits, we can multiply \( x \) by \( 1000 \) (which is \( 10^3 \)): \[ 1000x = 123.123123123... \]
Now we can subtract the original equation from this new equation: \[ 1000x - x = 123.123123123... - 0.123123123... \] This simplifies to: \[ 999x = 123 \]
Now, divide both sides by 999 to solve for \( x \): \[ x = \frac{123}{999} \]
Next, we simplify the fraction \( \frac{123}{999} \). Both the numerator and denominator can be divided by their greatest common divisor (GCD), which is 3: \[ \frac{123 \div 3}{999 \div 3} = \frac{41}{333} \]
Thus, the repeating decimal \( 0.123123123... \) as a fraction is: \[ \frac{41}{333} \]