Question

Change the decimal to a fraction: 0.123123123….

Answer 1

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:
\[
\frac{41}{333}
\]