Question

Change the decimal to a fraction: 0.123123123….

options:

123 over 999

123 over 1000

1 and 23 over 100

This decimal is Irrational and cannot be written as a fraction

12 and 13 over 10

Answer 1

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:

1. \( x = 0.123123123\ldots \)
2. \( 1000x = 123.123123123\ldots \)

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**