Question

Change the decimal to a fraction: 0.123123123….
Responses:

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

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:

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, 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.