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

To convert the repeating decimal \(0.123123123...\) into a fraction, we can denote it as \(x\):

\[
x = 0.123123123...
\]

Notice that the decimal repeats every three digits. To eliminate the repeating part, we can multiply \(x\) by \(1000\):

\[
1000x = 123.123123123...
\]

Now, we have two equations:

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

Next, we can subtract the first equation from the second:

\[
1000x - x = 123.123123123... - 0.123123123...
\]

This simplifies to:

\[
999x = 123
\]

Now, divide both sides by \(999\):

\[
x = \frac{123}{999}
\]

We can simplify the fraction \( \frac{123}{999} \). The greatest common divisor (GCD) of \(123\) and \(999\) is \(3\):

\[
\frac{123 \div 3}{999 \div 3} = \frac{41}{333}
\]

Therefore, the fraction that represents the repeating decimal \(0.123123123...\) is:

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

None of the options provided other than a direct representation of \( x \) and the simplified format in fractional form, so there isn't a correct choice listed in your responses.

However, if we look for approximately or closest fractions based on available options you provided, **none of them accurately represent the value** of \(0.123123123...\) as \( \frac{41}{333} \).