Question

Change the decimal to a fraction: 0.123123123.... (1 point)
123/1000
12/13/100
0 1 23/100
0 128/999
O This decimal is Irrational and cannot be written as a fraction

Answer 1

To convert the repeating decimal 0.123123123... into a fraction, let's denote it by \( x \):

\[
x = 0.123123123\ldots
\]

Notice that the repeating part "123" has a length of 3 digits. We can multiply \( x \) by 1000 (since "123" has three digits) to shift the decimal point three places to the right:

\[
1000x = 123.123123123\ldots
\]

Now, we can set up the equation by subtracting the original \( x \) from this new equation:

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

This simplifies to:

\[
999x = 123
\]

Next, we solve for \( x \):

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

Now, we can simplify the fraction \( \frac{123}{999} \) by finding the greatest common divisor (GCD) of 123 and 999, which is 3:

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

Therefore, the repeating decimal \( 0.123123123\ldots \) can be expressed as the fraction \( \frac{41}{333} \).

None of the provided answer options match this, but from the calculations, we can confidently say that \( 0.123123123\ldots \) is a rational number. If you meant to ask for the closest or something specific, please clarify.