Question
Question
Change the decimal to a fraction: 0.123123123….(1 point)
Responses
1231000
123 over 1000
123100
1 and 23 over 100
123999
123 over 999
This decimal is Irrational and cannot be written as a fraction
This decimal is Irrational and cannot be written as a fraction
121310
12 and 13 tenths
Skip to navigation
Answers
GPT-4o mini
To convert the repeating decimal \( 0.123123123\ldots \) to a fraction, we can use the following method:
Let \( x = 0.123123123\ldots \).
Since the repeating portion is "123" which has 3 digits, we multiply both sides of the equation by \( 1000 \) to move the decimal point three places to the right:
\[
1000x = 123.123123123\ldots
\]
Now we have two equations:
1. \( x = 0.123123123\ldots \)
2. \( 1000x = 123.123123123\ldots \)
Now, we can subtract the first equation from the second:
\[
1000x - x = 123.123123123\ldots - 0.123123123\ldots
\]
This simplifies to:
\[
999x = 123
\]
Now, solve for \( x \):
\[
x = \frac{123}{999}
\]
To simplify \( \frac{123}{999} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3:
\[
x = \frac{123 \div 3}{999 \div 3} = \frac{41}{333}
\]
Thus, the repeating decimal \( 0.123123123\ldots \) can be expressed as the fraction \( \frac{41}{333} \).
When checking the options given in your question, we can see that none of the provided options seem correct since the simplified answer \( \frac{41}{333} \) does not match any of them directly.
Therefore, the correct fraction for the repeating decimal \( 0.123123123\ldots \) is:
\[
\frac{41}{333}
\]
Let \( x = 0.123123123\ldots \).
Since the repeating portion is "123" which has 3 digits, we multiply both sides of the equation by \( 1000 \) to move the decimal point three places to the right:
\[
1000x = 123.123123123\ldots
\]
Now we have two equations:
1. \( x = 0.123123123\ldots \)
2. \( 1000x = 123.123123123\ldots \)
Now, we can subtract the first equation from the second:
\[
1000x - x = 123.123123123\ldots - 0.123123123\ldots
\]
This simplifies to:
\[
999x = 123
\]
Now, solve for \( x \):
\[
x = \frac{123}{999}
\]
To simplify \( \frac{123}{999} \), we can divide both the numerator and the denominator by their greatest common divisor (GCD), which is 3:
\[
x = \frac{123 \div 3}{999 \div 3} = \frac{41}{333}
\]
Thus, the repeating decimal \( 0.123123123\ldots \) can be expressed as the fraction \( \frac{41}{333} \).
When checking the options given in your question, we can see that none of the provided options seem correct since the simplified answer \( \frac{41}{333} \) does not match any of them directly.
Therefore, the correct fraction for the repeating decimal \( 0.123123123\ldots \) is:
\[
\frac{41}{333}
\]