To determine whether the fraction [tex]\(\frac{6}{9}\)[/tex] produces a repeating decimal, let's go through the following steps:
1. Simplify the Fraction:
Begin by simplifying the fraction [tex]\(\frac{6}{9}\)[/tex]. The greatest common divisor (GCD) of 6 and 9 is 3.
So, we can divide both the numerator and the denominator by 3:
[tex]\[
\frac{6 \div 3}{9 \div 3} = \frac{2}{3}
\][/tex]
2. Convert the Simplified Fraction to a Decimal:
Now, consider the simplified fraction [tex]\(\frac{2}{3}\)[/tex]. To convert [tex]\(\frac{2}{3}\)[/tex] to a decimal, we perform the division [tex]\(2 \div 3\)[/tex]:
[tex]\[
2 \div 3 = 0.6666\ldots
\][/tex]
The decimal representation [tex]\(0.6666\ldots\)[/tex] repeats indefinitely, which can be written as [tex]\(0.\overline{6}\)[/tex].
3. Identify the Repeating Nature:
Because the decimal representation is [tex]\(0.\overline{6}\)[/tex], it indicates that the digit 6 repeats indefinitely.
Given this analysis, we conclude that the fraction [tex]\(\frac{6}{9}\)[/tex] indeed produces a repeating decimal, represented as [tex]\(0.\overline{6}\)[/tex].
Thus, the answer to the question is:
A. True
