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Answer :
To determine which of the given fractions equals a repeating decimal, we need to analyze the decimal representation of each fraction. Let's examine each fraction one by one:
1. Fraction A: [tex]\(\frac{1}{25}\)[/tex]
To find the decimal representation, divide 1 by 25:
[tex]\[
\frac{1}{25} = 0.04
\][/tex]
Since [tex]\(0.04\)[/tex] is a terminating decimal (it stops after two decimal places), [tex]\(\frac{1}{25}\)[/tex] does not equal a repeating decimal.
2. Fraction B: [tex]\(\frac{13}{10}\)[/tex]
To find the decimal representation, divide 13 by 10:
[tex]\[
\frac{13}{10} = 1.3
\][/tex]
Since [tex]\(1.3\)[/tex] is a terminating decimal (it stops after one decimal place), [tex]\(\frac{13}{10}\)[/tex] does not equal a repeating decimal.
3. Fraction C: [tex]\(\frac{5}{9}\)[/tex]
To find the decimal representation, divide 5 by 9:
[tex]\[
\frac{5}{9} \approx 0.555\ldots \text{(repeating)}
\][/tex]
The decimal representation of [tex]\(\frac{5}{9}\)[/tex] is [tex]\(0.\overline{5}\)[/tex], which is a repeating decimal.
4. Fraction D: [tex]\(\frac{3}{5}\)[/tex]
To find the decimal representation, divide 3 by 5:
[tex]\[
\frac{3}{5} = 0.6
\][/tex]
Since [tex]\(0.6\)[/tex] is a terminating decimal (it stops after one decimal place), [tex]\(\frac{3}{5}\)[/tex] does not equal a repeating decimal.
After analyzing all the fractions, we see that Fraction C, [tex]\(\frac{5}{9}\)[/tex], is the one that equals a repeating decimal.
Therefore, the answer is:
C [tex]\(\frac{5}{9}\)[/tex]
1. Fraction A: [tex]\(\frac{1}{25}\)[/tex]
To find the decimal representation, divide 1 by 25:
[tex]\[
\frac{1}{25} = 0.04
\][/tex]
Since [tex]\(0.04\)[/tex] is a terminating decimal (it stops after two decimal places), [tex]\(\frac{1}{25}\)[/tex] does not equal a repeating decimal.
2. Fraction B: [tex]\(\frac{13}{10}\)[/tex]
To find the decimal representation, divide 13 by 10:
[tex]\[
\frac{13}{10} = 1.3
\][/tex]
Since [tex]\(1.3\)[/tex] is a terminating decimal (it stops after one decimal place), [tex]\(\frac{13}{10}\)[/tex] does not equal a repeating decimal.
3. Fraction C: [tex]\(\frac{5}{9}\)[/tex]
To find the decimal representation, divide 5 by 9:
[tex]\[
\frac{5}{9} \approx 0.555\ldots \text{(repeating)}
\][/tex]
The decimal representation of [tex]\(\frac{5}{9}\)[/tex] is [tex]\(0.\overline{5}\)[/tex], which is a repeating decimal.
4. Fraction D: [tex]\(\frac{3}{5}\)[/tex]
To find the decimal representation, divide 3 by 5:
[tex]\[
\frac{3}{5} = 0.6
\][/tex]
Since [tex]\(0.6\)[/tex] is a terminating decimal (it stops after one decimal place), [tex]\(\frac{3}{5}\)[/tex] does not equal a repeating decimal.
After analyzing all the fractions, we see that Fraction C, [tex]\(\frac{5}{9}\)[/tex], is the one that equals a repeating decimal.
Therefore, the answer is:
C [tex]\(\frac{5}{9}\)[/tex]
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