To determine which fractions have a repeating decimal equivalent, we need to find out if the fraction in its simplest form results in a non-terminating, repeating decimal. A fraction [tex]\(\frac{a}{b}\)[/tex] is a terminating decimal if and only if in its simplest form, the denominator [tex]\(b\)[/tex] contains no prime factors other than 2 or 5. Conversely, if [tex]\(b\)[/tex] contains any prime factors other than 2 or 5, the fraction will have a repeating decimal.
Let's analyze each fraction step-by-step:
1. Fraction: [tex]\( \frac{13}{65} \)[/tex]
- Simplify the fraction:
[tex]\[
\text{gcd}(13, 65) = 13, \quad \frac{13 \div 13}{65 \div 13} = \frac{1}{5}
\][/tex]
- The simplified denominator is 5, which contains only the prime factor 5.
- Therefore, [tex]\(\frac{13}{65}\)[/tex] is a terminating decimal.
2. Fraction: [tex]\( \frac{141}{47} \)[/tex]
- Simplify the fraction:
[tex]\[
\text{gcd}(141, 47) = 1, \quad \text{So, it is already in simplest form.}
\][/tex]
- The denominator 47 contains the prime factor 47.
- Since 47 is neither 2 nor 5, this fraction will have a repeating decimal.
3. Fraction: [tex]\( \frac{11}{12} \)[/tex]
- Simplify the fraction:
[tex]\[
\text{gcd}(11, 12) = 1, \quad \text{So, it is already in simplest form.}
\][/tex]
- The denominator 12 has prime factors [tex]\(2^2 \times 3\)[/tex].
- Since 12 contains a prime factor other than 2 or 5 (specifically 3), this fraction will have a repeating decimal.
4. Fraction: [tex]\( \frac{19}{3} \)[/tex]
- Simplify the fraction:
[tex]\[
\text{gcd}(19, 3) = 1, \quad \text{So, it is already in simplest form.}
\][/tex]
- The denominator 3 contains the prime factor 3.
- Since 3 is not 2 or 5, this fraction will have a repeating decimal.
In summary, fractions [tex]\(\frac{141}{47}\)[/tex], [tex]\(\frac{11}{12}\)[/tex], and [tex]\(\frac{19}{3}\)[/tex] have repeating decimal equivalents. Therefore, the fractions that have repeating decimals are:
[tex]\[ \frac{11}{12}, \frac{19}{3} \][/tex]
