Answer :
Sure, I'll provide a step-by-step explanation for each fraction.
### 13. [tex]\(\frac{2}{9}\)[/tex]
Repeating or Terminating: A fraction is repeating if its denominator in its simplest form has any prime factor other than 2 or 5. The prime factor of 9 is 3.
Decimal Expansion:
To find the decimal expansion, divide 2 by 9.
[tex]\[ \frac{2}{9} = 0.\overline{2} \][/tex]
So the decimal expansion is [tex]\( 0.222222... \)[/tex] which repeats.
### 14. [tex]\(\frac{5}{6}\)[/tex]
Repeating or Terminating: The prime factorization of 6 is [tex]\(2 \times 3\)[/tex]. Since it contains a prime factor other than 2 or 5, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 5 by 6.
[tex]\[ \frac{5}{6} = 0.833333... \][/tex]
So the decimal expansion is [tex]\( 0.833333... \)[/tex] which repeats.
### 15. [tex]\(\frac{2}{7}\)[/tex]
Repeating or Terminating: The prime factorization of 7 is a prime number itself, which is neither 2 nor 5. Thus, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 2 by 7.
[tex]\[ \frac{2}{7} = 0.\overline{285714} \][/tex]
So the decimal expansion is [tex]\( 0.285714285714... \)[/tex] which repeats.
### 16. [tex]\(\frac{14}{15}\)[/tex]
Repeating or Terminating: The prime factorization of 15 is [tex]\(3 \times 5\)[/tex]. Since it contains a prime factor other than 2 or 5, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 14 by 15.
[tex]\[ \frac{14}{15} = 0.933333... \][/tex]
So the decimal expansion is [tex]\( 0.933333... \)[/tex] which repeats.
### 17. [tex]\(\frac{15}{22}\)[/tex]
Repeating or Terminating: The prime factorization of 22 is [tex]\(2 \times 11\)[/tex]. Since it contains a prime factor other than 2 or 5, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 15 by 22.
[tex]\[ \frac{15}{22} = 0.681818... \][/tex]
So the decimal expansion is [tex]\( 0.681818... \)[/tex] which repeats.
### 18. [tex]\(\frac{1}{11}\)[/tex]
Repeating or Terminating: The prime factorization of 11 is a prime number itself, which is neither 2 nor 5. Thus, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 1 by 11.
[tex]\[ \frac{1}{11} = 0.\overline{09} \][/tex]
So the decimal expansion is [tex]\( 0.090909... \)[/tex] which repeats.
### 19. [tex]\(1 \frac{1}{9}\)[/tex]
Convert the mixed number to an improper fraction.
[tex]\[ 1 \frac{1}{9} = \frac{10}{9} \][/tex]
Repeating or Terminating: The prime factorization of 9 is 3. Thus, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 10 by 9.
[tex]\[ \frac{10}{9} = 1.\overline{1} \][/tex]
So the decimal expansion is [tex]\( 1.111111... \)[/tex] which repeats.
### 20. [tex]\(2 \frac{1}{3}\)[/tex]
Convert the mixed number to an improper fraction.
[tex]\[ 2 \frac{1}{3} = \frac{7}{3} \][/tex]
Repeating or Terminating: The prime factorization of 3 is a prime number itself, which is neither 2 nor 5. Thus, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 7 by 3.
[tex]\[ \frac{7}{3} = 2.\overline{3} \][/tex]
So the decimal expansion is [tex]\( 2.333... \)[/tex] which repeats.
### 13. [tex]\(\frac{2}{9}\)[/tex]
Repeating or Terminating: A fraction is repeating if its denominator in its simplest form has any prime factor other than 2 or 5. The prime factor of 9 is 3.
Decimal Expansion:
To find the decimal expansion, divide 2 by 9.
[tex]\[ \frac{2}{9} = 0.\overline{2} \][/tex]
So the decimal expansion is [tex]\( 0.222222... \)[/tex] which repeats.
### 14. [tex]\(\frac{5}{6}\)[/tex]
Repeating or Terminating: The prime factorization of 6 is [tex]\(2 \times 3\)[/tex]. Since it contains a prime factor other than 2 or 5, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 5 by 6.
[tex]\[ \frac{5}{6} = 0.833333... \][/tex]
So the decimal expansion is [tex]\( 0.833333... \)[/tex] which repeats.
### 15. [tex]\(\frac{2}{7}\)[/tex]
Repeating or Terminating: The prime factorization of 7 is a prime number itself, which is neither 2 nor 5. Thus, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 2 by 7.
[tex]\[ \frac{2}{7} = 0.\overline{285714} \][/tex]
So the decimal expansion is [tex]\( 0.285714285714... \)[/tex] which repeats.
### 16. [tex]\(\frac{14}{15}\)[/tex]
Repeating or Terminating: The prime factorization of 15 is [tex]\(3 \times 5\)[/tex]. Since it contains a prime factor other than 2 or 5, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 14 by 15.
[tex]\[ \frac{14}{15} = 0.933333... \][/tex]
So the decimal expansion is [tex]\( 0.933333... \)[/tex] which repeats.
### 17. [tex]\(\frac{15}{22}\)[/tex]
Repeating or Terminating: The prime factorization of 22 is [tex]\(2 \times 11\)[/tex]. Since it contains a prime factor other than 2 or 5, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 15 by 22.
[tex]\[ \frac{15}{22} = 0.681818... \][/tex]
So the decimal expansion is [tex]\( 0.681818... \)[/tex] which repeats.
### 18. [tex]\(\frac{1}{11}\)[/tex]
Repeating or Terminating: The prime factorization of 11 is a prime number itself, which is neither 2 nor 5. Thus, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 1 by 11.
[tex]\[ \frac{1}{11} = 0.\overline{09} \][/tex]
So the decimal expansion is [tex]\( 0.090909... \)[/tex] which repeats.
### 19. [tex]\(1 \frac{1}{9}\)[/tex]
Convert the mixed number to an improper fraction.
[tex]\[ 1 \frac{1}{9} = \frac{10}{9} \][/tex]
Repeating or Terminating: The prime factorization of 9 is 3. Thus, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 10 by 9.
[tex]\[ \frac{10}{9} = 1.\overline{1} \][/tex]
So the decimal expansion is [tex]\( 1.111111... \)[/tex] which repeats.
### 20. [tex]\(2 \frac{1}{3}\)[/tex]
Convert the mixed number to an improper fraction.
[tex]\[ 2 \frac{1}{3} = \frac{7}{3} \][/tex]
Repeating or Terminating: The prime factorization of 3 is a prime number itself, which is neither 2 nor 5. Thus, it is a repeating decimal.
Decimal Expansion:
To find the decimal expansion, divide 7 by 3.
[tex]\[ \frac{7}{3} = 2.\overline{3} \][/tex]
So the decimal expansion is [tex]\( 2.333... \)[/tex] which repeats.
