Answer :
Let's determine which of the given numbers are rational and which are not. We'll also provide reasons for our choices.
### Determination of Rational Numbers:
1. a. [tex]\(\frac{7}{9}\)[/tex]
- Rational: [tex]\(\frac{7}{9}\)[/tex] is a rational number because it can be expressed as the quotient of two integers [tex]\(7\)[/tex] and [tex]\(9\)[/tex], where [tex]\(9 \neq 0\)[/tex].
2. b. [tex]\(\frac{8}{19}\)[/tex]
- Rational: [tex]\(\frac{8}{19}\)[/tex] is a rational number because it can be expressed as the quotient of two integers [tex]\(8\)[/tex] and [tex]\(19\)[/tex], where [tex]\(19 \neq 0\)[/tex].
3. c. [tex]\(\frac{6}{12}\)[/tex]
- Rational: [tex]\(\frac{6}{12}\)[/tex] is a rational number because it can be expressed as the quotient of two integers [tex]\(6\)[/tex] and [tex]\(12\)[/tex], where [tex]\(12 \neq 0\)[/tex].
4. d. [tex]\(\frac{0}{7}\)[/tex]
- Rational: [tex]\(\frac{0}{7}\)[/tex] is a rational number because it can be expressed as the quotient of two integers [tex]\(0\)[/tex] and [tex]\(7\)[/tex], where [tex]\(7 \neq 0\)[/tex]. Any number with [tex]\(0\)[/tex] as the numerator and a non-zero denominator is rational.
5. e. [tex]\(\frac{16}{0}\)[/tex]
- Not Rational: [tex]\(\frac{16}{0}\)[/tex] is not a rational number because division by zero is undefined. Therefore, it cannot be expressed as a quotient of two integers with a non-zero denominator.
### Final Classification:
- Rational Numbers: [tex]\( \boxed{\frac{7}{9}} \)[/tex], [tex]\( \boxed{\frac{8}{19}} \)[/tex], [tex]\( \boxed{\frac{6}{12}} \)[/tex], [tex]\( \boxed{\frac{0}{7}} \)[/tex]
- Not Rational Number: [tex]\(\mathbf{\frac{16}{0}}\)[/tex]
### Reasons:
- [tex]\(\boxed{\frac{7}{9}}\)[/tex]: Rational: Can be expressed as a quotient of two integers.
- [tex]\(\boxed{\frac{8}{19}}\)[/tex]: Rational: Can be expressed as a quotient of two integers.
- [tex]\(\boxed{\frac{6}{12}}\)[/tex]: Rational: Can be expressed as a quotient of two integers.
- [tex]\(\boxed{\frac{0}{7}}\)[/tex]: Rational: Can be expressed as a quotient of two integers.
- [tex]\(\mathbf{\frac{16}{0}}\)[/tex]: Not Rational: Division by zero or cannot be expressed as a quotient of two integers.
### Determination of Rational Numbers:
1. a. [tex]\(\frac{7}{9}\)[/tex]
- Rational: [tex]\(\frac{7}{9}\)[/tex] is a rational number because it can be expressed as the quotient of two integers [tex]\(7\)[/tex] and [tex]\(9\)[/tex], where [tex]\(9 \neq 0\)[/tex].
2. b. [tex]\(\frac{8}{19}\)[/tex]
- Rational: [tex]\(\frac{8}{19}\)[/tex] is a rational number because it can be expressed as the quotient of two integers [tex]\(8\)[/tex] and [tex]\(19\)[/tex], where [tex]\(19 \neq 0\)[/tex].
3. c. [tex]\(\frac{6}{12}\)[/tex]
- Rational: [tex]\(\frac{6}{12}\)[/tex] is a rational number because it can be expressed as the quotient of two integers [tex]\(6\)[/tex] and [tex]\(12\)[/tex], where [tex]\(12 \neq 0\)[/tex].
4. d. [tex]\(\frac{0}{7}\)[/tex]
- Rational: [tex]\(\frac{0}{7}\)[/tex] is a rational number because it can be expressed as the quotient of two integers [tex]\(0\)[/tex] and [tex]\(7\)[/tex], where [tex]\(7 \neq 0\)[/tex]. Any number with [tex]\(0\)[/tex] as the numerator and a non-zero denominator is rational.
5. e. [tex]\(\frac{16}{0}\)[/tex]
- Not Rational: [tex]\(\frac{16}{0}\)[/tex] is not a rational number because division by zero is undefined. Therefore, it cannot be expressed as a quotient of two integers with a non-zero denominator.
### Final Classification:
- Rational Numbers: [tex]\( \boxed{\frac{7}{9}} \)[/tex], [tex]\( \boxed{\frac{8}{19}} \)[/tex], [tex]\( \boxed{\frac{6}{12}} \)[/tex], [tex]\( \boxed{\frac{0}{7}} \)[/tex]
- Not Rational Number: [tex]\(\mathbf{\frac{16}{0}}\)[/tex]
### Reasons:
- [tex]\(\boxed{\frac{7}{9}}\)[/tex]: Rational: Can be expressed as a quotient of two integers.
- [tex]\(\boxed{\frac{8}{19}}\)[/tex]: Rational: Can be expressed as a quotient of two integers.
- [tex]\(\boxed{\frac{6}{12}}\)[/tex]: Rational: Can be expressed as a quotient of two integers.
- [tex]\(\boxed{\frac{0}{7}}\)[/tex]: Rational: Can be expressed as a quotient of two integers.
- [tex]\(\mathbf{\frac{16}{0}}\)[/tex]: Not Rational: Division by zero or cannot be expressed as a quotient of two integers.
