Which of the following are rational numbers?

(i) $\dfrac{5}{15}$

(ii) $\dfrac{-6}{23}$

(iii) 17

(iv) -25

(v) 0

(vi) $\dfrac{8}{0}$

(vii) $\dfrac{0}{0}$

(viii) $\dfrac{0}{8}$

(ix) $\dfrac{-23}{-37}$

(x) $\dfrac{-1}{7}$

(i) $\dfrac{5}{15}$

⇒ It is in $\dfrac{p}{q}$ form where both 5 and 15 are integers and q ≠ 0.

Hence, it is a rational number.

(ii) $\dfrac{-6}{23}$

⇒ Both -6 and 23 are integers and the denominator is not zero.

Hence, it is a rational number.

(iii) 17

⇒ An integer that can be written as $\dfrac{17}{1}$.

Hence, it is a rational number.

(iv) -25

⇒ A negative integer that can be written as $\dfrac{-25}{1}$.

Hence, it is a rational number.

(v) 0

⇒ Zero is a rational number because it can be written as $\dfrac{0}{1}$.

Hence, it is a rational number.

(vi) $\dfrac{8}{0}$

⇒ Division by zero is undefined; the denominator q cannot be 0.

Hence, it is not a rational number.

(vii) $\dfrac{0}{0}$

⇒ The denominator is zero, which is not allowed in the definition of a rational number.

Hence, it is not a rational number.

(viii) $\dfrac{0}{8}$

⇒ The numerator can be 0 as long as the denominator is a non-zero integer.

Hence, it is a rational number.

(ix) $\dfrac{-23}{-37}$

⇒ Both are integers and the denominator is not zero (this simplifies to $\dfrac{23}{37}$).

Hence, it is a rational number.

(x) $\dfrac{-1}{7}$

⇒ Both are integers and the denominator is not zero.

Hence, it is a rational number.