Without actual division, show that each of the rational numbers given below is expressible as a repeating decimal :

(i) $\dfrac{23}{24}$

(ii) $\dfrac{79}{30}$

(iii) $\dfrac{100}{9}$

(iv) $\dfrac{205}{27}$

(v) $\dfrac{461}{60}$

(vi) $\dfrac{1003}{112}$

(vii) $\dfrac{127}{225}$

(viii) $\dfrac{219}{440}$

(i) $\dfrac{23}{24}$

The given rational number is $\dfrac{23}{24}$.

Its denominator is $24 = 2^3 \times 3$.

Thus, the denominator of $\dfrac{23}{24}$ has at least one prime factor, namely 3, other than 2 and 5.

∴ $\dfrac{23}{24}$ is expressible as a repeating decimal.

(ii) $\dfrac{79}{30}$

The given rational number is $\dfrac{79}{30}$.

Its denominator is $30 = 2 \times 3 \times 5$.

Thus, the denominator of $\dfrac{79}{30}$ has a prime factor, namely 3, other than 2 and 5.

∴ $\dfrac{79}{30}$ is expressible as a repeating decimal.

(iii) $\dfrac{100}{9}$

The given rational number is $\dfrac{100}{9}$.

Its denominator is $9 = 3^2$.

Thus, the denominator of $\dfrac{100}{9}$ has at least one prime factor, namely 3, which is different from 2 and 5.

∴ $\dfrac{100}{9}$ is expressible as a repeating decimal.

(iv) $\dfrac{205}{27}$

The given rational number is $\dfrac{205}{27}$.

Its denominator is $27 = 3^3$.

Thus, the denominator of $\dfrac{205}{27}$ has at least one prime factor, namely 3, which is different from 2 and 5.

∴ $\dfrac{205}{27}$ is expressible as a repeating decimal.

(v) $\dfrac{461}{60}$

The given rational number is $\dfrac{461}{60}$.

Its denominator is $60 = 2^2 \times 3 \times 5$.

Thus, the denominator has at least one prime factor, namely 3, other than 2 and 5.

∴ $\dfrac{461}{60}$ is expressible as a repeating decimal.

(vi) $\dfrac{1003}{112}$

The given rational number is $\dfrac{1003}{112}$.

Its denominator is $112 = 2^4 \times 7$.

Thus, the denominator has at least one prime factor, namely 7, other than 2 and 5.

∴ $\dfrac{1003}{112}$ is expressible as a repeating decimal.

(vii) $\dfrac{127}{225}$

The given rational number is $\dfrac{127}{225}$.

Its denominator is $225 = 3^2 \times 5^2$.

Thus, the denominator has at least one prime factor, namely 3, which is different from 2 and 5.

∴ $\dfrac{127}{225}$ is expressible as a repeating decimal.

(viii) $\dfrac{219}{440}$

The given rational number is $\dfrac{219}{440}$.

Its denominator is $440 = 2^3 \times 5 \times 11$.

Thus, the denominator has a prime factor, namely 11, other than 2 and 5.

∴ $\dfrac{219}{440}$ is expressible as a repeating decimal.