Convert the following decimal numbers in the form of $\dfrac{p}{q}$.
(i) 12.6
(ii) 0.0120
(iii) $3.0\overline{52}$
(iv) $1.2\overline{35}$
(v) $0.\overline{23}$
(vi) $2.0\overline{5}$
(vii) $2.12\overline{5}$
(viii) $3.12\overline{5}$
(ix) $2.\overline{1625}$

(i) Given:

12.6

⇒ $12.6 = \dfrac{126}{10} = \dfrac{63}{5}.$

Hence, $12.6 = \dfrac{63}{5}$.

(ii) Given: 0.0120

⇒ $0.0120 = \dfrac{120}{10000} = \dfrac{12}{1000} = \dfrac{3}{250}.$

Hence, $0.0120 = \dfrac{3}{250}$.

(iii) Given:

$3.0\overline{52}$

Let x = $3.0\overline{52}$ = 3.0525252…

Multiplying both sides by 10 (1 non-repeating digit) :

⇒ 10x = 30.5252…

Multiplying both sides by 1000 (1 non-repeating + 2 repeating digits) :

⇒ 1000x = 3052.5252…

Subtracting :

1000x - 10x = 3052.5252… - 30.5252…

990x = 3022

⇒ x = $\dfrac{3022}{990}$

⇒ x = $\dfrac{1511}{495}$

Hence, $3.0\overline{52} = \dfrac{1511}{495}$.

(iv) Given:

$1.2\overline{35}$

Let x = $1.2\overline{35}$ = 1.2353535…

Multiplying both sides by 10 (1 non-repeating digit) :

⇒ 10x = 12.353535…

Multiplying both sides by 1000 (1 non-repeating + 2 repeating digits) :

⇒ 1000x = 1235.353535…

Subtracting :

⇒ 1000x - 10x = 1235.353535… - 12.353535…

⇒ 990x = 1223

⇒ x = $\dfrac{1223}{990}$.

Hence, $1.2\overline{35} = \dfrac{1223}{990}$.

(v) Given, $0.\overline{23}$

Let x = $0.\overline{23}$ = 0.232323…

Multiplying both sides by 100 (2 repeating digits) :

⇒ 100x = 23.2323…

Subtracting :

⇒ 100x - x = 23.2323… - 0.2323…

⇒ 99x = 23

⇒ x = $\dfrac{23}{99}$.

Hence, $0.\overline{23} = \dfrac{23}{99}$.

(vi) Given, $2.0\overline{5}$

Let x = $2.0\overline{5}$ = 2.05555…

Multiplying both sides by 10 (1 non-repeating digit) :

⇒ 10x = 20.5555…

Multiplying both sides by 100 (1 non-repeating + 1 repeating digit) :

⇒ 100x = 205.5555…

Subtracting :

⇒ 100x - 10x = 205.5555… - 20.5555…

⇒ 90x = 185

⇒ x = $\dfrac{185}{90} = \dfrac{37}{18}$.

Hence, $2.0\overline{5} = \dfrac{37}{18}$.

(vii) Given, $2.12\overline{5}$

Let x = $2.12\overline{5}$ = 2.12555…

Multiplying both sides by 100 (2 non-repeating digits) :

⇒ 100x = 212.555…

Multiplying both sides by 1000 (2 non-repeating + 1 repeating digit) :

⇒ 1000x = 2125.555…

Subtracting :

⇒ 1000x - 100x = 2125.555… - 212.555…

⇒ 900x = 1913

⇒ x = $\dfrac{1913}{900}$.

Hence, $2.12\overline{5} = \dfrac{1913}{900}$.

(viii) Given, $3.12\overline{5}$

Let x = $3.12\overline{5}$ = 3.12555…

Multiplying both sides by 100 (2 non-repeating digits) :

⇒ 100x = 312.555…

Multiplying both sides by 1000 (2 non-repeating + 1 repeating digit) :

⇒ 1000x = 3125.555…

Subtracting :

⇒ 1000x - 100x = 3125.555… - 312.555…

⇒ 900x = 2813

⇒ x = $\dfrac{2813}{900}$.

Hence, $3.12\overline{5} = \dfrac{2813}{900}$.

(ix) Given, $2.\overline{1625}$

Let x = $2.\overline{1625}$ = 2.1625 1625…

Multiplying both sides by 10000 (4 repeating digits) :

⇒ 10000x = 21625.1625…

Subtracting :

⇒ 10000x - x = 21625.1625… - 2.1625…

⇒ 9999x = 21623

⇒ x = $\dfrac{21623}{9999}$.

Hence, $2.\overline{1625} = \dfrac{21623}{9999}$.