Prove that the following rational numbers are equal:

(i) $\dfrac{2}{3}$ and $\dfrac{4}{6}$

(ii) $\dfrac{5}{4}$ and $\dfrac{10}{8}$

(iii) $-\dfrac{3}{5}$ and $-\dfrac{6}{10}$

(iv) $\dfrac{9}{3}$ and $3$

Two rational numbers $\dfrac{a}{b}$ and $\dfrac{c}{d}$ are equal if ad = bc.

(i) Given, $\dfrac{2}{3}$ and $\dfrac{4}{6}$.

Cross-multiplying :

⇒ 2 × 6 = 12

⇒ 3 × 4 = 12

Since, 2 × 6 = 3 × 4.

Hence, $\dfrac{2}{3} = \dfrac{4}{6}$.

(ii) Given, $\dfrac{5}{4}$ and $\dfrac{10}{8}$.

Cross-multiplying :

⇒ 5 × 8 = 40

⇒ 4 × 10 = 40

Since, 5 × 8 = 4 × 10.

Hence, $\dfrac{5}{4} = \dfrac{10}{8}$.

(iii) Given, $-\dfrac{3}{5}$ and $-\dfrac{6}{10}$.

Cross-multiplying :

⇒ (-3) × 10 = -30

⇒ 5 × (-6) = -30

Since, (-3) × 10 = 5 × (-6).

Hence, $-\dfrac{3}{5} = -\dfrac{6}{10}$.

(iv) Given, $\dfrac{9}{3}$ and 3.

We can write 3 as $\dfrac{3}{1}$.

Cross-multiplying :

⇒ 9 × 1 = 9

⇒ 3 × 3 = 9

Since, 9 × 1 = 3 × 3.

Hence, $\dfrac{9}{3} = 3$.