Find four rational numbers equivalent to each of the following :

(i) $\dfrac{3}{10}$

(ii) $\dfrac{-5}{9}$

(iii) $\dfrac{6}{-13}$

(iv) 9

(v) -1

(i) $\dfrac{3}{10}$

We have:

$\dfrac{3}{10} = \dfrac{3 \times 2}{10 \times 2} = \dfrac{3 \times 3}{10 \times 3} = \dfrac{3 \times 4}{10 \times 4} = \dfrac{3 \times 5}{10 \times 5}$ [Multiply the numerator and denominator by 2, 3, 4, and 5]

$\Rightarrow \dfrac{3}{10} = \dfrac{6}{20} = \dfrac{9}{30} = \dfrac{12}{40} = \dfrac{15}{50}$

∴ Four rational numbers equivalent to $\dfrac{3}{10}$ are:

$\dfrac{6}{20}, \dfrac{9}{30}, \dfrac{12}{40} , \dfrac{15}{50}$

(ii) $\dfrac{-5}{9}$

We have:

$\dfrac{-5}{9} = \dfrac{(-5) \times 2}{9 \times 2} = \dfrac{(-5) \times 3}{9 \times 3} = \dfrac{(-5) \times 4}{9 \times 4} = \dfrac{(-5) \times 5}{9 \times 5}$ [Multiply the numerator and denominator by 2, 3, 4, and 5]

$\Rightarrow \dfrac{-5}{9} = \dfrac{-10}{18} = \dfrac{-15}{27} = \dfrac{-20}{36} = \dfrac{-25}{45}$

∴ Four rational numbers equivalent to $\dfrac{-5}{9}$ are:

$\dfrac{-10}{18}, \dfrac{-15}{27}, \dfrac{-20}{36} , \dfrac{-25}{45}$

(iii) $\dfrac{6}{-13}$

We have:

$\dfrac{6}{-13} = \dfrac{6 \times 2}{(-13) \times 2} = \dfrac{6 \times 3}{(-13) \times 3} = \dfrac{6 \times 4}{(-13) \times 4} = \dfrac{6 \times 5}{(-13) \times 5}$ [Multiply the numerator and denominator by 2, 3, 4, and 5]

$\Rightarrow \dfrac{6}{-13} = \dfrac{12}{-26} = \dfrac{18}{-39} = \dfrac{24}{-52} = \dfrac{30}{-65}$

∴ Four rational numbers equivalent to $\dfrac{6}{-13}$ are:

$\dfrac{12}{-26}, \dfrac{18}{-39}, \dfrac{24}{-52} , \dfrac{30}{-65}$

(iv) 9

Since $9 = \dfrac{9}{1}$, we have:

$\dfrac{9}{1} = \dfrac{9 \times 2}{1 \times 2} = \dfrac{9 \times 3}{1 \times 3} = \dfrac{9 \times 4}{1 \times 4} = \dfrac{9 \times 5}{1 \times 5}$ [Multiply the numerator and denominator by 2, 3, 4, and 5]

$\Rightarrow 9 = \dfrac{18}{2} = \dfrac{27}{3} = \dfrac{36}{4} = \dfrac{45}{5}$

∴ Four rational numbers equivalent to 9 are:

$\dfrac{18}{2}, \dfrac{27}{3}, \dfrac{36}{4} , \dfrac{45}{5}$

(v) -1

Since $-1 = \dfrac{-1}{1}$, we have

$\dfrac{-1}{1} = \dfrac{(-1) \times 2}{1 \times 2} = \dfrac{(-1) \times 3}{1 \times 3} = \dfrac{(-1) \times 4}{1 \times 4} = \dfrac{(-1) \times 5}{1 \times 5}$ [Multiply the numerator and denominator by 2, 3, 4, and 5]

$\Rightarrow -1 = \dfrac{-2}{2} = \dfrac{-3}{3} = \dfrac{-4}{4} = \dfrac{-5}{5}$

∴ Four rational numbers equivalent to -1 are:

$\dfrac{-2}{2}, \dfrac{-3}{3}, \dfrac{-4}{4}, \dfrac{-5}{5}$