Find the fourth proportional to :

(i) 3, 12, 15

(ii) $\dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}$

(iii) 1.5, 2.5, 4.5

(iv) 9.6 kg, 7.2 kg, 28.8 kg.

(i) Let the fourth proportion be x.

Then 3, 12, 15 and x are in proportion.

$\Rightarrow 3 : 12 :: 15 : x \\[0.5em] \Rightarrow \dfrac{3}{12} = \dfrac{15}{x} \\[0.5em] \Rightarrow x = \dfrac{15}{3} \times 12 \\[0.5em] \Rightarrow x = \dfrac{180}{3} \\[0.5em] \Rightarrow x = 60.$

Hence, the fourth proportion is 60.

(ii) Let the fourth proportion be x.

Then $\dfrac{1}{3}, \dfrac{1}{4}, \dfrac{1}{5}$ and x are in proportion.

$\Rightarrow \dfrac{1}{3} : \dfrac{1}{4} :: \dfrac{1}{5} : x \\[0.5em] \Rightarrow \dfrac{\dfrac{1}{3}}{\dfrac{1}{4}} = \dfrac{\dfrac{1}{5}}{x} \\[0.5em] \Rightarrow x = \dfrac{\dfrac{1}{5}}{\dfrac{1}{3}} \times \dfrac{1}{4} \\[0.5em] \Rightarrow x = \dfrac{3}{5} \times \dfrac{1}{4} \\[0.5em] \Rightarrow x = \dfrac{3}{20}.$

Hence, the fourth proportion is $\dfrac{3}{20}$.

(iii) Let the fourth proportion be x.

Then 1.5, 2.5, 4.5 and x are in proportion.

$\Rightarrow 1.5 : 2.5 :: 4.5 : x \\[0.5em] \Rightarrow \dfrac{1.5}{2.5} = \dfrac{4.5}{x} \\[0.5em] \Rightarrow x = \dfrac{4.5}{1.5} \times 2.5 \\[0.5em] \Rightarrow x = 3 \times 2.5 \\[0.5em] \Rightarrow x = 7.5.$

Hence, the fourth proportion is 7.5.

(iv) Let the fourth proportion be x.

Then 9.6 kg, 7.2 kg, 28.8 kg and x are in proportion.

$\Rightarrow 9.6 : 7.2 :: 28.8 : x \\[0.5em] \Rightarrow \dfrac{9.6}{7.2} = \dfrac{28.8}{x} \\[0.5em] \Rightarrow x = \dfrac{28.8}{9.6} \times 7.2 \\[0.5em] \Rightarrow x = 3 \times 7.2 \\[0.5em] \Rightarrow x = 21.6.$

Hence, the fourth proportion is 21.6 kg.