Find the third proportional to :

(i) 5, 10

(ii) 0.24, 0.6

(iii) ₹3, ₹12

(iv) $5\dfrac{1}{4}$ and 7.

(i) Let the third proportion be x, then 5, 10, x are in continued proportion.

$\Rightarrow \dfrac{5}{10} = \dfrac{10}{x} \\[0.5em] \Rightarrow x = \dfrac{10}{5} \times 10 \\[0.5em] \Rightarrow x = \dfrac{100}{5} \\[0.5em] \Rightarrow x = 20.$

Hence, the third proportion is 20.

(ii) Let the third proportion be x, then 0.24, 0.6, x are in continued proportion.

$\Rightarrow \dfrac{0.24}{0.6} = \dfrac{0.6}{x} \\[0.5em] \Rightarrow x = \dfrac{0.6}{0.24} \times 0.6 \\[0.5em] \Rightarrow x = \dfrac{0.36}{0.24} \\[0.5em] \Rightarrow x = 1.5.$

Hence, the third proportion is 1.5.

(iii) Let the third proportion be x, then 3, 12, x are in continued proportion.

$\Rightarrow \dfrac{3}{12} = \dfrac{12}{x} \\[0.5em] \Rightarrow x = \dfrac{12}{3} \times 12 \\[0.5em] \Rightarrow x = \dfrac{144}{3} \\[0.5em] \Rightarrow x = 48.$

Hence, the third proportion is ₹ 48.

(iv) Let the third proportion be x, then $5\dfrac{1}{4}$, 7, x are in continued proportion.

$5\dfrac{1}{4} = \dfrac{21}{4} \\[0.5em] \Rightarrow \dfrac{\dfrac{21}{4}}{7} = \dfrac{7}{x} \\[0.5em] \Rightarrow x = \dfrac{7}{\dfrac{21}{4}} \times 7 \\[0.5em] \Rightarrow x = \dfrac{7 \times 4 \times 7}{21} \\[0.5em] \Rightarrow x = \dfrac{28}{3} \\[0.5em] \Rightarrow x = 9\dfrac{1}{3}.$

Hence, the third proportion is $9\dfrac{1}{3}$.