Find the sub-duplicate ratio of :

(i) 9 : 16

(ii) $\dfrac{1}{4} : \dfrac{1}{9}$

(iii) 9a² : 49b²

(i) The sub duplicate ratio of 9 : 16 is,

$= \sqrt{9} : \sqrt{16} \\[0.5em] = 3 : 4$

Hence, the sub-duplicate ratio is 3 : 4.

(ii) The sub duplicate ratio of $\dfrac{1}{4} : \dfrac{1}{9}$ is,

$= \sqrt{\dfrac{1}{4}} : \sqrt{\dfrac{1}{9}} \\[0.5em] = \dfrac{1}{2} : \dfrac{1}{3} \\[0.5em] = \dfrac{\dfrac{1}{2}}{\dfrac{1}{3}} \\[0.5em] = \dfrac{3}{2} = 3 : 2$

Hence, the sub-duplicate ratio is 3 : 2.

(iii) The sub duplicate ratio of 9a² : 49b² is,

$= \sqrt{9a^2} : \sqrt{49b^2} \\[0.5em] = 3a : 7b$

Hence, the sub-duplicate ratio is 3a : 7b.