The ratio between the third proportional to 12 and 30 and mean proportion of 9 and 25, is :

1. 2 : 1

2. 5 : 1

3. 7 : 15

4. 9 : 14

Let, third proportional to 12 and 30 be t.

If 12 : 30 :: 30 : t, then,

$\Rightarrow \dfrac{12}{30} = \dfrac{30}{t} \\[1em] \Rightarrow t = \dfrac{30 \times 30}{12} = \dfrac{30^2}{12} \\[1em] \Rightarrow t = \dfrac{900}{12} \\[1em] \Rightarrow t = 75.$

Let Mean proportion between 9 and 25 be m.

$\Rightarrow \dfrac{9}{m} = \dfrac{m}{25} \\[1em] \Rightarrow m^2 = 9 \times 25 \\[1em] \Rightarrow m = \sqrt{9 \times 25} \\[1em] \Rightarrow m = \sqrt{225} \\[1em] \Rightarrow m = 15.$

Required ratio = t : m = 75 : 15 = 5 : 1.

Hence, option 2 is the correct option.