\begin{align} & {\text{Let }}x{\text{ be added to each term}}. \cr & {\text{According to question}}, \cr & \frac{{\left( {7 + x} \right)}}{{11 + x}} = \frac{3}{4} \cr & Or, \, 33 + 3x = 28 + 4x \cr & Or, \, x = 5 \cr\end{align}

Let the third proportional to 12 and 30 be x.

Then, \begin{align} & = 12:30::30:x \cr & \Rightarrow 12x = 30 \times 30 \cr & \Rightarrow x = \frac{{\left( {30 \times 30} \right)}}{{12}} = 75. \cr\end{align} ∴ Third proportional to 12 and 30 = 75

Mean proportional between 9 and 25 \begin{align} & = \sqrt {9 \times 25} = 15. \cr & \therefore {\text{Required ratio}} = 75:15 \cr & \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = 5:1. \cr\end{align}

	A	:	B	:	C
Annual Income	1	:	3	:	7
Let	x	:	3x	:	7x

Given A + C ⇒ x + 7x = 8x

⇒ 8x = 800000

⇒ x = 100000

⇒ Annual salary of B = 3x

⇒ 3 $\times$ 100000 = 300000 \begin{align} & \therefore {\text{Monthly salary}} \cr & {\text{ = }}\frac{{300000}}{{12}} \cr & = {\text{Rs}}.\, 25000 \cr\end{align}

\begin{align} & = {\text{A:B}} = 7:9\, {\text{and}} \cr & {\text{B}}:{\text{C}} = 5:4 \cr & = \left( {5 \times \frac{9}{5}} \right):\left( {4 \times \frac{9}{5}} \right) \cr & = 9:\frac{{36}}{5}. \cr & \Rightarrow {\text{A}}:{\text{B}}:{\text{C}} = 7:9:\frac{{36}}{5} \cr & \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = 35:45:36. \cr\end{align}

a + b = ck,

b + c = ak,

c + a = bk

Adding, we get : 2(a + b + c) = ak + bk + ck

⇒ k(a + b + c) or k = 2