\begin{align} & {4^{3.5}}:{2^5} \cr & = {\left( {{2^2}} \right)^{3.5}}:{2^5} \cr & = {2^{2 \times 3.5}}:{2^5} \cr & = {2^7}:{2^5} \cr & = {2^2}{.2^5}:{2^5} \cr & = {2^2}:1 \cr & = 4:1 \cr\end{align}

	A	:	B	:	C
	2	:	3	:	5
Let	2x	:	3x	:	5x

C - A = 12000

5x - 2x = 12000

3x = 12000

x = 4000 Monthly salary of B is = 3x = Rs. 12000

∴ Annual salary of B is

= 12 $\times$ 1200

=Rs. 144000

x/y = 5/2

Means x =5, y =2

Putting value of x and y in expression.

8*5+9*2 = 58.

8*5+2*2 = 44.

58:44 =29:22.

Ratio of students in 2012 in institutes A and B,

= 7 : 15.

Let number of students in institute A in 2012 = 700.

And Number of students in institutes B in 2012 = 1500.

25% increase in the number of students in 2013,

Now, number of students in Institute A,

= 700 + 25% of 700. = 875.

Number of students in B in 2013 as 26% students increased in B,

= 1500 + 26% of 1500 = 1890.

Current Ratio of the students,

= 875 /1890 = 25 : 54.

\begin{align} & \left( {3 + \sqrt 2 } \right):x:\left( {12 - \sqrt {32} } \right) \cr & a:b:c \cr & {\text{mean proportion}} \cr & \Rightarrow {b^2} = a \times c \cr & \Rightarrow {x^2} = \left( {3 + \sqrt 2 } \right) \times \left( {12 - \sqrt {32} } \right) \cr & \Rightarrow {x^2} = \left( {3 + \sqrt 2 } \right) \times \left( {12 - 4\sqrt 2 } \right) \cr & \Rightarrow {x^2} = \left( {3 + \sqrt 2 } \right) \times 4\left( {3 - \sqrt 2 } \right) \cr & \Rightarrow {x^2} = 4 \times {\left( 3 \right)^2} \times {\left( {\sqrt 2 } \right)^2} \cr & \Rightarrow {x^2} = 28 \cr & \Rightarrow x = 2\sqrt 7 \cr\end{align}