\begin{align} & a + b\sqrt 3 = \frac{1}{{2 - \sqrt 3 }} \cr & \Rightarrow \frac{1}{{2 - \sqrt 3 }} \times \frac{{2 + \sqrt 3 }}{{2 - \sqrt 3 }} \cr & \Rightarrow \frac{{2 + \sqrt 3 }}{{4 - 3}} \cr & \Rightarrow 2 + \sqrt 3 \cr\end{align} By rationalisation of denominator

⇒ a + b $\sqrt{3}$ = 2 + $\sqrt{3}$

⇒ Now compare the rational & irrational parts

∴ a = 2

b = 1

∴ a : b

2 : 1

Originally, let the number of boys and girls in the college be 7x and 8x respectively.

Their increased number is (120% of 7x) and (110% of 8x). \begin{align} & \Rightarrow \left( {\frac{{120}}{{100}} \times 7x} \right)\, {\text{and}}\, \left( {\frac{{110}}{{100}} \times 8x} \right) \cr & \Rightarrow \frac{{42x}}{5}\, {\text{and}}\, \frac{{44x}}{5} \cr & \therefore {\text{The}}\, {\text{required}}\, {\text{ration}} \cr & = \left( {\frac{{42x}}{5}:\frac{{44x}}{5}} \right) \cr & = 21:22. \cr\end{align}

\begin{align} & \Rightarrow 0.5{\text{A}} = 0.6{\text{B}} = 1.75{\text{C}} \cr & \Rightarrow \frac{5}{{10}} \times {\text{A}} = \frac{6}{{10}} \times {\text{B}} = \frac{{75}}{{100}} \times {\text{C}} \cr & \Rightarrow \frac{1}{2}{\text{A}} = \frac{3}{5}{\text{B}} = \frac{3}{4}{\text{C}} \cr & \Rightarrow 10{\text{A}} = 12{\text{B }} = 15{\text{C}} \cr & \Rightarrow {\text{A}}\, \, \, \, \, \, \, :\, \, \, \, \, \, {\text{B}}\, \, \, \, \, \, \, \, \, :\, \, \, \, \, \, {\text{C}} \cr & 12 \times 15:10 \times 15:10 \times 12 \cr & \Rightarrow 180\, \, \, :\, \, \, \, \, 150\, \, \, \, \, \, :\, \, \, \, 120 \cr & \Rightarrow 6x\, \, \, \, \, \, :\, \, \, \, \, \, 5x\, \, \, \, \, \, \, :\, \, \, \, \, 4x \cr & {\text{Total}} = 6x + 5x + 4x = 15x \cr & 15x = 1740 \cr & \Rightarrow x = \frac{{1740}}{{15}} = {\text{Rs}}{\text{. }}116 \cr & \therefore {\text{Share of C is }}4x \cr & = 4 \times 116 \cr & = {\text{Rs}}{\text{. 464}} \cr\end{align}

Let x = 5k and y = 7k

Then,

= x + y = 36 ⇒ 5k + 7k = 36

⇒ 12k = 36

⇒ k = 3

∴ x = 5k = 5 $\times$ 3 = 15

Let A = 2k, B = 3k, C = 5k

A = x% of (B + C)

⇒ 2k = x% of (3k + 5k) = x% of 8k \begin{align} & \Rightarrow \frac{x}{{100}} = \frac{{{\text{2k}}}}{{{\text{8k}}}} = \frac{1}{4} \cr & \Rightarrow x = \frac{{100}}{4} = 25. \cr\end{align}