Let the numbers be x and y \begin{align} & {\text{Then, }} \cr & = \frac{1}{4}{\text{ of }}\left( {60\% {\text{ of }}x} \right) \cr & = \frac{2}{9}{\text{ of }}\left( {20\% {\text{ of }}y} \right) \cr & \Rightarrow \left( {\frac{1}{4} \times \frac{{60}}{{100}} \times x} \right) = \left( {\frac{2}{5} \times \frac{{20}}{{100}} \times y} \right) \cr & \Rightarrow \frac{{3x}}{{20}} = \frac{{2y}}{{25}} \cr & \Rightarrow \frac{x}{y} = \frac{2}{{25}} \times \frac{{20}}{3} = \frac{8}{{15}} \cr & \Rightarrow \frac{{3x}}{{20}} = \frac{{2y}}{{25}} \cr & \Rightarrow \frac{x}{y} = \frac{2}{{25}} \times \frac{{20}}{3} = \frac{8}{{15}} \cr & \Rightarrow x:y = 8:15. \cr\end{align}

Required mean proportional \begin{align} & = \sqrt {\left( {3 + \sqrt 2 } \right)\left( {12 - \sqrt {32} } \right)} \cr & = \sqrt {\left( {3 + \sqrt 2 } \right)\left( {12 - 4\sqrt 2 } \right)} \cr & = \sqrt {36 - 8} \cr & = \sqrt {28} \cr & = 2\sqrt {7.} \cr\end{align}

\begin{align} & {\text{Seema's share}} \cr & = {\text{Rs}}{\text{.}}\left( {25000 \times \frac{3}{5}} \right) \cr & = {\text{Rs}}.15000. \cr & {\text{Meena's share}} \cr & = {\text{Rs}}{\text{.}}\left( {25000 \times \frac{2}{5}} \right) \cr & = {\text{Rs}}.10000. \cr\end{align} ∴ Required ration

= (15000 + 5000) : (10000 + 5000)

= 4 : 3

\begin{align} & 5.5{\text{a}} = 0.65{\text{b}} \cr & \Rightarrow \frac{{55}}{{10}}{\text{a}} = \frac{{65}}{{100}}{\text{b}} \cr & \Rightarrow 55{\text{a}} = \frac{{65}}{{10}}{\text{b}} \cr & \Rightarrow 550{\text{a}} = 65{\text{b}} \cr & {\text{a}}:{\text{b}} = 65:550 \cr & \, \, \, \, \, \, \, \, \, \, \, \, = 13:110 \cr\end{align}

	A	:	B
Income	9	:	7
Expense	4	:	3

Income - Savings + Expenditure \begin{align} & \therefore \frac{{9x - 200}}{{7x - 200}} = \frac{4}{3} \cr & \Rightarrow 27x - 600 = 28x - 800 \cr & \Rightarrow x = 200 \cr & {\text{Sum of weekly income}} \cr & = 9x + 7x = 16x \cr & = 16 \times 200 = {\text{Rs}}.\, 3200 \cr\end{align}