Let bucket contains 5x and 3x of liquids A and B respectively.

When 16 litres of mixture are drawn off, quantity of A in mixture left:

\begin{align} & \left[ {5x - \left( {\frac{5}{8}} \right) \times 16} \right] = \left( {5x - 10} \right) \cr & {\text{Similarly quantity of B in mixture left}}, \cr & \left[ {3x - \left( {\frac{3}{8}} \right) \times 16} \right] = \left( {3x - 6} \right) \cr & {\text{Now the ratio becomes, }} \cr & \frac{{\left( {5x - 10} \right)}}{{\left( {3x - 6} \right)}} = \frac{3}{5} \cr & \Rightarrow 25x - 50 = 9x - 18 \cr & \Rightarrow 16x = 32 \cr & \Rightarrow x = 2 \cr & {\text{So, quantity of liquid B initially}}, \cr & = 3 \times 2 = 6\, {\text{litres}} \cr\end{align}

Let the basic salaries of A and B be Rs. x and Rs. y respectively.

Then,

= x + 65% of x = y + 80% of y \begin{align} & \Rightarrow \frac{{165}}{{100}}x = \frac{{180}}{{100}}y \cr & \Rightarrow 165x = 180y \cr & \Rightarrow \frac{x}{y} = \frac{{180}}{{165}} = \frac{{12}}{{11}} \cr & \therefore {\text{Required ratio}} \cr & = 12:11 \cr\end{align}

Let the numbers be 17x and 45x.

Then, \begin{align} & = \frac{1}{5}{\text{of }}45x - \frac{1}{3}{\text{of }}17x = 15 \cr & \Leftrightarrow 9x - \frac{{17}}{3}x = 15 \cr & \Leftrightarrow \frac{{10x}}{3} = 15 \cr & \Leftrightarrow x = 15 \times \frac{3}{{10}} = \frac{9}{2}. \cr & \therefore {\text{Smaller number}} \cr & = 17x = \left( {17x \times \frac{9}{2}} \right) \cr & \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = \frac{{153}}{2} \cr & \, \, \, \, \, \, \, \, \, \, \, \, \, \, \, = {\text{76}}\frac{1}{2}. \cr\end{align}

\begin{align} & = \frac{{5a + 3b}}{{2a - 3b}} = \frac{{23}}{5} \cr & \Rightarrow 5\left( {5a + 3b} \right) = 23(2a - 3b) \cr & \Rightarrow 25a + 15b = 46a - 69b \cr & \Rightarrow 21a = 84b \cr & \Rightarrow \frac{a}{b} = 4. \cr & {\text{Hence}}, \, a:b = 4:1 \cr\end{align}

\begin{align} & = {\text{3A}} = {\text{5B and 4B}} = {\text{6C}} \cr & \Rightarrow \frac{{\text{A}}}{{\text{B}}} = \frac{5}{3}{\text{ and }}\frac{{\text{B}}}{{\text{C}}} = \frac{6}{4} = \frac{3}{2} \cr & \Rightarrow \frac{{\text{A}}}{{\text{C}}} = \left( {\frac{{\text{A}}}{{\text{B}}} \times \frac{{\text{B}}}{{\text{C}}}} \right) = \frac{5}{3} \times \frac{3}{2} = \frac{5}{2} \cr & \Rightarrow {\text{A}}:{\text{C}} = {\text{5}}:{\text{2}} \cr\end{align}