Let C gets x rupees

∴ B gets x + 6 rupees

∴ A gets x + 6 + 7 rupees

Total A + B + C ⇒ (3x + 19)

⇒ 3x + 19 = Rs. 76

⇒ 3x = 76 - 19

⇒ 3x = 57

⇒ x = 10

A gets = 19 + 13 = Rs. 32

B gets = 19 + 6 = Rs. 25 C gets = Rs. 19

∴ A : B : C

=32 : 25 : 19

We will use work equivalence method,

\begin{align} & \frac{{30}}{{18}} = \frac{{\left( {\frac{1}{7}} \right)}}{{\left( {\frac{6}{7}} \right)}} \times \frac{x}{2}; \cr & \frac{5}{3} = \left( {\frac{1}{6}} \right) \times \frac{x}{2} \cr & Or, \, x = \frac{{60}}{3} = 20\, {\text{days}} \cr\end{align}

\begin{align} & {\text{Required percentage}} \cr & = \left( {\frac{7}{{7 + 3}} \times 100} \right)\% \cr & = \left( {\frac{7}{{10}} \times 100} \right)\% \cr & = 70\% . \cr\end{align}

\begin{align} & = {x^2} + 4{y^2} = 4xy \cr & \Rightarrow {x^2} - 4xy + 4{y^2} = 0 \cr & \Rightarrow \left( {x - 2{y^2}} \right) = 0 \cr & \Rightarrow x = 2y \cr & \Rightarrow \frac{x}{y} = 2 \cr & \Rightarrow x:y = 2:1 \cr\end{align}

Originally, let the number of seats for Mathematics, Physics and Biology be 5x, 7x and 8x respectively.

Number of increased seats are (140% of 5x), (150% of 7x) and (175% of 8x). \begin{align} & = \left( {\frac{{140}}{{100}} \times 5x} \right)\, , \, \left( {\frac{{150}}{{100}} \times 7x} \right)\, {\text{and}}\, \left( {\frac{{175}}{{100}} \times 8x} \right) \cr & = 7x\, , \, \frac{{21x}}{2}\, {\text{and}}\, 14x. \cr & \therefore {\text{The}}\, {\text{required}}\, {\text{ratio}}\, = \, 7x:\frac{{21x}}{2}:14x \cr & \Rightarrow 14x:21x:28x \cr & \Rightarrow 2:3:4. \cr\end{align}