Solution:
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, $$ = \frac{{875}}{{1890}} = 25:54$$

Solution:
$$\eqalign{ \left( {\frac{1}{2}:\frac{2}{3}:\frac{3}{4}} \right) \times 12 \cr \Rightarrow 6:8:9 \cr \therefore {1^{{\text{st}}}}{\text{part's}}\,{\text{share}} \cr = {\text{Rs}}{\text{.}}\,\left( {\frac{6}{{23}} \times 782} \right) \cr = {\text{Rs}}{\text{.}}\,6 \times 34 \cr = {\text{Rs}}{\text{.}}\,204 \cr} $$

Solution:
$$\eqalign{ {\text{Let}}\,\frac{x}{y} = \frac{y}{z} = \frac{z}{u} = k. \cr {\text{Now, }}\frac{x}{u} = \frac{{64}}{{27}} \cr \Rightarrow \frac{x}{y} \times \frac{y}{z} \times \frac{z}{u} = \frac{{64}}{{27}} \cr \Rightarrow {k^3} = {\left( {\frac{4}{3}} \right)^3} \cr \Rightarrow k = \frac{4}{3}. \cr {\text{So}},\,x:y = y:z = z:u = 4:3. \cr \therefore \frac{x}{z} = \frac{x}{y} \times \frac{y}{z} = \frac{4}{3} \times \frac{4}{3} = \frac{{16}}{{9.}} \cr} $$

Solution:
Let the number are 5x, 7x and 12x
According to the question,
⇒ 5x + 12x = 7x + 50
⇒ 17x - 7x = 50
⇒ 10x = 50
⇒ x = 5
Sum of all three numbers
= 5x + 7x + 12x
= 24x
= 24 × 5
=120

Solution:
$$\eqalign{ {\text{Milk in the mixture}} \cr = \left( {\frac{3}{5} + \frac{7}{{10}} + \frac{{11}}{{15}}} \right){\text{units}} \cr = \frac{{61}}{{30}}{\text{units}}{\text{.}} \cr {\text{Water in the mixture}} \cr = \left( {\frac{2}{5} + \frac{3}{{10}} + \frac{4}{{15}}} \right){\text{units}} \cr = \frac{{29}}{{30}}{\text{units}}{\text{.}} \cr \therefore {\text{Required ratio}} \cr = \frac{{61}}{{30}}:\frac{{29}}{{30}} \cr = 61:29 \cr} $$

Solution:
The distance covered by policeman in 5 steps is equal to that of thief in 7 steps
⇒ 5P = 7T
P : T
7 : 5 (distance covered in each step)
⇒ And policeman goes 8 steps while thief moves 10 steps

Solution:
Each exterior angle of n sided polygon is
$${\text{ = }}\left( {\frac{{360}}{n}} \right)$$
And each interior angle of n sided polygon
$$\eqalign{ {\text{ = }}\frac{{\left( {n - 2} \right) \times 180}}{n} \cr \therefore \frac{{\frac{{\left( {n - 2} \right) \times 180}}{n}}}{{\frac{{360}}{n}}} = \frac{2}{1} \cr \Rightarrow \frac{{\left( {n - 2} \right)}}{2} = 2 \cr \Rightarrow n - 2 = 4 \cr \Rightarrow n = 6 \cr} $$