Answer: Option B. -> 12 years, 44 years
Let daughter's age be $$x$$ years.
Then, mother's age = (56 $$-\, x\,$$) years
$$\eqalign{
& \left( {56 - x} \right)+4 = 3\left( {x + 4} \right) \cr
& \Rightarrow 60 - x = 3x + 12 \cr
& \Rightarrow 4x = 48 \cr
& \Rightarrow x = 12 \cr} $$
∴ Daughter's age = 12 years
And the mother's age = (56 $$-$$ 12) = 44 years

Answer: Option A. -> 20 years
Let mother's age be 2x years, Then, son's age = x years
$$\eqalign{
& \therefore \left( {2x - 10} \right) = 3\left( {x - 10} \right) \cr
& \Rightarrow 2x - 10 = 3x - 30 \cr
& \Rightarrow x = 20 \cr} $$
Son's age = 20 years

Answer: Option C. -> 38 years
Let Rajan's age 8 years ago be x years, Then, present age = (x + 8) years
$$\eqalign{
& \therefore x + 8 = \frac{6}{5}x \cr
& \Rightarrow 5x + 40 = 6x \cr
& \Rightarrow x = 40 \cr} $$
Rajan's sister's age 8 years ago = (40 - 10) years = 30 years
∴ His sister's age now = (30+8) years = 38 years

Answer: Option C. -> 6 years
Mother's age when Reenu's brother was born = 36 year
Father's age when Reenu's brother was born = (38 + 4) years = 42 year
Required difference = (42 - 36) years = 6 years

Answer: Option C. -> 54 years
M → Mother, F → Father, S → Son and D → Daughter
F = 4S, D = $$\frac{1}{3}$$M, M = F - 6 and S = D - 3
$$\eqalign{
& \therefore M = 3D = 3\left( {S + 3} \right) \cr
& = 3S + 9 = \frac{3}{4}F + 9 = \frac{3}{4}\left( {M + 6} \right) + 9 \cr
& = \frac{3}{4}M + \frac{3}{4} \times 6 + 9 \cr
& \Rightarrow \left( {M - \frac{3}{4}M} \right) = \left( {\frac{9}{2} + 9} \right) \cr
& \Rightarrow \frac{1}{4}M = \frac{{27}}{2} \cr
& \Rightarrow M = \left( {\frac{{27}}{2} \times 4} \right) \cr
& \,\,\,\,\,\,\,\,\,\,\,\,\,\,\, = 54\,{\text{years}} \cr
& \therefore {\text{ The mother is 54 years old}}{\text{.}} \cr} $$

Answer: Option A. -> 18 years
Let the present age of the man be x years
Then,
$$\eqalign{
& \Rightarrow {\text{3}}\left( {x + 3} \right) - 3\left( {x - 3} \right) = x \cr
& \Rightarrow \left( {3x + 9} \right) - \left( {3x - 9} \right) = x \cr
& \Rightarrow x = 18 \cr} $$
∴ The present age of the man is 18 years

Answer: Option B. -> 2 : 1
Let the man's age be 7x years , Then son's age = 3x years
$$\eqalign{
& \therefore {\text{7}}x \times 3x = 756 \cr
& \Rightarrow 21{x^2} = 756 \cr
& \Rightarrow {x^2} = 756 \cr
& \Rightarrow {x^2} = 36 \cr
& \Rightarrow x = 6 \cr} $$
The ratio of their ages after 6 years
$$\eqalign{
& {\text{ = }}\left( {7x + 6} \right):\left( {3x + 6} \right) \cr
& = \left( {7 \times 6 + 6} \right):\left( {3 \times 6 + 6} \right) \cr
& = 48:24 \cr
& = 2:1 \cr} $$

Answer: Option B. -> 15
Let the age of Rahul, Sagar and Purav be x, y and z respectively
According to the given information
Age of Sagar - Age of Rahul = Age of Rahul - Age of Purav
$$\eqalign{
& \Rightarrow y - x = x - z \cr
& \Rightarrow 2x = y + z......(i) \cr} $$
Also y + z = 66 years
From (i) x = 33 years
Also as per equation (i) we have Purav's age + Sagar age = 66 years
by going through option (A) given Purav = 18, and Rahul = 33 years, Sagar = 48 years
Difference between Rahul's and Purav's age = 33 - 18 = 15 years

Answer: Option A. -> 34 years
Let present age of father , mother and son be x, y and z respectively
Sum of present ages of father and son = (Mother's present age + 8 years)
x + z = y + 8 years.................(i)
Mother's present age = Son's present age + 22 years)
⇒ y = z + 22 years..............(ii)
Put the value of y in equation (i) we get
x + z = z + 22 + 8
x + z = z + 30
x = 30 years
∴ Father's present age = 30 years
Age of father after four years = 30 + 4 = 34 years
∴ Required age of father = 34 years

Answer: Option D. -> 12
Let present ages of Simmi and Niti be a and b years respectively.
Ten years hence, the ratio between Simmi's age and Niti's age = 7 : 9
According to question
$$\frac{{a + 10}}{{b + 10}} = \frac{7}{9}$$
By cross multiplying we get
$$\eqalign{
& \Rightarrow 9a + 90 = 7b + 70 \cr
& \Rightarrow 7b - 9a = 20......(i) \cr
& {\text{Also}},\frac{{a - 2}}{{b - 2}} = \frac{1}{3} \cr} $$
By cross multiplying we get
$$\eqalign{
& \Rightarrow 3a - 6 = b - 2 \cr
& \Rightarrow 3a - b = 4.....(ii) \cr} $$
Multiplying equation (ii) by 3
$$9a - 3b = 12.....(iii)$$
Adding equation (ii) and (iii) we get
$$\eqalign{
& - 9a + 7b = 20 \cr
& 4b = 32 \cr
& b = 8{\text{ year}} \cr} $$
From equation (ii) we get
$$\eqalign{
& a = \frac{{4 + b}}{3} \cr
& \,\,\,\,\, = \frac{{4 + 8}}{3} \cr
& \,\,\,\,\,\, = \frac{{12}}{3} \cr
& \,\,\,\,\,\, = 4\,{\text{year}} \cr} $$
Since, Abhay is 4 years older to Niti
So, Abhay's present age = 8 + 4 = 12 years