The present ages of three persons are in the ratio of 4 : 7: 9. Eight years ago, the sum of their ages was 56. What are their present ages in years?

Let the present age of Divya and Namrata are 3x and 4x years respectively; then
According to question,
Ten years ago, Age of Divya = 3x - 10
Ten years ago, Age of Namrata = 4x - 10
Now According to question,
Ten years ago, the age of Divya was half of the age of Namrata
3x - 10 = (4x - 10)/2
? 6x - 20 = 4x - 10
? 2x =10
? x = 5
Hence, sum of their ages = 3x + 4x = 7x =7 x 5 = 35 years

Here, ratio of ages of Ram, Mohan and Anil = 4 : 5: 6
Let their present ages be 4x, 5x and 6x years respectively.
then 4x + 5x + 6x = 90
? 15x = 90
? x = 6
Hence, age of Mohan = 5 x 6 = 30 years

Let the present ages of Arun and Barun are x and x +14 years respectively; then
According to question, 7 years ago the ratio of their ages was 5 : 7
(x - 7)/(x + 14 - 7) = 5/7
? (x - 7)7 = 5(x+14 - 7)
? 7x - 49 = 5x + 35
? 2x = 84
? x = 42
Hence, age of Barun = 42 + 14 = 56 years

Lt us assume the present age of father is P and son age is Q.
According to question,
P + Q = 45.................. (1)
Five years ago, the father age = P - 5
Five years ago, the son age = Q - 5
According to question,
Five years ago, the product of their ages was four times the father's age at that time.
(P - 5) (Q - 5) = 4 (P - 5)
PQ - 5P - 5Q + 25 = 4P - 20
PQ - 5P - 5Q + 25 - 4P + 20 = 0
? PQ - 9P - 5Q + 45 = 0
? P(Q - 9) - 5(Q - 9) = 0
? (Q - 9) (P - 5) = 0
? (Q - 9) = 0 or (P - 5) = 0
? Q = 9 or P = 5
Put the vale of Q in equation (1), we will get.
if Q = 9 then P = 45 - 9 = 36 which is matched our answer.

If P = 5 then Q = 45 - 5 = 40 years which is not match our answer.

Method 2
Let the present ages of father and his son be x and (45 - x) years respectively;
then according to question,
(x - 5) (45 - x - 5) = 4 (45 - x -5)
? x - 5 = 4
? x = 9
Now, Father's age = 45 - x = 45 - 9 = 36 years

Method (1)
Let us assume the present age of Mayank is X and his mother age is Y.
Five years ago, Mayank's age = X - 5
Five years ago, Mayank's mother age = Y - 5
According to question,
Five years ago, Mayank's mother was three times as old as Mayank.
Y - 5 = 3 (X - 5)
? Y - 5 = 3X - 15
? 3X - Y = 15 - 5
? 3X - Y = 10................(1)
Again According to question,
After five years, Mayank's age = X + 5
After five years , Mayank's mother age = Y + 5
After five years she will be twice as old as Mayank.
Y + 5 = 2 ( X + 5 )
? Y + 5 = 2X + 10
? Y - 2X = 10 - 5
? Y - 2X = 5 ....................(2)
Add the equation (1) and (2), we will get
3X - Y + Y - 2X = 10 + 5
X = 15
Put the vale of X in equation (2), We will get
Y - 2 x 15 = 5
Y + 30 + 5 = 35 years
Mayank' Present Age is 15 years and mother age is 35 year.

Method 2
Let us assume the present age of Mayank be P, then the 5 years before Mayank's age = P - 5
According to Question
Age of his mother before 5 years = 3 x (P - 5 )
Present age of his mother = 3 x (P - 5 ) + 5
After 5 Years Mayank's age = P + 5
After 5 Years Mother' age = 3 x (P - 5 ) + 5 + 5
Now again according to question,
After five years she will be twice as old as Mayank, then
3 x (P - 5 ) + 5 + 5 = 2 (P + 5)
? 3P - 5 = 2P + 10
? 3P - 2P = 10 + 5
? P = 15
Mayank's present Age is 15 years.

Let the present ages of A and B are 5x and 3x years respectively. Then
A's age 4 years ago = 5x - 4
B's age 4 years hence = 3x + 4
According to question,
The ratio between A's age 4 years ago and B's age 4 years hence is 1 : 1
(5x - 4)/(3x + 4) = 1 : 1 = 1
? 5x - 4 = 3x + 4
? 2x = 8
? x = 4
present ages of A and B are 20 and 12 years.
The ratio between A's age 4 years hence and B's age 4 year ago is:
Hence, required ratio = (20 + 4) : (12 - 4) = 24 : 8 = 3: 1

Let the present ages of the man and his wife are 4x and 3x years respectively; then
After 4 years ages of man and his wife will be 4x + 4 and 3x + 4 years.
According to question,
After 4 years, this ratio of their ages will be 9 : 7,
(4x + 4)/ (3x + 4) = 9/7
? (4x + 4)7 = 9(3x + 4)
? 28x + 28 = 27x + 36
? x = 8
Hence, their present ages are 4 x 8 = 32 years and 3 x 8 = 24 years
If t years ago they were married, then
(32 - t) /(24 - t) = 5/3
? (32 - t)3 = 5(24 - t)
? 96 -3t = 120 - 5t
? 2t = 24
? t = 12 years

Let us assume the age of B be P years then age of A will be P + 2 and age of C will be P/2 years.
Then according to question,
P + 2 + P + P/2 = 27
? P + P + P/2 = 27 - 2
? ( 2P + 2P + P)/2 = 25
? 5P = 25 x 2
&there; P = 50/5 = 10 years
Age of B will be 10 years.

Let us assume 16 years ago, the age of Sudha and her grandfather were x and 8x years respectively; then
Present age of Sudha = x + 16
Present age of her grandfather = 8x + 16
After 8 years age of sudha from now = x + 16 + 8
After 8 years age of grandfather from now = 8x + 16 + 8
now according to question,
He would be 3 times of her age 8 years from now.
3(x + 16 + 8) = 8x + 16 + 8
? 3x + 72 = 8x + 24
? 5x = 48
? x = 48/5
Eight years ago , sudha age = x + 16 - 8 = x + 8
Eight years ago , grandfather age = 8x + 16 - 8 = 8x + 8
Hence, required ratio = (x + 8) / (8x + 8 )
Put the vale of x , we will get
? required ratio = (48/5 + 8) / ( 8 x 45/5 + 8)
? required ratio = ((48 + 40)/5) / ( (8 x 45 + 40)/5)
? required ratio = (88/5) / ( 400/5)
? required ratio = 88 / 400
? required ratio = 11 : 50

Let us assume the A age is P , B age is Q and C age is R years.
Sum of the age of A and B age is P + Q years and B and C ages is Q + R Years.
According to question,
The sum of ages of A and B is 12 years more than the sum of ages of B and C
Here, (P + Q) = (Q + R) + 12
? (P + Q) - (Q + R) = 12
? P - R = 12
? P = R + 12
Hance, C is 12 years younger than A.