The sum of the ages of a father and his son is 45 years. Five years ago, the product of their ages was four times the father's age at that time. What are their present ages?

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.

Suppose age of son = P years
Age of the man = 4P years
Age of man's wife = 4P ? 3 years
According to the question,
After 3 years, age of son + 3 years = 15
P + 3 = 15
? P = 15 ? 3 = 12 years
Now, age of wife = 4P ? 3 years
age of wife = 4 × 12 ? 3 years
age of wife = 48 ? 3 years
age of wife = 45 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

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 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

Let the present age of three persons are 4x, 7x and 9x years respectively.
Eight years ago, Ages of three person were 4x - 8, 7x - 8, 9x - 8 years.
According to question,
4x - 8 + 7x - 8 + 9x - 8 = 56
? 20x - 24 = 54
? 20x = 80
? x = 4
Hence, their present ages are
4 x 4 = 16 years,
7 x 4 = 28 years,
9 x 4 = 36 years respectively .

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