Important Formulas on Problems on Ages
1. If the current age is ‘x’, then ‘n’ times the age is ‘nx’.
2. If the current age is x, then age years later/hence = (n).
3. If the current age is ‘x’, then age ‘n’ years ago = (n).
4. The ages in a ratio ‘b’ will be ‘ax’ and ‘bx’.
5. If the current age is x, then 1/n of the age is x/n.

Example. 01
A father was 4 times as old as his son 8 years ago. Eight years hence, father will be twice as old as his son. Find their present ages.
Solution:
Let son's age 8 years ago be x years.
Thus, father's age at that time = 4x years
After 8 years, son's age = (x + 8) + 8 = (x+16) years
After 8 years, father's age = (4x + 8) + 8 = (4x+16) years
So, According to Question, 2(x + 16) = 4x + 16 or x = 8
Therefore, The present age of the son = x + 8 = 16 years
The present age of the father = 4x + 8 = 32 + 8= 40 years
Example. 02
Father is aged three times more than his son. After 8 years, he would be two and a half times of his son's age. After further 8 years, how many times would he be of his son's age?
Solution:
Let Son's present age be ‘x’ years.
Then, father's present age =(+ 3x) years = 4years.
Therefore, (4+ 8) = (2/5) (+ 8)
=> 8+ 16 = 5+ 40
=> 3= 24 => = 8.
After further 8 years, Son's age will be (x + 16) = 24 years.
And father's age will be (4x +16) = 48 years.
Hence, the required ratio is (4x +16)/(x+16) = 48/24 = 2.

Example. 03
A man is 24 years older than his son. In two years, his age will be twice the age of his son. The present age of his son is:
(A) 14 years
(B) 18 years
(C) 20 years
(D) 22 years

Solution:
Let the son's present age be x years. Then, man's present age = (x + 24) years.
∴ (x + 24) + 2 = 2(x + 2)
=> x + 26 = 2x + 4
=> x = 22.

SOLVED EXAMPLE  SET 1
Question No. 01
A father was 4 times as old as his son 8 years ago. Eight years hence, father will be twice as old as his son. Find their present ages.
Solution:
Let son's age 8 years ago be x years.
Thus, father's age at that time = 4x years
After 8 years, son's age = (x + 8) + 8 = (x+16) years
After 8 years, father's age = (4x + 8) + 8 = (4x+16) years
So, According to Question, 2(x + 16) = 4x + 16 or x = 8
Therefore, The present age of the son = x + 8 = 16 years
The present age of the father = 4x + 8 = 32 + 8= 40 years
Question No. 02
Father is aged three times more than his son. After 8 years, he would be two and a half times of his son's age. After further 8 years, how many times would he be of his son's age?
Solution:
Let Son's present age be ‘x’ years.
Then, father's present age =(x + 3x) years = 4x years.
Therefore, (4x + 8) = (2/5) (x + 8)
=> 8x + 16 = 5x + 40
=> 3x = 24 => x = 8.
After further 8 years, Son's age will be (x + 16) = 24 years.
And father's age will be (4x +16) = 48 years.
Hence, the required ratio is (4x +16)/(x+16) = 48/24 = 2.
Question No. 03
The sum of ages of 5 children born at the intervals of 3 years each is 50 years. What is the age of the youngest child?
Solution:
Let the ages of children be x, (x + 3), (x + 6), (x + 9) and (x + 12) years.
Then, x + (x + 3) + (x + 6) + (x + 9) + (x + 12) = 50
=> 5x = 20
=> x = 4.
Age of the youngest child = x = 4 years.
Question No. 04
‘A’ is twice as old as ‘B’ was two years ago. If the difference in their ages be 2 years, find A's age.
Solution:
Let B's age 2 years ago be x years
Therefore, A's present age = 2x years
Also 2x - (x + 2) = 2 or x=4
So, A's age = 2x = 2 × 4 = 8 years
Question No. 05
A father said to his son, "I was as old as you are at the present at the time of your birth". If the father's age is 38 years now, the son's age five years back was:
Solution:
Let the son's present age be x years. Then, (38 - x) = x
=> 2x = 38
=> x = 19.
Son's age 5 years back (19 - 5) = 14 years.
Question No. 06
A is two years older than B who is twice as old as C. If the total of the ages of A, B and C be 27, the how old is B?
Solution:
Let C's age be x years.
Then, B's age = 2x years. A's age = (2x + 2) years.
Therefore, (2x + 2) + 2x + x = 27
=> 5x = 25
=> x = 5.
Hence, B's age = 2x = 10 years.
Question No. 07
Present ages of ‘A’ and ‘B’ are in the ratio of 5 : 4 respectively. Three years hence, the ratio of their ages will become 11 : 9 respectively. What is B's present age in years?
Solution:
Let the present ages of ‘A’ and ‘B’ be 5x years and 4x years respectively.
Then, (5x + 3)/ (4x + 3) = 11/9
=> 9(5x + 3) = 11(4x + 3)
=> 45x + 27 = 44x + 33
=> 45x - 44x = 33 - 27
=> x = 6.
B's present age = 4x = 24 years.
Question No. 08
The age of a father 10 years ago was thrice the age of his son. Ten years hence, the father's age will be twice that of his son. What is the ratio of their present ages?
Solution:
Let the present ages of father and son be ‘x’ and ‘y’ years respectively.
Then (x - 10) = 3 (y - 10) or 3y - x = 20 ------ (1)
And (x+10) = 2 (y + 10) or x - 2y = 10 ------- (2)
Adding eq. (1) + (2) => y = 30
Substituting the value of y = 30 in equation (1) we get x = 70
Ratio of their ages = 70: 30 or 7:3
Question No. 09
A man is 24 years older than his son. In two years, his age will be twice the age of his son. The present age of his son is:
Solution:
Let the son's present age be x years.
Then, man's present age = (x + 24) years.
Therefore, (x + 24) + 2 = 2(x + 2)
=> x + 26 = 2x + 4
=> x = 22.
Hence, the present age of his son is 22 years.
Question No. 10
Six years ago, the ratio of the ages of ‘A’ and ‘B’ was 6 : 5. Four years hence, the ratio of their ages will be 11 : 10. What is B's age at present?
Solution:
Let the ages of ‘A’ and ‘B’ 6 years ago be 6x and 5x years respectively.
Then, [(6x + 6) + 4[/ [(5x + 6) + 4] =11/10
=> 10(6x + 10) = 11(5x + 10)
=> 5x = 10
=> x = 2.
Therefore, B's present age = (5x + 6) = 16 years.

SOLVED EXAMPLE  SET 2

Question No. 01
Ratio of A's age to B's age is equal to 4:3. A will be 26 years old after 6 years. How old is B now?
Solution:
A's present age = (26 - 6) = 20 years
B's present age = 20 × (3/4) = 15 years
Question No. 02
The sum of the present ages of a father and his son is 60 years. Six years ago, father's age was five times the age of the son. After 6 years, son's age will be:
Solution:
Let the present ages of son and father be ‘x’ and (60 -x) years respectively.
Then, (60 - x) - 6 = 5(x - 6)
=> 54 - x = 5x - 30
=> 6x = 84
=> x = 14.
Son's age after 6 years = (x+ 6) = 20 years.
Question No. 03
At present, the ratio between the ages of ‘A’ and ‘B’ is 4 : 3. After 6 years, A's age will be 26 years. What is the age of ‘B’ at present?
Solution:
Let the present ages of ‘A’ and ‘B’ be ‘4x’ years and ‘3x’ years respectively.
Then, 4x + 6 = 26
=> 4x = 20
=> x = 5.
B's age = 3x = 15 years.
Question No. 04
A is younger than B by 7 years. If their ages are in the respective ratio of 7 : 9, how old is A?
Solution:
Let B's age be ‘x’ years.
Then, A's age = (x - 7) years.
=> (x - 7)/x =7/9
=> (9x - 63) = 7x
=> 2x = 63
=> x = 31.5
Hence, A's age =(x - 7) = 24.5 years.
Question No. 05
The ratio of the ages of father and son at present is 6:1. After 5 years the ratio will become 7:2. The present age of the son is:
Solution:
Let their present ages be 6x and x years respectively.
Then 6x + 5)/(x + 5) = 7/2 = 2 (6x + 5) = 7 (x + 5) or, x=5
Therefore, Present age of the son = 5 years.
Question No. 06
The present ages of three persons in proportions 4 : 7 : 9. Eight years ago, the sum of their ages was 56. Find their present ages (in years).
Solution:
Let their present ages be 4x, 7x and 9x years respectively.
Then, (4x - 8) + (7x - 8) + (9x - 8) = 56
=> 20x = 80
=> x = 4.
Their present ages are 4x = 16 years, 7x = 28 years and 9x = 36 years respectively.
Question No. 07
Alisha's father was 38 years of age when she was born while her mother was 36 years old when her brother, four years younger to her was born. What is the difference between the ages of her parents?
Solution:
Mother's age when Alisha's brother was born = 36 years.
Father's age when Alisha's brother was born = (38 + 4) years = 42 years.
Required difference = (42 - 36) years = 6 years.
Question No. 08
A person's present age is two-fifth of the age of his mother. After 8 years, he will be one-half of the age of his mother. How old is the mother at present?

Solution:
Let, the mother's present age be ‘x’ years.
Then, the person's present age = [(2/5) x] years.
Therefore,   [(2/5)(x + 8)] = [(1/2)(x + 8)]
=> 2(2x + 40) = 5(x + 8)
=>x = 40.
Hence, at present mother’s age is 40 years.
Question No. 09
Three years ago the average age of ‘A’ and ‘B’ was 18 years. With ‘C’ joining them now, the average becomes 22 years. How old is C now?
Solution:
(A+B)'s total present age = (2 x 18+3+3) = 42 years ----- (1)
(A+B+C)'s total present age = 22 x 3 = 66 years -------- (2)
Substituting Eq. (2) from (1), C's age = 66-42 = 24 years
Question No. 10
The age of father 10 years ago was thrice the age of his son. Ten years hence, father's age will be twice that of his son. The ratio of their present ages is:
Solution:
Let the ages of father and son 10 years ago be ‘3x’ and ‘x’ years respectively.
Then, (3x + 10) + 10 = 2[(x + 10) + 10]
=> 3x + 20 = 2x + 40
=> x = 20.

Required ratio = (3x + 10) : (x + 10) = 70 : 30 = 7 : 3.